Continuing the discussion of peculiar devices a little bit:
1) Students often fail to make the (important!) distinction of
-- unitarity versus
-- reciprocity.
Example: The projection operator
Px = [ 1 0 ]
[ 0 0 ]
is perfectly reciprocal, but non-unitary. It is reciprocal
because you can send a beam through it left-to-right or
right-to-left and get the same result; it has no "front"
or "back" marked on it, unlike a diode.
Example: In microwave technology, there exists a critter called
a "circulator" i.e. a three-port device such that
-- a signal entering on port A comes out port B, while
-- a signal entering on port B comes out port C, and
-- a signal entering on port C comes out port A.
This is blatantly non-reciprocal but it is nicely unitary. Hint:
check that the conjugate transpose (in this case just the transpose)
works as the inverse. That's the definition of unitarity.
Circ = [ 0 0 1 ]
[ 1 0 0 ]
[ 0 1 0 ]
In part of Carl's PDF the reciprocity property is called
"reversibililty". I think I prefer the term reciprocity.
2) Any device that actually exists is unitary. If it's non-unitary,
it doesn't exist. If you are given a non-unitary description of
your device, you need a better description, stat.
Beware: Be on the lookout for bogus non-unitary descriptions
whenever there is absorption going on. People tend to get this
wrong with depressing regularity.
If you see
a) an alleged violation of Liouville's theorem,
b) an alleged perpetual motion machine of the 2nd kind, i.e.
an alleged violation of the 2nd law of thermo,
c) an alleged Heisenberg microscope, i.e. an alleged violation
of the uncertainty principle, or
d) an alleged violation of the fluctuation/dissipation theorem,
then look around to see which components have been mis-described
in non-unitary terms.
3) Do not be alarmed by non-reciprocal behavior. Some people
wig out when they see this the first time, and think it violates
some deep law of physics, but it doesn't.
In the optics realm, there are all sorts of non-reciprocal
devices, many based on the Faraday rotation as Carl mentioned.
In the microwave realm, there are circulators as mentioned
above. There is also a two-port device called an isolator
which lets signals flow from port A to port B and not vice
versa, but it's just a circulator with a termination on the
"C" port.
(Of course if you mis-describe the termination, you wind
up with something that is non-reciprocal *and* non-unitary,
guaranteed to cause mass confusion.)
In the lumped-circuits realm, there is the 2wire/4wire hybrid
which is basically 19th-century technology but without it
20th-century long-distance telephony wouldn't have worked.
Of course lots of things do exhibit reciprocity:
-- a simple lens made of ordinary glass is reciprocal
-- a simple mirror made of ordinary silver or aluminum is reciprocal
-- a Yagi antenna is reciprocal, i.e. it works equally well as
a receiving antenna or a transmitting antenna.
-- etc.
... but remember, reciprocity is not a law of nature. It's not
even particularly hard to build non-reciprocal devices.
You might really, really want a circulator in your antenna circuit
somewhere, so that power from your transmitter doesn't explode the
dainty little circuits on the front end of your receiver.