A homodyne detector integrated onto a photonic chip for measuring quantum states and generating random numbers

A non-ideal homodyne detector is subject to a number of sources of noise. Photodiodes exhibiting dark current or a non-optimal quantum efficency, as well as electronic noise in the amplification circuit all contribute to measurement noise, which limits the detection efficiency. These manifestations of noise can all be modelled as optical loss in the channel of the signal field [21] and quantified by means of individual efficiencies. The global detection efficiency is given by the product of all of these inidividual contributions, including loss in the signal waveguide and in the beam-splitter.

In our device, we identified three sources of inefficiency: the electronic noise generated by the detection circuit, the optical loss in the MMI and the inefficiency of the photodiodes. The quantum efficiencies of the photodiodes were characterised by means of two effective responsivities taking into account loss in the splitter, we obtained a value of (0.78 ± 0.06) A W⁻¹ for one photodiode and (0.80 ± 0.07) A W⁻¹ for the other, corresponding to an estimated quantum efficiency of ${\eta }_{{\rm{pd}}}=0.64\pm 0.05$. Additional information on how these values have been measured are reported in the supplementary information.

The electronic noise is a gaussian-distributed random quantity which can be measured directly in the absence of a LO. With an optical signal present, the electronic output will be gaussian-distributed, with a variance given by the sum of the variances of electrical signal and noise. So the variance of the noise-free signal can be estimated from

$$\begin{eqnarray}&&{\sigma }_{{\rm{SN}}}^{2}={\sigma }_{O}^{2}-{\sigma }_{{\rm{EN}}}^{2},\end{eqnarray} \tag{ 2 }$$

where ${\sigma }_{O}$ is the standard deviation of the raw output of the detector, ${\sigma }_{{\rm{SN}}}$ is the standard deviation of the shot-noise contribution—the fundamental quantum noise of the light field—and ${\sigma }_{{\rm{EN}}}$ is the standard deviation of the electronic technical noise contribution.

Figure 2(a) shows a plot of the variance of the signal measured by our detector for different powers of the LO. The line of best fit through the noise-subtracted variances on a bi-logarithmic scale is a line of slope 1.00 ± 0.02. This confirms the linear dependence on LO power, which agrees with the expected manifestation of quantum vacuum fluctuations as gaussian-distributed white noise.

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The ratio between the variance of the raw output of the detector measured at the highest LO power used (4.5 ± 0.4 mW) and zero LO power is ∼11 dB. This quantity is named shot noise clearance (SNC) and is related to the efficiency of the homodyne detector by [20]

$$\begin{eqnarray*}&&{\eta }_{{\rm{SNC}}}=1-\displaystyle \frac{{\sigma }_{{\rm{EN}}}^{2}}{{\sigma }_{O}^{2}}=0.93,\end{eqnarray*}$$

which, combined with the photodiodes contribution, leads to a total detector efficiency of

$$\begin{eqnarray*}&&\eta ={\eta }_{{\rm{pd}}}\ast {\eta }_{{\rm{SNC}}}=0.59\pm 0.05.\end{eqnarray*}$$

This value is already sufficient to characterise the quantum features of optical states [22, 23].

The bandwidth of a homodyne detector defines the speed at which it can be maximally operated and the maximum spectral width that the signal field can have in order to be measured efficiently. The measured spectral response of our detector is shown in figure 2(b) and the 3 dB bandwidth is ∼150 MHz. The most significant limiting factors for this value are the parasitic capacitance of the PCB and the internal capacitances of the electronic components involved, mainly the operational amplifier (OPA847) [24]. See supplementary data for a more detailed analysis. Enhancements in the bandwidth could be achieved by taking advantage of different amplification schemes, such as the one proposed in [25] which allows to reach 300 MHz of bandwidth, with a similar SNC to the one demonstrated here.

Coherent states (displaced and Gaussian states in general) are amongst the main resources for CV quantum computing and CV quantum communications. For example, in CV QKD a sender shares a secret key with a receiver by encoding two randomly selected real variables x and y in a displaced coherent state described in the phase space by $| x+{\rm{i}}{y}\rangle$. These states are sent to the receiver who performs homodyne measurements on the quadratures in order to extract the secret key. Therefore homodyne detectors capable of characterising displaced gaussian states are one of the main tools when performing CV based QKD.

