Generalized length-gauge time-dependent Liouville equation.

The length-gauge (LG) time-dependent Liouville equation is a major workhorse to simulate ultrafast dynamics. The equation features the commutator between the density operator ρ and the position operator r, nk|[r, ρ]|mk ' , where |nk and |mk ' are Bloch states for bands n and m at crystal momenta k and k ' , respectively. This commutator has two irreconcilable versions: Kohn and Luttinger (KL) (Phys. Rev. 108, 590 (1954)) have two partial derivatives with respect to k and k ' , but Aversa and Sipe (AS) (Phys. Rev. B 52, 14636 (1995)) have one partial derivative and one Dirac delta. For several decades, this outstanding discrepancy has received little attention. Here, we show that the generalized LG time-dependent Liouville equation has two partial derivatives, but no Dirac delta, regardless of whether the electric field is homogeneous or not. Physically, the k-off-diagonal matrix elements of the position operator provide a channel by which the electric field generates k-off-diagonal matrix elements for ρ in LG. The discrepancy emerges when the position matrix is assumed to be diagonal in k. Consequently the AS' equation is a limiting case of our generalized Liouville equation, where the relative momentum Q between two states is ignored.Our numerical test reveals that the k-diagonal formalism yields qualitatively different results and underestimates both energy absorption and polarization by a factor of 7. Our study establishes a general formalism for the length-gauge time-dependent Liouville equation and opens the door for future research on the ultrafast time scale, while prior results based on the k-diagonal only equation may need to be revisited.