At the start of my Ph.D., I spent a couple of months working out the gravity of a laser pulse. While I never tried to publish this (I should have), it did entail making some fun videos in the Wolfram language.

On the left you can see the `gravitomagnetic' (Lense-Thirring) part of the gravitational field of six separated photons, emitted from a stationary point with spin \[|\Psi\rangle\propto|\uparrow\rangle|\downarrow\rangle|\uparrow\rangle|\downarrow\rangle|\uparrow\rangle|\downarrow\rangle\] You could localise this with enough LSZ out-states. The field exists on a surface abreast of the source, since the speed of gravity is equal to the speed of light \(c\). This reasoning is illustrated below by real and apparent positions of a moving body relative to an obsever.
So, how could you detect such a field? The gravitomagnetic part is sourced by the photon spin. It couples to the momentum of other matter sources, which feel a kick as the field passes. Thus, the photons will carve out a `wake' as they pass through an ideal gas with some momentum distribution, shown on the right for the dipole \(|\Psi\rangle\propto|\uparrow\rangle|\downarrow\rangle\). The same is shown below for the monopole \(|\Psi\rangle\propto|\uparrow\rangle\), both perpendicular and parallel to the photon path. The gas should be very cold, because the wake would be quickly erased by hot particles.

To my knowledge, none of this has any current application. Plausibly, it could someday be used as an inefficient test of the weak Lense-Thirring effect, but even with the most powerful lasers this is not yet possible. Quel chagrin…!