Photonic Boson Sampling in a Tunable Circuit

References

Acknowledgments

We thank Robert Fickler for help with characterisation, Marcelo de Almeida, Devon Biggerstaff, and Geoffrey Gillett for experimental assistance, and Alex Arkhipov, Michael Bremner, and Terry Rudolph for discussions. This work was supported in part by: the Australian Research Council’s Federation Fellow program (FF0668810), Centre for Engineered Quantum Systems (CE110001013), and Centre for Quantum Computation and Communication Technology (CE110001027); the University of Queensland Vice-Chancellor’s Senior Research Fellowship program; the National Science Foundation’s Grant No. 0844626, and a Science and Technology Centre grant; a DARPA Young Faculty Award grant; a TIBCO Chair; and a Sloan Fellowship.

Experimental Boson Sampling: Supplementary Material

I. Four-photon source and BosonSampling circuit

A mode-locked Ti:Sapphire (Coherent 900 HP) laser with a repetition rate of $76$ MHz, $100$ fs pulses and an average output power of ${\sim}3.8$ W at $820$ nm is frequency doubled in a $2$ mm long bismuth borate (BiBO) nonlinear crystal to give ${\sim}1.5$ W centred at $410$ nm. Photon-pairs are produced via spontaneous parametric downconversion: the two pairs are produced by forward and backward passes of a $2$ mm long beta-barium-borate (BBO) nonlinear crystal cut for type-I phase matching. Single photons pass through spectral filterers (FWHM $5$ nm) before being coupled into single mode optical fibres. At $100\%$ pump power the forward and backward passes of the source produce approximately $290$ kHz and $180$ kHz two-fold coincidences, with a maximum four-fold rate of $1.20$ kHz. The discrepancy in two-fold coincidences between forward and backward passes is due to differing focussing conditions in each case.

Single photons are directed into the BosonSampling circuit using a combination of calcite beam-displacers and waveplates The forward pass of the downconversion source is used for the two-photon measurements and the source is run at $20\%$ of the maximum pump power to reduce the ratio of higher-order photon emission. With an average input coupling efficiency of $64\%$ into all modes of the circuit, we obtain an average two-photon coincidence rate of $260$ Hz across all modes. When injecting three photons into the circuit, the remaining fourth photon is sent directly to a single photon detector to serve as a trigger for its twin. Again, running at $20\%$ maximum pump power we obtain an average four-fold coincidence rate of $185$ mHz. For all measurements in the main text we use an additional $2$ nm filter at the input of the BosonSampling circuit, this reduces the effect of group-velocity mismatch between interfering photons from independent downconversion events.

II. Measured unitary matrix and calculation of permanents

In this section we present a sample measured unitary evolution and demonstrate how we generate sub-matrices whose permanents are used to calculate the probability amplitudes of bosonic scattering events.

Changes in laboratory conditions cause slight changes to the in the optical fibres network we use as our BosonSampling circuit. For this reason we measure the evolution $U_{\textrm{exp}}$, of the linear optical network after each of Bob’s experimental runs, ensuring that Alice generates the most representative scattering probabilities. Below we give the unitaries that were measured after we completing the two- and three-photon non-classical interference measurements respectively,

For our example calculation we will input photons with the configuration $S{=}(1,0,1,0,0,0)$ and look at the output modes given by $T{=}(0,1,0,0,1,0)$. First we generate the sub-matrix $U_{T}$ by selecting $t_{j}$ copies of the $j^{\textrm{th}}$ column of $U^{2\textrm{photon}}_{\textrm{exp}}$ Scheel (2004); Aaronson and Arkhipov (2011) to give,

Then we generate $U_{ST}$ by selecting $s_{i}$ copies of the $i^{\textrm{th}}$ row of $U_{T}$ to give

The probabilities of observing the even $T$ for completely indistinguishable and distinguishable photons are given by $P^{Q}_{T}$ and $P^{C}_{T}$ respectively,

The estimated errors for predicted visibilities given in the main text were calculated as the standard deviation of the predicted visibilities from 10 separate unitary characterisations.

III. Calculation of visibility using coherent-state inputs

We calculate interference visibilities by injecting $n$ equal-amplitude coherent states with normalised electric field amplitudes $E_{i}{=}e^{i\theta_{i}}$, into the input modes $\{i,...,m\}$ of the linear optical network. The input vector ${\bf E}{=}(E_{1},...,E_{i}...,E_{n})$ is transformed under $U$ such that the electric field at output mode $j$ is given by

When the input coherent states overlap with a zero time delay the $2n$-order correlation in electric field between detectors at output modes $\{j,...,m\}$ is given by the phase averaged cross-correlation function Metcalf et al. (2012); Mandel (1983),

Conversely, if input coherent states are delayed by significantly more than their coherence lengths no interference can occur between them, leaving the cross-correlation function as an incoherent sum of input fields at the output modes $j$,

where $E^{(i)}_{j}$ is the electric field at output mode $j$ given the input at mode $i$. The interference visibility is then defined analogously to Eq.(3) in the main text,

IV. Effects of non-ideal photon sources from downconversion

Spontaneous parametric downconversion, as the name implies, is a probabilistic phenomenon which outputs photon states of the form

where: $\left|n_{a}n_{b}\right>$ gives the photonic occupation numbers $n_{a}$ and $n_{b}$ in the spatial modes $a$ and $b$ respectively Broome et al. (2011); the probability amplitude of creating one photon-pair per pump pulse is $\eta$, which incorporates the nonlinear interaction strength, interaction time of the pump field with the nonlinear crystal and the optical coupling efficiency of the down-converted modes, and $\eta{\ll}1$. Triggering using one photon of a pair removes the vacuum component, but since $\eta^{i}{>}0$ there is a possibility that two or more photons remain in the signal modes—these are the so-called higher-order terms.

Higher-order terms can never be entirely removed from downconversion light with linear optics alone. Their detrimental effect can be reduced though by keeping $\eta$ at a necessary minimum, while keeping the single-photon rate constant through temporal Broome et al. (2011) or spatial Ma et al. (2011) multiplexing of multiple downconversion sources.