Kicked rotor in Wigner phase space - The University of Texas at Austin

484 M. Bienert et al.: Kicked rotor in Wigner phase space
containing the expansion coefficients
Sl (z) = 1
π
2π
−π
dx e ilx −izV (x)
e
of the state vector map. This sum determines the spread of the momentum at the main resonance.
When wesubstitutethedefinition, Eq. (27), of theexpansion coefficients Sl(z) into thesum and exchange
integration and summation wearriveat
I(z) = 1
4π2 π
−π
dx ′
π
−π
dx
l
l 2 e il(x+x′ )
(27)
e −iz[V (x)+V (x′ )] . (28)
The representation
e ilx =2π
δ(x +2πν) (29)
l
ν
of a comb of delta functions allows us to express the term in the square brackets in Eq. (28) as the second
derivativeof thedelta function, that is
l 2 e il(x+x′
)
= −2π
l
ν
∂ 2
∂x 2 δ(x + x′ +2πν).
Thederivativecan beshifted to theother x–dependent part e −izV (x) of theintegrand. When weintegrate
over the remaining delta function we find
I(z) =− 1
π
2π
−π
−izV (−x) d2
dx e
dx2 e−izV (x) .
Here we have recognized that the integration only extends over a single interval of 2π which reduces the
summation over ν to theterm ν =0.
Thesymmetry V (−x) =−V (x) of the potential yields
I(z) = 1
π
dx z
2π
2
−π
2 d
V (x)
dx
+ iz d2
V (x) .
dx2 Dueto theanti-symmetry of thepotential thesecond term of theintegral does not contributeand weobtain
theresult
I(z) =
∞
l=−∞
l 2 S 2 l (z) =z 2 ·〈F 2 〉.
Herewehaveintroduced theaverage
〈F 2 〉≡ 1
π
2 d
dx V (x)
2π dx
−π
of thesquareof theforceF = −dV/dx acting on theparticle.