Solving 4th order polynomial

Question

I have this 4x4 matrix

$$ \hbar\left( \begin{array}{cccc} n \omega _0+\frac{\omega _1}{2}+\frac{\omega _2}{2} & \sqrt{n+1} g_2 & \sqrt{n+1} g_1 & 0 \\ \sqrt{n+1} g_2 & (n+1) \omega _0+\frac{\omega _1}{2}-\frac{\omega _2}{2}& T & \sqrt{n+2} g_1 \\ \sqrt{n+1} g_1 & T & (n+1) \omega _0+\frac{\omega _2}{2}-\frac{\omega _1}{2} & \sqrt{n+2} g_2 \\ 0 & \sqrt{n+2} g_1 & \sqrt{n+2} g_2 & (n+2) \omega _0-\frac{\omega _1}{2}-\frac{\omega _2}{2} \\ \end{array} \right) $$

Subscript[H, n]:= \[HBar]{{Subscript[\[Omega], 1]/2 + Subscript[\[Omega], 2]/2 + n*Subscript[\[Omega], 0], Subscript[g, 2]*Sqrt[n+1], Subscript[g, 1]*Sqrt[n+1],0},{Subscript[g, 2]*Sqrt[n+1], Subscript[\[Omega], 1]/2 - Subscript[\[Omega], 2]/2 + (n+1)*Subscript[\[Omega], 0], T, Subscript[g, 1]*Sqrt[n+2]},{ Subscript[g, 1]*Sqrt[n+1], T, -Subscript[\[Omega], 1]/2 + Subscript[\[Omega], 2]/2 + (n+1)*Subscript[\[Omega], 0], Subscript[g, 2]*Sqrt[n+2]},{0, Subscript[g, 1]*Sqrt[n+2], Subscript[g, 2]*Sqrt[n+2], -Subscript[\[Omega], 1]/2 - Subscript[\[Omega], 2]/2 + (n+2)*Subscript[\[Omega], 0]}}

which is Hermitian, implying that all its eigenvalues are real. Also, since the characteristic polynomial is of 4th order, there must be analytical solution for the eigenvalues. However, when I request Mathematica for the eigenvalues:

eigenvalues = Eigenvalues[Subscript[H, n]];

I get this mess:

