High-resolution Interferometric Diagnostics for Ultrashort Pulses

8. QUANTUM-PATH INTERFEROMETRY IN HIGH-HARMONIC GENERATIONTo visualize the control-field sensitivity vectors, it is useful to plot them as points in controlfieldsensitivity space (α 1 ,α 2 ,...). One can connect such a diagram with the multi-dimensionalFourier transform of (ω; ¯θ ) taken along θ 1 ,θ 2 ,...:˜(ω;ᾱ)=(2π) L/2 j=(2π) L/2 j (j ) (ω)e iS(j )ω,D δ(α1 −S (j )ω,1 ,α 2 −S (j )ω,2 ,...) (j ) (ω)e iS(j )ω,D δ(ᾱ − ¯S (j )ω,C ) (8.12)where L is the number of control fields. In this control-field-sensitivity space, each quantum trajectoryis Dirac-delta function located at ¯S (j )ω,C .8.4 ExampleI shall consider quasi-monochromatic control fields — that is, pulses which are significantly longerthan the drive field. The control-field frequency is considered to be fixed, and one may parameterizeits remaining degrees of freedom by the (real-valued) amplitude of the in-phase and quadraturecomponents (with respect to the drive field), denoted θ I and θ Q respectively. I use this rectangularrepresentation because the resulting control field is a linear combination of basis functions,and it thus fits naturally into the perturbative formalism. Of course, in the laboratory it may bemore convenient to scan the amplitude and phase of the control field, but this provides the sameinformation and may be digitally converted into rectangular coordinates.Mathematically, the control field is C (t )=E 0θ I cos(ω C t )+θ Q cos(ω C t − π 2 ) . (8.13)The control field amplitudes (ω Q ,ω I ) are scaled by the peak amplitude of the drive field E 0 . I willconsider the effect of this control field on two drive fields: a monochromatic field and a few-cyclepulse.192