Re: less than sqrt(hbar)

[John Denker <jsd@AV8N.COM>]

Ludwik Kowalski wrote:
> 1) It terms of volts, what is the smallest measurable DOP?

I have no idea what a DOP is.

In any case... there is no nontrivial lower bound on the
amount of voltage noise at the input to a phase-sensitive
amplifier.  The lower bound is zero.  The laws of physics
allow you to approach zero as closely as your engineering
skill will take you.

> I see a reference to vacuum fluctuations.
> How can such fluctuations be explained at
> the level of an introductory physics course?

If you can visualize thermal fluctuations, you can
visualize zero-point fluctuations.

Draw a graph of
    E == energy = .5 kT coth(hbar omega / kT)
as a function of T.

The graph is easy to draw:  at low temperature, it is
asymptotically a horizontal straight line with E-intercept
of .5 hbar omega, as you can easily verify by expanding
coth() to first order.  Meanwhile at high temperature, it
is asymptotically a straight line with slope .5 kT, as
you can again easily verify.  So draw two straight lines,
with a little bit of rounding instead of a corner, and
you're done.

Think about this for a while and you will convince yourself
of various amusing facts, including
 -- Zero-point fluctuations are just the low-temperature
  limit of thermal fluctuations.
 -- You can reduce thermal fluctuations by building a
  better refrigerator, but no fridge on earth will get
  the fluctuations below .5 hbar omega, no matter how
  cold you get.
 -- The same physics that enforces the 2nd law of thermodynamics
  also enforces the Heisenberg uncertainty principle.  If
  you can build a Heisenberg microscope you can build a
  perpetual motion machine and vice versa.

Another fun fact:  the crossover between thermal behavior
and quantum behavior occurs at 21 GHz per kelvin.  That's
K/hbar in the appropriate units.  Among other things, that
means that almost all optics is in the nonthermal regime,
and almost all electronics is in the thermal regime.