M1L7i.txt

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Our next unit is partially motivated
to show you what all this entanglement can do for us.
So I mentioned that entanglement is a resource.
It's something useful, like energy is a resource.
And so maybe ultimately there should be a stock market
and you can buy so and so many bits of entanglement.
And you have to pay for it, because there's
something you can do.
And let me sort of introduce what can be done with it.
If you have one photon at frequency omega,
and you have an observation time,
you do a measurement over time T,
then the precision at which you measure the frequency
is fundamentally limited by, you can see time energy uncertainty
by the Fourier theorem that the uncertainty in the frequency
of a single measurement is 1 over the time T
you had to detect or to measure the frequency of the photon.
But now we have n photons, and that
means that the uncertainty in the measurement, which
is now small delta omega is delta omega
for a single photon.
But you know when you do n measurements and average n
measurements, you gain by the square root of n.
And this is regarded as the fundamental short noise
limit of measurements.
But now, assume we can do something fancy.
We can make a super photon.
We take our n photons of frequency omega
and make one big photon of frequency n omega.
Now we have only one photon, the frequency uncertainty
of the measurement is delta omega.
But this is now the uncertainty of the n times more energetic
photon.
So therefore, if we're interested in the quantity
omega, which is the frequency of the single photon,
we have now made an improvement over the standard shot noise
limit by square root n.
So that's just the Gedankenexperiment.
If you can take n photons, I've told you
how we've talked about the optical limit
parametric oscillator how we can pump a crystal.
And in your homework assignment, you do a nice calculation
for the Hamiltonian, one big photon in,
it breaks into two photons.
The reverse process is frequency dot doubling.
So if you would not measure n photons individually,
but first [? ate ?] up their frequency
by making a photons of n times the frequency
and then look at a single photon, you have now--
you measure all the photons together,
and therefore your accuracy improves
by a factor of n and not just by a factor of square root n.
So that tells us something if we do something with the photons.
If we entangle them, there is a possibility
to vastly improve the standard limit of measurements.
And so people who are really interested in it
are people who push the limits of precision, people
who build atomic clocks, and want
to get the last little bit of accuracy which is possible.
So they've already exhausted all technical possibilities,
and the next thing is now, well, maybe entanglement.