The detector's capability in performing homodyne tomography was demonstrated using the full arrangement displayed in figure 1(a). A 1550 nm continuous wave laser with 2.5 μs coherence time (Tunics T100S-HP) was split at a fibre beam-splitter with 1% reflectivity. The reflected beam was further attenuated by a variable optical attenuator, phase-modulated by means of a fibre phase shifter and then injected into the chip. The transmitted beam was used as a LO. Quadrature measurements in a phase interval of length π were acquired by driving the phase shifter with a triangle wave sweeping the interval at a frequency of 200 kHz. The entire set of data was acquired within a time interval of 40 μs, significantly shorter than the time scale of phase instabilities of the optics external to the SOI chip (∼150 μs). Quadratures were sampled at 145 MHz, meaning that the state we measured is the projection of the original state on a 7 ns long temporal mode.

The Wigner function of the state was then reconstructed using an iterative maximum-likelihood reconstruction algorithm taking into account the reduced efficiency of the detector and the uncertainty on the coupling losses [26]. The characterisation was performed for three different amplitude values of the coherent state, α: 0.45, 1.04, 1.40. The Wigner functions for these states are reported in figure 4. The quantum state fidelities obtained in the three cases were respectively ${{ \mathcal F }}_{0.45}=99.57 \% \pm 0.31 \%$, ${{ \mathcal F }}_{1.04}=99.31 \% \pm 0.40 \%$ and ${{ \mathcal F }}_{1.40}=99.13 \% \pm 0.67 \%$ . The errors on the fidelities take into account the uncertainty on the measured efficiency of the detector and the uncertainty on the coupling losses experienced by the measured coherent states.

Random numbers are a key resource for quantum cryptography, as well as classical cryptography and having application in more general computational simulation and fundamental science. However true randomness cannot be generated with a classical computer—currently used pseudo-random numbers generated with software can in-principle be predicted. In contrast, quantum random number generators (QRNGs) rely on the outcomes of inherently non-deterministic quantum processes to generate random numbers that cannot be predicted [27–31]. Examples of compact QRNGs have been recently demonstrated [32, 33]. To the best of our knowledge, our report is of the first experimental demonstration in the SOI platform. The quadrature measurements $\hat{Q}$ for the vacuum states are non-deterministic and follow a Gaussian probability distribution,

$$\begin{eqnarray}&&P(\hat{Q})=\displaystyle \frac{1}{\sqrt{\pi }}{{\rm{e}}}^{\tfrac{-{\hat{Q}}^{2}}{{\hslash }}},\end{eqnarray} \tag{ 3 }$$

as shown in figure 3. They were obtained by injecting the LO beam into the top waveguide, while blocking the bottom waveguide (figure 1(c)). To extract the random bits, the voltage output of the homodyne detector was read by an oscilloscope, in windows of 10⁵ samples. The range of measurements of the vacuum states were divided into 2⁸ equally spaced bins, and each bin was labelled with an 8 bit string, similarly to figure 3. Thus each measurement outcome corresponded to the generation of an 8 bit number. To be compatible with randomness extraction hardware we used equally spaced bins, but this means the bits strings associated with the central bins were more likely to appear, skewing the randomness of the random bits. Moreover, correlations in the electronic background noise could be used by an adversary. For this reason a further step of randomness extraction from the raw data was required. We implemented the Toeplitz hashing algorithm [34] as a randomness extractor with a desktop computer (details in Methods section). The output of the Toeplitz algorithm was a sample of bits characterised by a uniform distribution, where the residual correlations between the raw random data have been removed. To determine at which speed to operate our QRNG, we measured the autocorrelations of the bit-strings at different sampling rates. In figure 5(a) we plotted the autocorrelation for the raw data. While increasing the sampling rate up to 1 Gbit s⁻¹ clearly introduces correlations, sampling at 200 Msamples s⁻¹ does not show higher correlation than, for example, the 125 Msamples s⁻¹ sampling rate. This is because the quantum noise is well above the electronic noise level up to 200 MHz, as can be observed in figure 2(b). Thus, a sampling rate of 200 Msamples s⁻¹ was chosen. It can be also observed that the hashed data do not present any significant correlation, as shown in figure 5(b).

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Moreover, we estimated the amount of certified randomness of the generated bits by calculating the min-entropy [34], obtaining ${{H}}_{\infty }=5.9\,{\rm{bit}}/{\rm{sample}}$ (see Methods). Finally the calculated generation rate was 1.2 Gbps, obtained as the product between the calculated min-entropy of 5.9 bits/sample and the sampling rate of 200 Msamples s⁻¹. Here we notice that since we acquired the data with an oscilloscope and used a software based Toeplitz algorithm, the randomness extraction was performed off-line. However, this estimation gives information about the capabilities of the detector itself and the obtained generation rate is the direct result of the combination of SNC and bandwidth of our homodyne detector. Hardware based randomness extractors could be used to improve the generation rate [29]. We then tested the generated random bits with the NIST SP 800-22 statistical tests provided in [35]. Our QRNG passed all the tests provided. In table 1 we report the results for the NIST SP 800-22 statistical tests. Figure 6 shows the results for the uniformiity tests on the P-values (see Methods for more details).

