Reciprocity and `conductance' of atom waves

We can ask if the conductance (7.1) computed using transmission of left-side reservoir plane wave states through the QPC is equal to that using right-side reservoir states. Since the two directions correspond to opposite signs of , then in order to have linear response (well-defined constant around ) we would hope that they are equal. That the angular average of transmission cross section is equal from the left and right sides is not immediately apparent in a general asymmetric system. For instance, consider Fig. 7.4b which has a small acceptance angle from the left but a large from the right, therefore very different forms of the transmission cross sections and .

If we assume classical motion then we can imagine a map from a Poincaré
Section (PS) at a vertical slice at to another PS
at .
At each PS we consider only rightwards-moving () particles, and
take .
A certain area of phase space
is transmitted and is mapped
to an *equal area* in phase space [88].
Time-reversal invariance holds since we consider magnetic field ,
so we can negate the momenta (now considering ) and find that
the *same* phase space area is transmitted right-to-left.
When it is realised that the angle-averaged cross section is proportional
to the transmitted phase space area on a PS, then the symmetry
of the angle-averaged classical cross sections follows.

This is not obvious either for quantum cross sections, but it is also true.
Comparing (7.11) with (7.14) gives

So in Fig. 7.4b is it now clear that the ratio of acceptance angles is balanced by the ratio of effective areas.

We now discuss a case in which non-thermal occupation
of incoming states is possible: the
rapidly developing field of coherent matter-wave optics, in which
potentials are defined by microfabricated structures
([76,191,16] and references therein).
There is a recent proposal [191] for observation of
quantization of atomic flux
through a micron-sized 3D QPC defined by the Zeeman effect potential of
a magnetic field.
The device is illuminated by a beam of atoms passing through a vacuum,
whose angular distribution is an experimental parameter
(for instance, a collimated oven source or a dropped cloud of cold atoms
[116]).
The atomic flux transmitted (per unit , at wavevector ) will be
where
is the flux incident per unit wall area,
and we define the atomic `conductance' by

For an integer number of quantum channels, the 2D quantization of in units of [see Eq.(7.19] becomes in 3D the quantization of in units of [191,20,179]. As stated in Ref. [191], this accurate flux quantization requires the incident beam width to be much larger than the QPC acceptance angle.

Eq.(7.25) is the matter-wave equivalent of
Eq.(7.11), with the
important difference that it has a general weight function.
Non-uniformity of this weight function leads to a key result:
that *asymmetry* of the conductance is possible given
identical illumination on either side, even though the (center of mass)
motion is time-reversal invariant.
For example, if the incident flux used to illuminate
the horn QPC of Fig. 7.4b
is narrow in angular spread,
then the left-to-right conductance will be much larger
than the right-to-left conductance.
This contrasts with the 2DEG case where the conductance is
always symmetric.

Finally, it is interesting to note that for the non-thermal
incident (reservoir) distributions
discussed above, the
Landauer formula has the modified form