Point contact coupled to a resonator

Fig. 7.3 illustrates a QPC-plus-reflector system whose conductance has been experimentally measured [112]. The circular arc reflector and the vertical wall together form a cavity which can support long-lived resonances; the energy of these resonances can be swept by sweeping the reflector gate voltage. The classical condition for stability of the cavity modes is that the arc center must lie at, or to the left of, the wall () [112]. The cavity modes are coupled to the left terminal via the QPC, and to the right terminal via leakage of the modes out through the cavity top and bottom. The system is interesting because it is `open' in the sense that it has no Coulomb blockade [20], but `closed' in the sense that the dwell time is much greater than the ballistic time (the resonances are long-lived). It has also been studied recently in our laboratory using microwave measurements [94,93,95].

The actual potential in a mesoscopic experiment differs from
the illustration:
it has soft walls (on the scale
),
it may have deviations from the circle due to
lithographic error, and it has modulations of the background potential
due to elastic disorder [112].
However, we will not be interested in details of the
resonator on the right-hand side.
Rather, we will adopt the view of a 2D scattering-theorist
`looking' from the left-hand side.
In this section we discuss the maximum
conductance of this system, when the `bare' QPC (*i.e. *without the reflector)
is in the tunneling regime (conductance ).

We use the slit model from the previous section to model the QPC.
This simplifies the treatment of the left-hand side scattering problem,
and we do not believe it alters the
basic conclusion.
As before, we consider the incident plus reflected wave Eq.(7.1)
when the QPC is closed to be the `unscattered'
wave.
This we expand in Bessel functions,

The first term in the expansion is the incoming plus outgoing p-wave, which in the tunneling limit, will dominate in our consideration of the absorption (as shown in Appendix K).

Now we open the slit, and replace
in the above by
,
where
follows the usual definition of partial-wave phase shift [128].
The closed slit corresponds to .
An open slit leading into a closed resonator
(imagine extending the arc in Fig. 7.3 to seal off the
entire right side), in the case of infinite dephasing length,
corresponds to real, and would appear from the left side
as an elastic dipole scatterer.
An open slit with an open resonator
corresponds to complex with positive imaginary part,
and would appear as a general inelastic dipole scatterer.
Therefore transmission though the QPC appears,
to an observer on the left side, to be *absorption* of incident waves.
is interpreted as an `inelastic' cross section (since exiting
the right-hand terminal is equivalent to leaving in a new channel),
and
as an `elastic' one.
can be
found from integrating the net incoming flux [as in Eq.(7.5)]
of the total wavefunction on the *left* side.
Substitution into (7.4) then gives
.
For
the maximal cross section is reached,

The associated maximum conductance is found easily
using (7.21) and (7.1) to be

How do we know that it is possible to build a resonant geometry
which corresponds to
?
The reflector can be described by , the amplitude with which it returns
an outgoing p-wave back to the QPC as an incoming p-wave.
If
, where the p-wave transmission of the QPC
is [*e.g. *see Eq.(7.17)], then the p-wave channel
becomes a 1D Fabry-Perot resonator with mirrors of matched
reflectivity.
Sweeping the round-trip phase then produces peaks of complete transmission
(corresponding to complete p-wave absorption on the left side).
The ratio of peak separation to peak width is the quality factor
.
Such peaks, with heights much greater than the bare tunneling QPC conductance,
were observed in the experiments
of Katine *et al.*[112].
However, Eq.(7.22) has not yet been tested quantitatively because
of the difficulty of matching the Fabry-Perot reflectivities in a real
2DEG experiment.
Note that the maximum conductance (7.22) also
follows immediately from
the Landauer formula, when we realize that there can be complete transmission
of the incoming channel state (in Section 7.2).

An interesting possibility arises when we realize (Appendix K)
that higher channels are still *slightly* transmitted by the bare QPC,
when , even though they are increasingly evanescent.
If the resonator has a high enough reflectivity for these modes, then
additional Fabry-Perot conductance peaks will be produced [205,38].
The peaks may be extremely narrow, but can carry a full quantum of conductance
because they can transmit another incoming channel.
By careful arrangement of the cavity, one or more of these
peaks could be brought into conjunction with an already-existing
peak at the Fermi energy.
(For instance, the and resonances are in different
symmetry classes in Fig. 7.3
so there can be an exact level crossing).
Therefore, we have the surprising result that, in theory, a
conductance of can pass through
an arbitrarily small QPC hole if resonances (from different channels)
coincide at the Fermi energy.
However, due to their extremely small width, such large conductance
peaks are unlikely to be observable in a real mesoscopic tunneling QPC
due to finite dephasing length and finite-temperature smearing [20].

Finally, we should not overlook the fact that our expressions for partial cross sections are a factor of 4 greater than those conventionally arising in 2D scattering theory from a radial potential [128,132,3,43], because we are measuring cross section on the reflective boundary of a semi-infinite half plane. For instance, the maximum inelastic partial cross section for a single channel in free space is [128,132,3,43], compared to our maximum `inelastic' cross section per channel Eq.(7.21). Similarly, the maximum elastic result in free space is , compared to our maximum (normal-incidence) `elastic' cross section per channel . This latter case occurs when .