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 . 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
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  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
The atomic flux transmitted (per unit , at wavevector ) will be
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. , 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