Luminescence-induced noise in single photon sources based on BBO crystals

First, we evaluated spectral (time-integrated) properties of both SPDC-generated photons and luminescence. The typical measurement outcome is shown in figure 3. In this figure, we show the spectra of our signal in three distinct configurations, all measured with the pump wavelength set to 267 nm. First, we set the pump-beam polarization to satisfy phase matching conditions. In this regime, we observe a double peak blob of the combined SPDC and luminescence signal. In the second configuration, we set the orthogonal pump beam polarization (by rotating HWP) so that the phase-matching conditions are no longer satisfied. As expected, the SPDC is turned off and only the luminescence signal remains. Finally, we reset the pump-beam polarization to satisfy the phase matching conditions and inserted a linear polarizer (LP) in front of the fiber coupler to filter out the linearly polarized SPDC. Since the luminescence photons have mixed polarization states, we still observe about one half of the luminescence passing through the linear polarizer. This set of three measurements allows us to identify spectra belonging to the SPDC and luminescence, respectively. We also conclude that there is a non-negligible spectral overlap between the two.

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As stated above, the BBO crystal is always rotated to satisfy degenerate phase-matching conditions. Therefore, the SPDC central wavelength doubles that of the pump beam. On the other hand, luminescence spectra as a function of the central pump wavelength are to be analyzed. We have therefore acquired luminescence spectra for various pump-beam central wavelengths. The data are presented in figure 4. Note that for the sake of better readability, the spectral intensities in the plot are normalized to their peak. Figure 4 shows that, apart from 240 nm, the luminescence spectra are skewed towards longer wavelengths, do not depend on the pumping wavelength and have maxima at about 430 nm. We conclude this analysis by stating that in the range of 240–290 nm of the central-pump wavelength, the luminescence spectra overlap with SPDC. The overlap is more significant when using shorter pump wavelengths and is expected to reach a maximum for about 220 nm of pump wavelength, which is unfortunately unavailable to our OPA.

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To complete the spectral analysis, we have also investigated the influence of the pump power on luminescence spectra. The pump wavelength was fixed at 267 nm while the pump power ranged from 10–100 mW. The observed results do not indicate any shift in the luminescence spectra as a function of pump power. Moreover, as expected, the luminescence intensity scales linearly with the pump power (data not shown).

All the spectral measurements indicate that there is an unavoidable spectral overlap between luminescence and SPDC when pumping with wavelengths between 240 and 290 nm. The luminescence will thus inevitably generate noise in the SPDC signal. There are generally two possible courses of action to remove this luminescence-induced noise from the SPDC spectra. The first option is to shift the pumping wavelength above 290 nm so that the luminescence and SPDC have only a negligible spectral overlap. This approach would, however, shift the SPDC central wavelength. The second option is to use edge or interference filters to cut out as much luminescence as possible. This way, one can certainly reduce the amount of noise in the SPDC signal, but only to the extent given by their spectral overlap. Also note that typical interference filters with bandwidths of about 5 nm have non-unit transmissivity even in their transmission maxima and, therefore, introduce signal losses.

We conclude this subsection stating that simple spectral filtering is not sufficient to completely remove the luminescence noise from the SPDC signal and can only be used to correct this issue partially and usually also at the expense of signal loss.

The photons originating from the SPDC process are generated almost instantaneously as the pump pulse propagates through the BBO crystal, since they are described by electron transitions to virtual levels. Luminescence, on the other hand, has its typical exponential decay due to a life-time of electrons on excited energy levels. Time-resolved spectroscopy allows us to monitor this aspect of our signal as well. To separate the SPDC signal from luminescence, we have selected corresponding regions of interest as schematically illustrated in figure 1. By using this procedure, we are able to depict typical temporal profiles of combined and separate SPDC and luminescence emissions integrated over the spectrum (see figure 5).

