Trajectory differences

As well as the action difference, there are other differences between the two trajecto-ries. The longer trajectory revolves around the periodic orbit an additionalr times, so its time will be longer by approximately r times the period of the orbit

T_ζ0 ≈Tζ+rTγ (5.4.8)

For chaotic cavities with leads attached, the stability amplitude of a trajectory is proportional to √ ¹

|M21|, where M₂₁ is an element of the stability matrix M of the whole trajectory (Richter and Sieber, 2002). As stability matrices are multiplicative, we can separate the stability matrix of the trajectoryζinto a product of the matrices corresponding to its initial part up to the first piercing point, the part where it follows the periodic orbit ktimes, and then its final part to the exit lead

Mζ ≈MFM_γ^˜kMI (5.4.9)

Here we have approximated the stability matrix of the trajectory as it follows the periodic orbit ktimes by the stability matrix of the periodic orbit itself. The same approximation holds for the longer trajectory ζ⁰, except we have an additional r repetitions of the orbit. We can express the stability matrix of the periodic orbit using its eigenbasis as

M_γ^k = Λ^k_γPu+ Λ^−k_γ Ps (5.4.10) where Pu is the projector onto the unstable eigenvector and Ps onto the stable one. For large k the unstable part dominates and we can get the leading order approximation to the stability amplitudes

Aζ⁰ ≈ Aζ

|Λγ|^r2 (5.4.11)

For the Maslov type indices, we can use the geometric interpretation from Creagh et al. (1990) to obtain a simple relation betweenνζ⁰ andνζ. Along a periodic orbit, the Maslov index is the number of times the stable and unstable manifolds rotate by half a turn (plus twice the number of reflections on hard walls with Dirichlet boundary conditions). After each loop along the periodic orbit, the manifolds are back where they started and the Maslov index must be an integer. As long as both ζ and ζ⁰ are close enough to the periodic orbit they will rotate with the manifolds of the orbit and ζ⁰ will pick up r times the Maslov index of the orbit γ over the

trajectory ζ. Outside of the encounter, both trajectories should be close and have the same index, giving

ν_ζ0 =ν_ζ+rµγ (5.4.12)

All of these approximations become more accurate the larger k becomes, and the closer the trajectories get to the periodic orbit. As was the case previously, the semiclassical contribution comes from the region of small stable and unstable sepa-rations, which in this case means near the periodic orbit itself, and this provides a justification of the use of these approximations.

We now have the elements we need to include trajectory pairs, that follow a trapped periodic orbit γ as described above, into the correlation function of the scattering matrix (equation (5.1.6))

C^γ,r() ≈ X

a,b

e^−¯hirSγe^iπ2rµγe^2¯ihrTγ TH|Λγ|^r2

X

ζ(a→b)

|A_ζ|²e^{¯hisu(1−Λ−rγ )}e^i¯hTζ + (r→ −r) (5.4.13)

The (r → −r) denotes a term derived from exchanging the two trajectories ζ and ζ⁰, which gives a similar contribution, but with r replaced by−r.

5.4.3 Probability of encounters

We can perform this sum in a similar way to the sums in section 5.3.1 by thinking of the approach as an encounter with the orbit. We then split the trajectory into three parts, a section from the lead (channel a) to a coordinate≈calong the stable manifold of the periodic orbit, the encounter with the orbit following a hyperbola, and then a final section where the trajectory leaves, after its unstable coordinate grows larger than the constant c, and travels back to the lead (channel b). We let the three parts have timest1,t^γenc and t2 respectively, with total time

Tζ =t₁+t^γ_enc+t₂ (5.4.14)

Trajectories, if long enough, will cover the available phase space uniformly because of ergodicity. Strictly speaking we consider trajectories as uniform over the energy shell (of the closed system), and we then weight each trajectory with its average probability of survival. We can then define a weight function that is the density that a trajectory approaches the periodic orbit closely. This means that the trajectory pierces the Poincar´e section of the orbit within a small region dudsof a point (u, s), where both coordinates are bounded by the small constant c. With uniformity, the probability that the trajectory pierces the Poincar´e section between times t₁ and t₁+ dt₁ is given by

dudsdt₁

Ω (5.4.15)

The time of the first link is free to vary between 0 andT_ζ−t^γ_enc so the total density of an encounter is

Z Tζ−t^γenc

0

dt₁duds

Ω (5.4.16)

However, in the fixed Poincar´e section we would count the trajectoryktimes, once for each time that it winds round the periodic orbit, so we should divide by this number to compensate for overcounting. kitself is given by the number of iterations in the Poincar´e section where the point (u, s) remains bounded by the constant c

k≈ 1 λγTγ

ln c

|s|+ 1 λγTγ

ln c

|u| (5.4.17)

The encounter time is approximatelyk times the period of the periodic orbit, and so it is given by

t^γ_enc≈kTγ ≈ 1 λ_γ ln c

|su| (5.4.18)

This equation has the same form that we had previously for self-encounters, and in fact this allows us to use similar methods with only minor modifications. Impor-tantly we can therefore use the semiclassical result of equation (2.2.25). The weight function, once we correct for the overcounting by a factor ofk≈ ^t_T^γenc_γ , is given by

wγ,T(s, u) =

Z T−t^γenc

0

dt1

T_γ

Ωt^γenc (5.4.19)

where T is the time of the trajectory Tζ. The coordinates of the weight function are the stable and unstable separations between the piercing point and the periodic orbit.