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The design implemented on the optical chip does not allow a direct measurement of the insertion loss in the grating coupler used to inject the LO. For this reason two other grating couplers have been placed in line with the LO input, making it possible to couple light in and out of all of the couplers simultaneously by means of a single fibre array. The two test grating couplers are connected by a single-mode waveguide that is two times as long as the one connecting the grating coupler to the MMI in the LO channel, meaning that the loss measured in this test structure provides us with a doubled estimation of the loss experienced by the LO before reaching the homodyne detector. The error on this estimation has been obtained by measuring insertion loss in a further 20 equivalent test structures and calculating its standard deviation. The obtained value of the transmissivity, including its error is 0.31 ± 0.03.

For reconstructing the density matrix and the Wigner function we employed the maximum likelihood algorithm. The quantum fidelity between the experimental data and the ideal density matrix was calculated by setting ${\alpha }_{{\rm{sim}}}={\alpha }_{{\rm{\exp }}}\pm {\rm{\Delta }}P$, where ${\rm{\Delta }}P$ takes into account the uncertainty on the coupling losses and the efficiency of the detector. The fidelity was taken as the mean of the fidelities of 100 different sets of simulated data. The standard deviation was obtained as the standard deviation of these fidelities. The calculated fidelities are respectively ${{ \mathcal F }}_{0.45}=99.57 \% \pm 0.31 \%$, ${{ \mathcal F }}_{1.04}=99.31 \% \pm 0.40 \%$ and ${{ \mathcal F }}_{1.4}=99.13 \% \pm 0.67 \%$ .

The Toeplitz hashing algorithm takes a k-bit string of raw bits obtained by binning the random data from the oscilloscope and multiplying it by a k × j Toeplitz matrix, giving as a result an unbiased j-bit random string [34]. Here j is given by the length of the input sequence of bits times the ratio between the ${{H}}_{\infty }$ and number of bits used (8 bits in our case). Hence, in order to extract pure random bits, for each sequence we estimated the min-entropy which describes the amount of extractable randomness from the quantum signal distribution. It is defined as

$$\begin{eqnarray}&&{{H}}_{\infty }=-{\mathrm{log}}_{2}(\mathop{\max }\limits_{x\in \{0,1\}{}^{n}}\Pr [X=x]),\end{eqnarray} \tag{ 4 }$$

where X corresponds to the quantum signal shot-noise distribution over ${2}^{n}$ bins, and ${\rm{\Pr }}[X=x]$ is the probability to obtain a particular value for X. In homodyne detection however, we do not have direct information about the quantum signal distribution because it is always mixed with some classical noise. We thus estimated the true quantum variance under the assumption of a Gaussian distribution, using equation (2). For each sequence we calculated min-entropy of $\sim 5.9\,{\rm{bits}}/{\rm{sample}}$ and then built a Toeplitz matrix, using a pseudo-random seed of k + j-1 bits as in [34]. An alternative approach could be to substitute part of this pseudo-random seed with a certified random string, obtained by previous experiments. Finally the raw sequence of bits was multiplied by the Toeplitz matrix to obtain the unbiased random sequence.

We applied the NIST SP 800-22 test to a sequence of 10⁹ random bits. This provides 15 different tests. For each test the total string of random bits was divided into 1000 blocks. All the tests were applied to each block (with the exceptions of the random excursion and random excursion variant tests, which use approximately 600 blocks), and a P-value was extracted for each single test. These P-values describe the probability to obtain a more biased string of bits than the one obtained, under the assumption that the bits are the outcomes of a perfect QRNG. In order to assess the randomness of the data, there are two requirements specified by NIST SP 800-22 test. First, the proportion of single tests with a P-value greater than 0.01, reported in the second column of table 1, must be above 0.98. Second, by definition of P-value, the P-values obtained from all the single tests must be uniformly distributed. Thus, a second set of P-values was calculated to assess the uniformity of the distributions original P-values. These final P-values, one for each of the 15 tests, must be above 0.01 to confirm the randomness of the data. In figure 6 we plotted the P-values distributions for the different tests. As can be observed, the P-values are uniformly distributed, indicating the randomness of the experimental data. In the third column of table 1 the P-values for the uniformity tests are reported.