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The temporal properties of luminescence can be characterized by their exponential decay time τ (time period in which the intensity reaches $1/{\rm e}$ of its initial level). We measured time-resolved spectra at different settings of the streak-camera time window, from ns to μs. By applying exponential fit to the data, we have identified several distinct decay processes with different decay times (see in figure 6). The first decay process is a fast one with a typical decay time of ${{\tau }_{1}}\;=0.73$ ns. The second and third processes are considerably slower, having decay times of ${{\tau }_{2}}\;=\;$ 1.85 μs and ${{\tau }_{3}}\;=$ 9.95 μs, respectively. The values obtained for all the decay times were established by fitting data from repeated measurements with a relative uncertainty of about 7%. Moreover, we have established that these decay times do not depend on the pump wavelength. As an example, see figure 7, which shows the decay time τ₂ as a function of pump wavelength. Similarly, no dependence of the decay times on pumping power has been observed.

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Considering the above-determined facts, we can assess the potential of temporal filtering to remove the luminescence generated noise. In quantum-optics experiments, there is usually some sort of detection window used to filter real signals from noise or detector dark counts. Typical widths of such detection windows are on the order of units of nanoseconds, which roughly coincides with the fastest observed decay time τ₁. Therefore, a detection window cannot serve to filter out the fastest decaying luminescence signal. On the other hand, it can, under some circumstances, be used to filter out the slower luminescence emission. Doing so would require using pulsed pumping and reducing the overall repetition rate in the experiment (to about 100 kHz) to allow for the luminescence to extinguish before the new detection window is opened. This strategy cannot be adopted with continuous pumping since it would only limit the amount of luminescence-added noise, without removing it completely.

So far, we have determined both spectral and temporal properties of luminescence in BBO crystals. The results of the two previous subsections indicate that some reduction of luminescence-induced noise can be achieved using spectral and temporal filtering. Complete removal of luminescence noise is, however, impossible. In this subsection, we quantify the influence of luminescence on single-photon-source quality by analyzing a typical heralded single-photon source.

In order to study such a model, let us consider typical parameters of heralded single photon sources: rate of single photon generation ${{R}_{S}}\approx 100\;{\rm kHz}$ (number of signal photons detected per 1 s) and detection window ${{t}_{w}}\approx 10\;{\rm ns}$ (window in which we expect the signal photon to appear). These values are based on recent experiments in the field of linear-optical quantum information processing performed in our laboratory [18, 28, 30]. Note that for the purposes of subsequent quantitative models, we have considered a pump wavelength of 267 nm (fourth harmonics of Nd-YAG laser) and ideal binary detectors with unity quantum efficiency. The probability of generating a pair of SPDC photons can now be expressed as ${{P}_{S}}={{R}_{S}}{{t}_{w}}\approx 1\times {{10}^{-3}}$. Under these assumptions, we can readily neglect the simultaneous generation of two SPDC photon pairs that appear with a probability of $P_{S}^{2}\approx 1\times {{10}^{-6}}\;$. Similarly, we can define the luminescence–photon rate R_L and luminescence probability ${{P}_{L}}={{R}_{L}}{{t}_{w}}$ to be used in forthcoming calculations. Assuming that the luminescence rate is of the same order as the SPDC rate or lower, we can neglect the probability of simultaneous generation of two luminescence photons within one detection window.

Ideally, a perfect heralded single-photon source yields a signal photon in the pure Fock state $|1\rangle \langle 1|$ every time the idler photon is detected. Due to luminescence, such a source will, however, yield a mixed state in the form of

$$\begin{eqnarray}&&\hat{\rho }=\frac{1}{N}\left( {{p}_{0}}|0\rangle \langle 0|+{{p}_{1}}|1\rangle \langle 1|+{{p}_{2}}|2\rangle \langle 2| \right),\end{eqnarray} \tag{ 1 }$$

where the normalization N will be calculated later. The vacuum term $|0\rangle \langle 0|$ appears due to 'false heralding' by a luminescence photon in the idler mode instead of in the genuine SPDC idler photon. Since luminescence photons do not originate in pairs, the signal mode is in a vacuum state even if the detector in idler mode clicks. Considering that this situation happens every time a luminescence photon is detected without being accompanied by SPDC photons within one detection window, we obtain

