In Part 1 we obtained a complex superposition of states because electrons were represented in Cartesian coordinates (Bloch states) while the light was written in cylindrical coordinates. In fact, the excited (superposition) state reflects the cylindrical symmetry of the electro-magnetic field.
Since we would like to have a simpler representation of the excited states, we follow here a different path. We abandon the use of Bloch states in an attempt to make more compatible the light's and electron's representations, and express the envelope part of the electron's wavefunction by the cylindrical state [1,2]:
$$\large \psi_{bkm}(\mathbf r)={\cal N}J_m(k_r r)e^{im\phi}e^{ik_z z}u_b(\mathbf r)$$
where $b$ is the energy-band index, $\mathbf k$ represents the quantum numbers $\{k_z,k_r\}$ with $k_z=2 n/H$ and $k_r=r_{ms}/L$. $H$ and $L$ are the height and radius of the cylinder, respectively, $r_{ms}$ is the sth root of the Bessel function of order $m$, the normalization constant N, and $n$ is an integer.
The matrix element of the interaction Hamiltonian $h_I^{(+)}=-(q/m_e)\mathbf A^{(+)}(\mathbf r)\cdot\mathbf p$ giving the absorption of a photon is
$$\large \langle c\mathbf k' m'|h_I^{(+)}| v\mathbf k m\rangle = \xi e^{-i\omega t}\delta_{\ell,m'-m}\delta_{q_z,k_z'-k_z}$$
where $\xi$ contains the microscopic matrix element $\mathbf p_{cv}$ and an integral on the Bessel functions of the light beam and initial and final electron envelop states. We already see that we benefited from the coordinates change: In contrast to what happens when we use Cartesian states for electrons, the matrix element shows now a clear one-to-one transition between states with different orbital quantum number: $m'-m=\ell$.
We then use the matrix elements to construct a second-quantization Hamiltonian, and write the Heisenberg equations of motion $i\hbar d/dt\rho_{b'\alpha',b\alpha}=[\rho_{b'\alpha',b\alpha},H]$ for $\rho_{b'\alpha',b\alpha}=a^\dagger_{b'\alpha'} a_{b\alpha}$, where $a^\dagger/a$ creates and annihilates electrons in states $\psi_{bkm}(\mathbf r)$ ($\alpha=\{k,m\}$).
Low-excitation regime
The equations of motion for $\rho_{b'\alpha',b\alpha}$ are a system of coupled differential equations for populations ($\alpha'=\alpha$ and $b'=b$) and coherences, that can be solved perturbatively. To lowest order in the vector potential amplitude A, the interband coherence is
$$
\large \rho_{v\alpha',c\alpha}^{(1)} (t) = -\frac{1-e^{-i[\Delta\varepsilon_{\alpha\alpha'}-\hbar\omega]t/\hbar}}{\Delta\varepsilon_{\alpha\alpha'}-\hbar\omega}
\langle c\alpha|h_I^{(+)}| v\alpha'\rangle \,,~~~~~~~(1)
$$
where $\Delta\varepsilon_{\alpha\alpha'}=\varepsilon_{c\alpha}-\varepsilon_{v\alpha'}$ is the energy difference between states in the valence and conduction bands. For populations, for example
$$
\large \rho_{c\alpha,c\alpha}^{(2)} (t) = \frac{2}{\hbar} \sum_{\alpha'} \frac{|\langle c\alpha|h_I^{(+)}| v\alpha'\rangle|^2}{(\Delta\varepsilon_{\alpha\alpha'}-\hbar\omega)^2}
\{1-\cos[(\Delta\varepsilon_{\alpha\alpha'}-\hbar\omega)t/\hbar]\} \,. ~~~~~~~(2)
$$
Notice that Eqs. (1) and (2) are linear and quadratic in the vector potential amplitude $A$, respectively. These solutions are building blocks for the construction of mean values of other operators, such as the orbital angular momentum.
Transfer of orbital angular momentum
From optics we know that twisted light carries orbital angular momentum in the $z$-direction. We then investigate the mean value of
$$
\large L_z(t) = \sum_{b'b\alpha'\alpha} \langle b'\alpha'|(-i\hbar\partial_\phi)|b\alpha\rangle \rho_{b'\alpha',b\alpha} \,.~~~~~~~(3)
$$
We have already obtained the solutions for $\rho_{b'\alpha',b\alpha}$ that we can now insert in Eq. (3). The immediate result is that we get a linear (from interband coherences) and a quadratic (from intraband coherences and populations) contributions to the angular momentum. This is important, since then populations and coherences not only differ in strength but also in the rate at which they evolve in time.
The interband-coherence contribution is
$$
\large L_z^{(1)}(t) = \sum_{\mathbf k' m'\mathbf k m} 2{\cal R}
[{\cal L}_{v\mathbf k' m',c\mathbf k m} \rho_{v\mathbf k' m',c\mathbf k m}^{(1)}]
$$
with ${\cal L}_{v\mathbf k' m',c\mathbf k m} \propto \delta_{\pm 1,m-m'}$ and $\cal R$ the Real part. Since $ \rho_{v\mathbf k' m',c\mathbf k m}^{(1)}\propto \delta_{\ell,m-m'}$ we learn that the transfer of orbital angular momentum to electrones to first order in the vector potential only happens for TL with $\ell=1$.
In contrast, the population contribution is not limited to $\ell=1$ and is
$$
\large L_z^{(pop)}(t) = \sum_{\mathbf k m} \hbar m
[\rho_{v\mathbf k m,v\mathbf k m}^{(2)}+\rho_{c\mathbf k m,c\mathbf k m}^{(2)}]
$$
Remember that TL connects the valence-band state $m $ to the conduction-band state $m+\ell$. An unbalanced population in the conduction band is produced. This asymmetry between populations in both bands with respect to the quantum number $m$ is responsible for the net orbital angular momentum to electrons, which we call rotational photon drag -see Fig. 1.
 |
| Figure 1: The absorption of photons causes a biased population in the conduction band, which is responsible for the net orbital angular momentum gained by the semiconductor. The plot is simplified and only shows the quantum number ´m´. |
References
[1] G. F. Quinteiro and P. I. Tamborenea, Twisted-light-induced optical transitions in semiconductors: Free-carrier quantum kinetics, Phys. Rev. B 82, 125207 (2010).
[2] "The choice of boundary condition, whenever one is dealing with problems that are not explicitly concerned with effects of the metallic surface, is to a considerable degree at one's disposal and we should expect its bulk properties to be unaffected by the detailed configuration of its surface", , chapter 2 of SOLID STATE PHYSICS, Ashcroft and Mermin.
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