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

Fortschr. Phys. 51, No. 4–5 (2003) 483
from theWigner map. Moreover, wecalculatetheintegral of Sl(κ; x) over one spatial period. This quantity
establishes a crucial connection between Sl(κ) and Sl(κ; x).
We start the discussion by first noting that both functions are Fourier coefficients. Indeed, the coefficients
Sl result from the expansion of exp[−iκV (x)] whereas Sl comefrom exp[−iκV(x, y)] where V(x, y) ≡
V (x + y) − V (x − y). For anti-symmetric potentials V (−x) =−V (x) wefind V(0,y)=2V (y) which
establishes the connection
Sl (κ;0)=Sl (2κ)
between the two expansion coefficients.
Moreover, it is also important that due to the anti-symmetry V (−x) =−V (x) of thepotential and the
resulting anti-symmetry V(x, −y) =−V(x, y) of the generalized potential both functions Sl and Sl are
purely real.
Theintegral
Il ≡
π
−π
dx Sl(κ; x)
over the expansion coefficients Sl(κ; x) brings out a deep connection with Sl. Indeed, when we substitute
theexpression Eq. (26) for Sl into thedefinition of Il and use the Fourier representations, Eq. (8), of
exp[−iκV (x + y)] and exp[iκV (x − y)] wearriveat
Il =
r
s
SrSs
π
1
dx e
2π
−i(r−s)x
π
dy e i(l−r−s)y .
Theintegration over x yields the condition r = s and thus
Il =
Hence, we find
and
r
I2s+1 =
I2s =
π
−π
S 2 r
π
−π
π
−π
−π
dy e i(l−2r)y .
dx S2s+1(κ; x) =0
dx S2s(κ; x) =2πS 2 s (κ).
−π
For all odd values l theintegral of Sl over one period vanishes, whereas for all even values this integral
produces thesquareof thecoefficients Sl of thestatespacemapping. This featureguarantees that the
interference terms in phase space vanish when integrated over position.
B Momentum spread
In this Appendix weevaluatethesum
I(z) ≡
∞
l=−∞
l 2 S 2 l (z)