High-resolution Interferometric Diagnostics for Ultrashort Pulses

6.2 Theory of the single-atom responsewhere d(v) =〈v|x|g 〉 is the atomic dipole matrix element for a bound-free transition. I have assumedthat the orbital is symmetric so that 〈g |x|g 〉 = 0 and also that the relevant continuum statesare highly energetic so that 〈v|g 〉≈0. At this stage I assume that a solution for g (t ) can be foundor approximated by other means, such as the Ammosov, Delone, and Kraïnov (ADK) formula fortunnel ionization [370]. Equation (6.6) can be transformed into an ordinary differential equationby writing f (v,t )=b(p,t ), where p = v − A(t ) is the canonical momentum. Upon performing thissubstitution, (6.6) is solved to yield tb(p,t )=−i dt b ion (t b ,p)e iS(t ,t b,p)−∞(6.7)whereandS(t ,t b ,p)=− t p + A(tdt ) 22t b+ I p(6.8) ion (t b ,p)=g (t b ) (t b ) · d(p + A(t b )). (6.9)Equation (6.7) can be interpreted as follows: electrons of canonical momentum p are ionized intothe continuum at time t b with amplitude ion (t b ,p), which is the product of the ground stateamplitude, the electric field strength and the amplitude of the corresponding dipole transition.Because the canonical momentum p is an invariant of the motion of a free charged particle in antime-varying electric field, the Volkov states |p〉 are eigenstates of the field Hamiltonian. They havetime-varying eigenvalues and their evolution in the field is represented by a phase S(t ,t b ,p). Theionization potential enters (6.8) because the oscillations of the ground state are factored out in(6.5).The harmonic emission results from rapid oscillations of the atomic dipole (t ), computedusing the ansatz (6.5) and the solution (6.7). This gives(t )=−i td 3 p dt b ion (t b ,p)e iS(t ,tb,p) rec (t ,p)+c.c. (6.10)−∞137