Following our previous post and Ref. [1], we want to use the equation for the interband coherence
(iℏddt−Δ)ρvk′,ck(t)+∑i≠0Viρvk′−i,ck−i(t)=⟨ck|[−(Q/m)A(r,t)⋅p]|vk′⟩ (1)
(see Below-bandgap excitation. Part 1) with a twisted-light vector potential. Then, the matrix elements in Cartesian coordinates are
⟨ck|[−(Q/m)A(r,t)⋅p]|vk′⟩=ξ(t)eiθℓLqr
where k′=k−q0 and q0=qrcos(θ)ˆx+qrsin(θ)ˆy+qzˆz with variable θ [2]. Following the procedure for plane waves, we find
[ℏω−10(Eg+ℏ2q202|m∗v|)+iℏ2q0|m∗v|⋅∇+ℏ22μ∇2+V(r)]ρq0(ω,r)=L3ξ(ω)eiθℓLqrδ(r)(2).
Notice the new factor in the RHS.
Once again we can solve by completing squares and using the unitary transformation U(r)=exp[i(μ/|m∗v|)q0⋅r]:
ρvk−q0,ck(ω)=L3ξ(ω)eiθℓLqr×∑νψ∗ν(r=0)ℏω−Eg−ℏ2q202M−Eνψν(k−μ|m∗v|q0)(3).
Notice the new phase factor.
Let us focus first on the center-of-mass (COM) motion of the exciton-like particle, signaled by ℏ2q20/(2M) in the denominator of Eq. (3). Thanks to the square, all interband coherences (differing in their q0 or more specifically in the angle θ) share the same COM momentum given by {qr,qz}. For a plane wave, ℏqz is clearly the momentum of the photon traveling along z; for twisted light, qz is the component of the wave-vector in the z direction. If we calculated the linear momentum of the field P=ϵ0E×B in the paraxial approximation qr<qz we would observe that the radial and angular components of P are proportional to qr, and so qr is the transverse wave-vector. The excitons are then absorbing the momentum of the twisted light beam, which contributes the kinetic energy ℏ2q20/(2M) that adds to Eg. To extend our understanding, recall how we expanded the twisted-light vector potential in terms of plane-waves (end of Absorption in Bulk. Part 1): we could then say that each plane-wave with a particular linear momentum transfers it to the COM of an exciton (exciton-like).
The second interesting feature is the argument of ψν in Eq. (3). We know that a pure exciton is represented by ψν(k); them this tells that twisted light creates state that are not exactly excitons, or in other words, the relative-motion wave particle is perturbed.
Finally, and also in connection to the finding in Absorption in Bulk. Part 1 related to the expansion of A(r,t) we see that Eq. (3) contains a particular phase factor for each value of θ.
Macroscopic quantities
Equation (3) is the building-block of several macroscopic quantities. I show as an example the polarization [1]. Because the twisted light field has spatial variation, one must consider a local polarization
P(R,t)=2(LRL)3∑k,θℜ{eiq0⋅Rdvcρvk−q0,ck(t)},
corresponding to a macroscopic cell located at R=(R,Φ,Z) with linear size LR; ℜ is the real part. Now it is evident that the response of the system to the twisted-light beam takes into account a superposition of states differing in the value of θ -for plane waves, the polarization still retains the sum over k. From the polarization one can derive the susceptibility which clearly shows the absorption lines, in this case shifted by the COM term.
References
[1] I recall that, in problems of twisted light-matter interaction, one should not neglect of the photon's momentum. We therefore have abandoned the "vertical transition" approximation.
[2] Notice the connection between the vector q0 of electrons and the vector q(η) of the light entering the superposition of planes waves that form a twisted-light vector potential - see the end of the previous post Absorption in Bulk. Part 1.
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