In[22]:= eigenvalues[[2]]
Out[22]= 1/2 \[HBar] Root[#1^4+32 Subsuperscript[g, 1, 4]+48 n Subsuperscript[g, 1, 4]+16 n^2 Subsuperscript[g, 1, 4]-64 Subsuperscript[g, 1, 2] Subsuperscript[g, 2, 2]-96 n Subsuperscript[g, 1, 2] Subsuperscript[g, 2, 2]-32 n^2 Subsuperscript[g, 1, 2] Subsuperscript[g, 2, 2]+32 Subsuperscript[g, 2, 4]+48 n Subsuperscript[g, 2, 4]+16 n^2 Subsuperscript[g, 2, 4]+64 T Subscript[g, 1] Subscript[g, 2] Subscript[\[Omega], 0]+160 n T Subscript[g, 1] Subscript[g, 2] Subscript[\[Omega], 0]+64 n^2 T Subscript[g, 1] Subscript[g, 2] Subscript[\[Omega], 0]-32 n T^2 Subsuperscript[\[Omega], 0, 2]-16 n^2 T^2 Subsuperscript[\[Omega], 0, 2]-32 Subsuperscript[g, 1, 2] Subsuperscript[\[Omega], 0, 2]-112 n Subsuperscript[g, 1, 2] Subsuperscript[\[Omega], 0, 2]-112 n^2 Subsuperscript[g, 1, 2] Subsuperscript[\[Omega], 0, 2]-32 n^3 Subsuperscript[g, 1, 2] Subsuperscript[\[Omega], 0, 2]-32 Subsuperscript[g, 2, 2] Subsuperscript[\[Omega], 0, 2]-112 n Subsuperscript[g, 2, 2] Subsuperscript[\[Omega], 0, 2]-112 n^2 Subsuperscript[g, 2, 2] Subsuperscript[\[Omega], 0, 2]-32 n^3 Subsuperscript[g, 2, 2] Subsuperscript[\[Omega], 0, 2]+32 n Subsuperscript[\[Omega], 0, 4]+80 n^2 Subsuperscript[\[Omega], 0, 4]+64 n^3 Subsuperscript[\[Omega], 0, 4]+16 n^4 Subsuperscript[\[Omega], 0, 4]+#1^3 (-8 Subscript[\[Omega], 0]-8 n Subscript[\[Omega], 0])+16 T Subscript[g, 1] Subscript[g, 2] Subscript[\[Omega], 1]-16 T^2 Subscript[\[Omega], 0] Subscript[\[Omega], 1]-24 Subsuperscript[g, 1, 2] Subscript[\[Omega], 0] Subscript[\[Omega], 1]-16 n Subsuperscript[g, 1, 2] Subscript[\[Omega], 0] Subscript[\[Omega], 1]+8 Subsuperscript[g, 2, 2] Subscript[\[Omega], 0] Subscript[\[Omega], 1]+16 Subsuperscript[\[Omega], 0, 3] Subscript[\[Omega], 1]+32 n Subsuperscript[\[Omega], 0, 3] Subscript[\[Omega], 1]+16 n^2 Subsuperscript[\[Omega], 0, 3] Subscript[\[Omega], 1]+4 T^2 Subsuperscript[\[Omega], 1, 2]+12 Subsuperscript[g, 1, 2] Subsuperscript[\[Omega], 1, 2]+8 n Subsuperscript[g, 1, 2] Subsuperscript[\[Omega], 1, 2]-12 Subsuperscript[g, 2, 2] Subsuperscript[\[Omega], 1, 2]-8 n Subsuperscript[g, 2, 2] Subsuperscript[\[Omega], 1, 2]-4 Subsuperscript[\[Omega], 0, 2] Subsuperscript[\[Omega], 1, 2]-16 n Subsuperscript[\[Omega], 0, 2] Subsuperscript[\[Omega], 1, 2]-8 n^2 Subsuperscript[\[Omega], 0, 2] Subsuperscript[\[Omega], 1, 2]-4 Subscript[\[Omega], 0] Subsuperscript[\[Omega], 1, 3]+Subsuperscript[\[Omega], 1, 4]+16 T Subscript[g, 1] Subscript[g, 2] Subscript[\[Omega], 2]-16 T^2 Subscript[\[Omega], 0] Subscript[\[Omega], 2]+8 Subsuperscript[g, 1, 2] Subscript[\[Omega], 0] Subscript[\[Omega], 2]-24 Subsuperscript[g, 2, 2] Subscript[\[Omega], 0] Subscript[\[Omega], 2]-16 n Subsuperscript[g, 2, 2] Subscript[\[Omega], 0] Subscript[\[Omega], 2]+16 Subsuperscript[\[Omega], 0, 3] Subscript[\[Omega], 2]+32 n Subsuperscript[\[Omega], 0, 3] Subscript[\[Omega], 2]+16 n^2 Subsuperscript[\[Omega], 0, 3] Subscript[\[Omega], 2]+8 T^2 Subscript[\[Omega], 1] Subscript[\[Omega], 2]-8 Subsuperscript[\[Omega], 0, 2] Subscript[\[Omega], 1] Subscript[\[Omega], 2]+4 Subscript[\[Omega], 0] Subsuperscript[\[Omega], 1, 2] Subscript[\[Omega], 2]+4 T^2 Subsuperscript[\[Omega], 2, 2]-12 Subsuperscript[g, 1, 2] Subsuperscript[\[Omega], 2, 2]-8 n Subsuperscript[g, 1, 2] Subsuperscript[\[Omega], 2, 2]+12 Subsuperscript[g, 2, 2] Subsuperscript[\[Omega], 2, 2]+8 n Subsuperscript[g, 2, 2] Subsuperscript[\[Omega], 2, 2]-4 Subsuperscript[\[Omega], 0, 2] Subsuperscript[\[Omega], 2, 2]-16 n Subsuperscript[\[Omega], 0, 2] Subsuperscript[\[Omega], 2, 2]-8 n^2 Subsuperscript[\[Omega], 0, 2] Subsuperscript[\[Omega], 2, 2]+4 Subscript[\[Omega], 0] Subscript[\[Omega], 1] Subsuperscript[\[Omega], 2, 2]-2 Subsuperscript[\[Omega], 1, 2] Subsuperscript[\[Omega], 2, 2]-4 Subscript[\[Omega], 0] Subsuperscript[\[Omega], 2, 3]+Subsuperscript[\[Omega], 2, 4]+#1^2 (-4 T^2-12 Subsuperscript[g, 1, 2]-8 n Subsuperscript[g, 1, 2]-12 Subsuperscript[g, 2, 2]-8 n Subsuperscript[g, 2, 2]+20 Subsuperscript[\[Omega], 0, 2]+48 n Subsuperscript[\[Omega], 0, 2]+24 n^2 Subsuperscript[\[Omega], 0, 2]+4 Subscript[\[Omega], 0] Subscript[\[Omega], 1]-2 Subsuperscript[\[Omega], 1, 2]+4 Subscript[\[Omega], 0] Subscript[\[Omega], 2]-2 Subsuperscript[\[Omega], 2, 2])+#1 (-48 T Subscript[g, 1] Subscript[g, 2]-32 n T Subscript[g, 1] Subscript[g, 2]+16 T^2 Subscript[\[Omega], 0]+16 n T^2 Subscript[\[Omega], 0]+40 Subsuperscript[g, 1, 2] Subscript[\[Omega], 0]+80 n Subsuperscript[g, 1, 2] Subscript[\[Omega], 0]+32 n^2 Subsuperscript[g, 1, 2] Subscript[\[Omega], 0]+40 Subsuperscript[g, 2, 2] Subscript[\[Omega], 0]+80 n Subsuperscript[g, 2, 2] Subscript[\[Omega], 0]+32 n^2 Subsuperscript[g, 2, 2] Subscript[\[Omega], 0]-16 Subsuperscript[\[Omega], 0, 3]-80 n Subsuperscript[\[Omega], 0, 3]-96 n^2 Subsuperscript[\[Omega], 0, 3]-32 n^3 Subsuperscript[\[Omega], 0, 3]+8 Subsuperscript[g, 2, 2] Subscript[\[Omega], 1]-16 Subsuperscript[\[Omega], 0, 2] Subscript[\[Omega], 1]-16 n Subsuperscript[\[Omega], 0, 2] Subscript[\[Omega], 1]+8 Subscript[\[Omega], 0] Subsuperscript[\[Omega], 1, 2]+8 n Subscript[\[Omega], 0] Subsuperscript[\[Omega], 1, 2]+8 Subsuperscript[g, 1, 2] Subscript[\[Omega], 2]-16 Subsuperscript[\[Omega], 0, 2] Subscript[\[Omega], 2]-16 n Subsuperscript[\[Omega], 0, 2] Subscript[\[Omega], 2]+8 Subscript[\[Omega], 0] Subsuperscript[\[Omega], 2, 2]+8 n Subscript[\[Omega], 0] Subsuperscript[\[Omega], 2, 2])&,2]

Why doesn't Root[] evaluate? How do I get the eigenvalues? Thanks a lot.