= Solution
For a periodic single-electron <Hamiltonian operator>, <Bloch's theorem> gives
$$
\psi_{b\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{b\mathbf k}(\mathbf r),\quad u_{b\mathbf k}(\mathbf r+\mathbf R)=u_{b\mathbf k}(\mathbf r),\quad \psi_{b\mathbf k}(\mathbf r+\mathbf R)=e^{i\mathbf k\cdot\mathbf R}\psi_{b\mathbf k}(\mathbf r).
$$
Two bands at the same $\mathbf k$ have the same translation <eigenvalues> but different cell-periodic <eigenfunctions>, generally different energies and different orbital/nodal character. They may be chosen orthogonal even at a band degeneracy. The basis choices are treated in the two scoped branches below.
For a slowly varying external potential $U(\mathbf r)$, the semiclassical wavepacket <Hamiltonian> is $\epsilon_b(\mathbf k)+U(\mathbf r)$. Hamilton's equation for the conjugate variables $\mathbf r,\hbar\mathbf k$ gives $\hbar\dot{\mathbf k}=-\nabla U=\mathbf F$. Equivalently the electric perturbation changes the translation phase at the rate $q\mathbf E\cdot\mathbf R/\hbar$. This is the single-band acceleration theorem, requiring a slowly varying field and negligible interband tunnelling/scattering; it refers to <crystal momentum>, not free-space mechanical momentum.
Let $e>0$ and $K=eE_0/(\hbar\omega)$. Integrating from the specified initial condition $\mathbf k(0)=0$ gives, with the first field alone,
$$
\boxed{k_x=K(\cos\omega t-1),\qquad k_y=0}.
$$
With both components switched on,
$$
\boxed{k_x=K(\cos\omega t-1),\qquad k_y=-K\sin\omega t}.
$$
This is a <circle> centred at $(-K,0)$, not at the band minimum. Since $v_i=\hbar k_i/m_i^*$, integration gives
$$
\boxed{x-x_0=\frac{eE_0}{m_x^*\omega^2}(\sin\omega t-\omega t),\quad y-y_0=\frac{eE_0}{m_y^*\omega^2}(\cos\omega t-1)}.
$$
The <switch-on drift in a circularly driven parabolic band> accompanies unequal transverse scales: it is a stretched cycloid. In a frame moving with that drift, the oscillation is an <ellipse> with semiaxis ratio $m_y^*/m_x^*\gg1$. A closed real-space <ellipse> would require different initial momentum. Indeed, after one period $x-x_0=-2\pi eE_0/(m_x^*\omega^2)\ne0$, an explicit check against that interpretation.
Writing $\theta=\omega t$, the kinetic energy above the band minimum is
$$
\epsilon-\epsilon_\Gamma=\frac{e^2E_0^2}{2\omega^2}\left[\frac{(1-\cos\theta)^2}{m_x^*}+\frac{\sin^2\theta}{m_y^*}\right].
$$
For $m_x^*<m_y^*$ it is largest at $\theta=\pi$, with value $2e^2E_0^2/(m_x^*\omega^2)$. Equating this to the optical-phonon energy gives the emission threshold estimate
$$
\boxed{E_{0,\max}\simeq\frac\omega e\sqrt{\frac{m_x^*\hbar\omega_{\rm LO}}2}=5.80\times10^5\ \mathrm{V\,m^{-1}}}.
$$
This treats the optical branch as approximately dispersionless and assumes a momentum-allowed final state, a cold initial <electron>, and negligible <phonon> absorption. It is the amplitude of each Cartesian field component; the circular field has constant magnitude $E_0$. A steady periodic orbit centred at $\mathbf k=0$ would instead give twice this amplitude, but fails the stated switch-on condition.
\Image[/past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2016-driven-electron-orbits.png]
{title=Shifted circular crystal-momentum orbit and drifting anisotropic real-space path}
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