Parametric correlation of transmission amplitudes

M¨uller et al. (2007) evaluate semiclassically many quantities like the average con-ductance and higher order correlation functions. They also include the effect of a symmetry breaking parameter. However we will consider a non-symmetry breaking parameter and will first introduce an artificial quantity, the average of a parametric

correlation function of the transmission amplitudes, which we shall define as

to mimic the form of the conductance. The parametric dependence of this quantity would not be directly observable, but the calculation will be useful later when we examine the parametric conductance variance. Semiclassically we have the following double sum over trajectories

In the following, we evaluate this along the lines of Heusler et al. (2006). We note that for the trajectory pairs we will consider in this Section, the topological indices cancel (νζ = ν_ζ0) so we can drop the corresponding exponential factor for convenience.

Firstly the diagonal approximation ζ=ζ⁰ gives DG˜^diag(E, x)E

For trajectories of a fixed periodTζ the parametric factor is approximated by using the Gaussian noise assumption as in equation (3.1.14)

D

e^σ¯ixhQζE

≈e^−βTζ (5.2.4)

The sum over all trajectories connecting channela tob can be performed by using the analogue of the Hannay–Ozorio de Almeida sum rule for open systems (Richter and Sieber, 2002) by turning the sum into an integral over the trajectory time

X

Figure 5.3: An example of a trajectory with two self-encounters and its partner.

where µ is the classical escape rate of the system, or the inverse of the average time delay (= _T^M

H). The exponential term represents the average probability that a trajectory remains in the system for the timeT_ζ. The diagonal approximation then takes the form of the following integral over all trajectory timesT

DG˜^diag(E, x)E

=X

a,b

Z _∞

0

dT 1

T_He^−µTe^−βT = M₁M₂

M+B (5.2.6)

whereB =βT_H and the sum over channels a and b simply gives a factor of M1M2

because of the respective number of choices of each channel.

The off-diagonal terms are found by considering trajectory pairs that are cor-related and differ only in encounter regions, in a similar way to the calculation for periodic orbit pairs. An example of a trajectory with two self encounters and its partner is shown in Figure 5.3. For open systems we remember that we need to use the exposure time rather than the trajectory time, to reflect the slightly reduced probability of escape due to the encounter regions. For a trajectory pair with en-counters described by the vectorv, the contribution can be written, using the open sum rule, as

DG˜v(E, x)E

= M1M2N(v) T_H

Z dT

Z

d˜sd˜uwT(˜s,u)e˜ ^h¯i˜s˜ue^−µTexpe^σ¯ixhQζ (5.2.7)

whereN(v) is the number of trajectory structures corresponding to each vectorv.

The main difference from periodic orbits is that there is an extra link (givingL+ 1 in total) because both ends of the trajectory are free. The weight function changes because of this freedom, and we also no longer overcount by a factor of L because the start and end are fixed in the leads. The restriction on the links is that they all have positive duration, and in terms of an integral the weight is given by

wT(˜s,u) =˜

we see that now it is simply written as an L-fold integral over different link times t_i, i= 1. . . L, while the last link time is fixed by the total trajectory time

When we perform the sum over trajectories of different lengths we integrate over the trajectory time T, which can be re-expressed as an integral over the last link time. With the weight factor, the contribution now includes integrals over all the links times ti. Heusler et al. (2006) then rewrite the contribution of a correlated trajectory pair as integrals over the link times and the encounter regions. To do this, we need to decompose the term from the survival probability (cf equation (4.1.13)) as follows and the parametric correlations (cf equation (3.1.22))

D

into terms from the links and encounters. This then gives the following contribution to the average parametric correlation function of the transmission amplitudes

DG˜v(E, x)E

We can now see the real advantage of using the correlation function as we can separate the above integral into a product over the links and the encounters. The integral over the links is easily performed and gives a factor of _M+B^TH for each link.

Each encounter integral can be expanded to first order in the encounter time and gives a factor of ^M+l2αB

T_H^lα−1 . In total all the Heisenberg times cancel and we can view the result as having a factor of (M+B)⁻¹for each link and (M+l²_αB) for each encounter.

These diagrammatic type rules show the power of separating the contribution into links and encounters (which can also be seen in Appendix B) as they allow us to effectively read off the contribution of trajectories corresponding to any vectorv.

The number of trajectory structures corresponding to each vectorv is the same as for periodic orbits, because of a one-to-one relation between them. In fact, by joining the ends of each trajectory structure (connecting the parts of the trajectory and its partner that leave through the exit lead to the start of the trajectories in the entrance lead) we obtain a periodic orbit structure. Equivalently, we can start with any periodic orbit structure and cut one of its links. By moving the cut ends of the link to the leads we create a trajectory structure (the link must be traversed in the same direction in both orbits, so with time reversal symmetry we must choose either the partner orbit or its time reversal to ensure that this is the case). This effectively creates L trajectory structures, which is why we no longer overcount byL in the weight function. When we sum over the possible trajectories in both symmetry classes we obtain the following result for the situation without time reversal symmetry

and for the situation with time reversal symmetry

If we setB = 0, we recover the first few terms of the non-parametric result (Heusler et al., 2006).