[Phys-L] unitarity ... was: another Gedanken from PTSOS

[John Denker <jsd@av8n.com>]

On 05/16/2012 07:22 PM, Bernard Cleyet wrote:

> What if you were able to get 2 beams to travel in the same direction,
> one on top of the other.  Say, with a partially silvered mirror.  If
> you could get two out-of-phase waves to travel in the same direction,
> one behind the other, you could achieve total cancellation.  The
> beams would never diverge, so there would be no point where the
> energy reappears - it would effectively be gone forever.

This is one member of a famous family of problems.  In various guises,
this problem has shown up several times in the history of physics.

The short answer is that the half-silvered mirror has _four_ ports 
and all four must be taken into account.


             B |
               |
               |
               |/
 A _ _ _ _ _ _ /_ _ _ _ _ _ C
              /|
             / |
               |
               |
             D |


If we treat A and B as inputs then C and D are outputs.  You can choose
the relative phase of A and B such that there is indeed zero energy at
C ... in which case all the energy comes out at D.

In other words, you cannot simply write
       C =  transmitted + reflected
       D =  reflected + transmitted 
without regard for phases.  

Instead, in the simplest case, you get
       C = +transmitted + reflected
       D = -reflected + transmitted 
where that minus sign really matters.  Conservation of energy requires it.

========================
Connections:

1) The same physics shows up at RF and at microwave frequencies in 3dB splitters.

It is easy to buy a splitter that "looks" like a three-port device, but as we
have just seen, it *must* be a four-port device.  One of the output ports is 
terminated in a 50-Ω resistor internally.

2) The same physics shows up in 20dB directional couplers.

In general, you can write things in terms of the S-matrix:

           [ C ]     [  cos θ   sin θ ]  [ A ]
           [   ]  =  [                ]  [   ]
           [ D ]     [ -sin θ   cos θ ]  [ B ]

for some mixing angle θ.  So this is connected to a rotation in some abstract space.

3) There is more at stake here than conservation of energy ... namely conservation of
phase space.  The S-matrix must be *unitary* which is an even stricter requirement.
Conservation of energy is a mere corollary of unitarity.

Some famous people have gotten this wrong.  Indeed whole groups of exceedingly smart
people have gotten this wrong, and fooled themselves for years at a time.

In particular, consider the case where one of the _input_ ports (say B) is terminated 
in a 50-Ω resistor internally.  Then we have nominally one input port and two output
ports.  If you treat it as a 3-port device, energy is conserved ... but if you look 
closely you find that 
 -- the S-matrix is not unitary
 -- the 2nd law of thermodynamics is being violated, 
 -- the Heisenberg uncertainty principle is being violated,
 -- Liouville's theorem is being violated,
 -- the optical "brightness theorem" is being violated,
 -- etc.
which is not a N-way coincidence but rather N ways of saying the same thing.

To get out of this mess, you have to treat it as a four-port device.  In particular,
you have to consider the thermal fluctuations and quantum fluctuations that are
emitted by the resistor on port B ... which (after splitting) appear in the output 
ports.  That is, the output ports contain contributions from input A *and* input B,
even if input B is coming from a cold load.

To say the same thing another way, "cold load" is not equivalent to "zero voltage".
There will be nonzero fluctuations in the voltage.

This famously showed up in connection with quantum-nondemolition back-action-evading 
measurement devices, which in turn showed up in connection with gravity-wave antennas.
There is more to this story ... but a crucial starting point is realizing that
  a) you need to account for all the ports, and 
  b) you need to make sure the equation of motion is unitary.

To say the same thing another way:  The /correct/ equation of motion is unitary.  If 
you write down something that is not unitary, it's not the correct equation of motion.