High-resolution Interferometric Diagnostics for Ultrashort Pulses

8.3 Action-shift induced by a weak control-fieldI shall calculate the first order change in the action of a quantum path caused by the control field.The action is written asS (j ) (ω) ≈ S (j )D + θS(j )C . (8.2)The aim is to calculate S (j )C, assuming that a set of control-field-free solutions (with subscript D)are already known. All of the other variables — for example the birth and return times of eachquantum path, and the vector potential — may be written in a similar fashion.To begin, one calculates the total derivative of the action (6.14) with respect to θ :S (j )ω,C = dS ωdθ= − t(j )rt (j )b∂ t (j )r∂θ⎧⎨⎩p(j ) + A(t ) ∂∂θA(t (j )r )+p (j ) 22p(j ) + A(t ) dt − ∂ t (j )b∂θ⎫⎬+ I p⎭⎧⎨⎩A(t (j )b )+p(j ) 22+ I p⎫⎬⎭ +(j )t r+ ω∂∂θ . (8.3)The second and third summands arise from the dependence of the action integral on its start- andend-points. The second summand is identically zero due to the birth-time saddle-point condition(6.17), whilst the third and fourth summands cancel due to the recollision-time saddle-pointcondition (6.18). Equation (8.3) therefore reduces to (j ) tS (j )rω,C = − p(j ) + A(t ) ∂ A(t )∂θt (j )bdt −∂ p(j)∂θ t(j )rt (j )bp(j ) + A(t ) dt . (8.4)The second integral is identically zero due to the momentum saddle-point condition (6.15). Writingthe total field as a sum of drive and control fields, so that A(t )=A D (t )+θ A C (t ), one obtainsthe result (j ) tS (j )rω,C = −t (j )bp(j ) + A D (t ) A C (t ) dt . (8.5)Equation (8.5) is invariant to the gauge of the control field vector potential — the addition of aconstant to A C (t ) does not change its result because of the momentum saddle-point condition.189