Attosecond-resolution Hong-Ou-Mandel interferometry

A new Hong-Ou-Mandel interferometer protocol achieves few-attosecond (nanometer) photon path delay resolution.


Dynamic range of the measurement procedure
The dynamic range may be defined as the maximum interval in τ such that the estimator is single-valued. Phase sensitive interferometers (such as the Mach-Zehnder) suffer from a 'phase-wrapping' problem, where a multitude of physical time delays result in the same relative phase in the interferometer. Those physical time delays cannot then be mutually distinguished, unless some prior information is available. The dynamic range of a Mach-Zehnder interferometer is set by the wavelength of light that is used. One can use a uniform prior distribution to exclude parameters outside a certain interval: this can restore the uniqueness of estimates, at the expense of having enough prior information (represented by the inverse width of the interval). A higher dynamic range implies that less prior information is required to avoid ambiguities. In principle the dynamic range of our Gaussian HOM interferometer is infinite because of the long tails of f (t). In practice, however, one should cap the dynamic range to a region where F is above some threshold value. A conservative estimate would be on the order of 1 or 2 σ, which is typically 10-100× larger than the wavelength of light in a phase dependent (e.g. MZ) interferometer.

Activating phase fringes
As we alluded to in the main text, it is straightforward for our setup to be converted to a phase sensitive one by including a single optical element ( λ /2 waveplate before the PBS). Here we will show the full effect of an arbitrary rotation of the waveplate on the coincidence probability, and the significantly increased Fisher information that results.
Consider misalignment between the downconversion crystal and the PBS. Assume that horizontal (H) and vertical (V ) polarisations are defined by the PBS, and that the downconversion source is rotated around the pump propagation axis by an angle θ. Let us model the SPDC state as: where the creation operators now carry polarisation labels. The PBS has the action of transforming V photons to mode 2 with a phase shift of i due to reflection. It has no effect on H photons: Next the HWP takes V to H and H to V but only in mode 2 (this should also be well aligned with the PBS): We now drop the polarization labels since all photons have the same value now. Next we introduce the path length change of τ , which gives rise to the phase shift e iφ = e i2πτ /λ in mode 2.
Now the final step is the beamsplitter (whose action is shown in the main text), The next step is to combine the correct amplitudes to get a coincidence probability. There are amplitudes for a † 3 a † 4 and a † 4 a † 3 : These operators commute, because of the orthogonal spatial modes: therefore a † 3 a † 4 = a † 4 a † 3 and we should add these terms coherently, allowing the HOM interference effect to operate. There are two more amplitudes (that cannot interfere) for b † 3 a † 4 and b † 4 a † 3 (where once again operators commute): Computing these terms and making the substitutions β 2 = 1 − η 2 , η 2 → αe −τ 2 /σ 2 and φ → 2πcτ /λ = 2πτ ν (where ν 120.

320.
Fisher Information (ps) P c τ (ps) θ = 45 θ = 3 θ = 0 tuned to increase the obtainable precision albeit at the expense of introducing potential instabilities and reducing the dynamic range. These plots correspond to α = 0.63, σ = 0.033 ps, and ν = 371 THz (no loss is modelled, so γ = 0). The curves in the lowermost panel have been alternately cutaway either side of the origin (for clarity). All curves are symmetric around τ = 0.
fig. S1. Predicted Fisher information for a HOM with added phase-dependent fringes. The visibility of the fringes can be is the frequency of the light), we have Note the cos 2 function, indicative of N=2 N00N interferometry, which is easily switched on in our setup -this could provide higher information and a way to remove ambiguities whilst not sacrificing the dynamic range of the HOM setup by rotating the polarisation of the input photon pair before the PBS thereby switching between the two schemes.
It should be noted that in this case the same level of stabilisation is required as classical interferometry. When θ → 0, we recover the usual HOM dip (see Figure S1). The full information is: In the case of θ → 0, this expression reduces to the lossless version of the Fisher information used in the main paper. In the case of θ → 45 • , it simplifies to σ 2 (cos(4πντ ) + 3) .

List of itting arameters esults
The ----S1 Summary of all measurements and parameters used in the fitting procedure. The final row corresponds to an average of the absolute value of each column. The factor 1 × 10 9 is denoted bn for billions. n = 1 or 1.5 is the refractive index of the sample material (air or glass respectively). The single datapoint relating to the glass wedges is highlighted with horizontal lines. Average and best case values are shown in bold face.