$$\begin{eqnarray*}&&{{p}_{0}}={{P}_{L}}(1-{{P}_{S}}).\end{eqnarray*}$$

The only desired term $|1\rangle \langle 1|$ appears with a success probability p₁ corresponding to the SPDC photon being generated in the signal mode without the luminescence photon

$$\begin{eqnarray*}&&{{p}_{1}}={{P}_{S}}(1-{{P}_{L}}).\end{eqnarray*}$$

And finally, one observes the state $|2\rangle \langle 2|$ if both the SPDC and luminescence photon are emitted in the signal mode

$$\begin{eqnarray*}&&{{p}_{2}}={{P}_{S}}{{P}_{L}}.\end{eqnarray*}$$

Note that ${{p}_{0}}+{{p}_{1}}+{{p}_{2}}\ne 1$ since we have not taken into account the remaining case of neither SPDC nor luminescence photons being emitted. This case corresponds to the source not heralding the signal photon presence, and thus it does not contribute the effective outgoing state, but only reduces the success probability. In order to correctly normalize the density matrix $\hat{\rho }$, we set

$$\begin{eqnarray*}&&N={{p}_{0}}+{{p}_{1}}+{{p}_{2}}={{P}_{S}}(1-{{P}_{L}})+{{P}_{L}}.\end{eqnarray*}$$

The quality of a generated quantum state is typically expressed by the fidelity F, which in our case takes the form

$$\begin{eqnarray}&&F=\frac{{{p}_{1}}}{{{p}_{0}}+{{p}_{1}}+{{p}_{2}}}=\frac{{{P}_{S}}(1-{{P}_{L}})}{{{P}_{S}}(1-{{P}_{L}})+{{P}_{L}}}.\end{eqnarray} \tag{ 2 }$$

Time-resolved spectroscopy allows us to separate the SPDC and luminescence signal by performing intensity integration over suitable regions of interest. This way, we determine the SPDC-to-luminescence count ratio, or signal-to-noise ratio

$$\begin{eqnarray}&&{\rm SNR}=\frac{{{C}_{S}}}{{{C}_{L}}}=\frac{{{R}_{S}}}{{{R}_{L}}},\end{eqnarray} \tag{ 3 }$$

where C_S and C_L denote SPDC and luminescence photon counts, respectively. It also follows from equation (3) that realistic detector efficiency η will not affect the resulting SNR since both R_S and R_L will be rescaled by the same factor η. The fidelity can be now expressed using SNR in the form of

$$\begin{eqnarray*}&&F=\frac{{\rm SNR}-{{t}_{w}}{{R}_{S}}}{1+{\rm SNR}-{{t}_{w}}{{R}_{S}}}\approx \frac{{\rm SNR}}{1+{\rm SNR}},\end{eqnarray*}$$

where the last approximation holds for low-generation probability.

We have calculated the expected value of fidelity for several different scenarios: (i) no spectral or time filtering of luminescence, (ii) spectral filtering only by a long-pass edge filter with cutoff at 460 nm (e.g. continuous pump), and (iii) both spectral and temporal filtering. For all calculations, we have assumed the above-mentioned typical values of t_w and R_S while using experimentally acquired values of SNR. Values are summarized in table 1. The values of C_S and C_L were obtained by integrating over the respective regions using the experimentally acquired time-resolved spectra. To obtain CL without filtering, the entire fluorescence signal was summed, while for spectrally filtered C_L, we have summed only over the region where fluorescence spectrally overlaps with SPDC. Finally, to obtain spectrally and time-filtered counts, we have counted only the fluorescence signal from the region with spectral overlap with SPDC, which is on top of that occurring during the same detection window of 1 ns as the SPCD. As we can see, without simultaneous spectral and temporal filtering, fidelity is significantly reduced. Only when both the spectral and temporal filtering are applied, the fidelity approaches its theoretical limit F = 1. Spectral filtering alone can achieve about half of this improvement.