In the dual-slit experiment, can the time it takes for a photon to arrive reveal the path it took?

It seems like this experiment is possible:

1. Fire a photon, and start a timer
2. The photon travels through the slit(s)
3. The photon strikes a plate -- the timer is stopped.

Based on where on the plate the photon struck, the time it took, and the speed of light, it seems possible to deduce through which slit the photon passed.

Or, would the times correspond with a direct line from the source to destination?

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This question becomes more interesting if you diffract the clock itself through the slits. Only identical clocks interfere, so whenever there is a measure of time difference along the two paths, they do not interfere. – Ron Maimon Jan 9 '12 at 15:13

In order to do this you need photons that can be timed very precisely, i.e. $\Delta t$ needs to be small: smaller than $c/d$ where $d$ is the slit seperation. According to Heisenberg uncertainty, this means that the energy uncertainty must be big for $\Delta E\cdot\Delta t>\hbar$ to remain satisfied. If $\Delta E$ is big, then the wavelength uncertainty will also be big; namely $$\Delta\lambda \approx \frac{\Delta\nu}{c} = \frac{\Delta E}{2\pi\hbar c} > \frac{d}{2\pi}$$ But if the wavelength uncertainty is in the order of the slit seperation, there won't be a proper interference pattern as this requires the light to be coherent on the length scales of the experiment!