and therefore introducing $\tilde{a} \propto \sqrt{\alpha} \pi_x + \frac{i}{\sqrt{\alpha}}\pi_y$ and $\tilde{a}^\dagger \propto  \sqrt{\alpha} \pi_x - \frac{i}{\sqrt{\alpha}}\pi_y$, we can immediatealy write down the single particle orbitals for this case: $$\label{singleparticleorbitalg}
\phi_l^{\alpha} (x,y) = \left( \frac{x}{\sqrt{\alpha}} + iy\sqrt{\alpha} \right)^l e^{-\frac{1}{4\ell_B^2} \left( \frac{x^2}{\alpha} + \alpha y^2 \right)}.$$ Notice that the probability density $|\phi_l|^2$ is no longer localized on a circle, but rather an ellipse for $\alpha \neq 1$. Therefore, on a single particle level, the effect of mass anisotropy is to stretch or squeeze the one-body orbitals along certain directions, possibly rotating the principal axis (for $\phi \neq 0$).

As we mentioned in Sec. \[sec\_introduction\], certain semiconductor materials are likely to have non-trivial metric defined by the anisotropy. Alternatively, the effective mass tensor can be experimentally tuned by tilting the magnetic field [@xia2011]. Tilting is known to produce complicated effects because it induces the coupling between electronic subbands and LL mixing, and a detailed analysis will be presented elsewhere. However, with a normal confinement (perpendicular to the Hall surface) given by a harmonic well, the Hamiltonian with a strong perpendicular magnetic field $B$ and a non-zero in-plane magnetic field $B_\parallel$ can be solved exactly, yielding two characteristic harmonic oscillator frequencies [@demler]: $$\begin{aligned}
\omega_{1,2}^2=\frac{1}{2}[(\omega_z^2+\omega_c^2)\pm(\omega_z^2-\omega_c^2)\sec(2\bar{\theta})],\end{aligned}$$ 