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 $\delta V$, then in order to have linear response (well-defined constant $G$ around $\delta V = 0$) 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 $\sigma_{{\mbox{\tiny T}}}^{{\mbox{\tiny L$\rightarrow$R}}}(k,\phi)$ and $\sigma_{{\mbox{\tiny T}}}^{{\mbox{\tiny R$\rightarrow$L}}}(k,\phi)$.

If we assume classical motion then we can imagine a map from a Poincaré Section (PS) $(y,p_y)$ at a vertical slice at $x = -x_0$ to another PS $(y',p_y')$ at $x = +x_0$. At each PS we consider only rightwards-moving ($p_x > 0$) particles, and take $x_0 > L$. A certain area of phase space $(y,p_y)$ is transmitted and is mapped to an equal area in phase space $(y',p_y')$ [88]. Time-reversal invariance holds since we consider magnetic field $B{=}0$, so we can negate the momenta (now considering $p_x < 0$) 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

$$\int_{-\pi/2}^{\pi/2} \! \! d\phi \, \sigma_{{\mbox{\tiny T... ...hi) \; = \; \lambda_{{\mbox{\tiny F}}}\mbox{Tr}( t^\dag t) ,$$ (7.23)

where $t$ is measured from left to right states. It is instructive to derive this directly; this is done in Appendix L, giving an independent verification of (7.1). This relation ties together the cross section and Landauer views of conductance. Time-reversal invariance and flux conservation together imply ^7.7that $\mbox{Tr}( t^\dag t)$ is unchanged by swapping the labelling of the leads [20,187], thus we immediately have from (7.23) the reciprocity of angle-integrated quantum cross section

$$\int_{-\pi/2}^{\pi/2} \! \! d\phi \, \sigma_{{\mbox{\tiny T... ..._{{\mbox{\tiny T}}}^{{\mbox{\tiny R$\rightarrow$L}}}(k,\phi) .$$ (7.24)

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 $k$, at wavevector $k$) will be $F(k) = G_{{\mbox{\tiny atom}}}(k) \, J_{{\mbox{\tiny0}}}(k)$ where $J_{{\mbox{\tiny0}}}(k)$ is the flux incident per unit wall area, and we define the atomic `conductance' by

$$G_{{\mbox{\tiny atom}}}(k) \; \equiv \; \int d\Omega \, w(k,\Omega) \, \sigma_{{\mbox{\tiny T}}}(k,\Omega) .$$ (7.25)

As before, the quantum transmission cross section is $\sigma_{{\mbox{\tiny T}}}(k,\Omega)$, but now there is a weighting function $w(k,\Omega)$ which defines the angular distribution of the incident beam ^7.8. The weight has the normalization $\int d\Omega \, w(k,\Omega) \, \cos(\theta) = 1$. (All integrals over solid angle $\Omega \equiv (\theta,\phi)$ are over a range of $2\pi$ appropriate for the half-sphere). Following the analogy of Ref. [191], $F(k)$ plays the role of current, $J_{{\mbox{\tiny0}}}(k)$ that of bias voltage. However, the name `conductance' does not imply any definite chemical potential difference as in the 2DEG case. For classical particles, the `conductance' of an aperture of area $A_{{\mbox{\tiny eff}}}$ in a thin wall is simply $G_{{\mbox{\tiny atom}}}(k) = A_{{\mbox{\tiny eff}}}$, regardless of the incident angular distribution. Thus $G_{{\mbox{\tiny atom}}}(k)$ gives the effective area $A_{{\mbox{\tiny eff}}}$ of a QPC, in an analogous fashion to $a_{{\mbox{\tiny eff}}}$ in 2D [see Eq.(7.19)].

For an integer number of quantum channels, the 2D quantization of $a_{{\mbox{\tiny eff}}}$ in units of $\lambda/2$ [see Eq.(7.19] becomes in 3D the quantization of $A_{{\mbox{\tiny eff}}}$ in units of $\lambda^2/\pi$ [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

$$G \; \propto \; \mbox{Tr} (t^\dag t \rho)$$ (7.26)

where $\rho$ is the density matrix of the incident beam.