Perfect State Transfer from a Localised Two-Excitation State to a Dicke State via Static Spin-Network Hamiltonians

RESUMO

I construct a family of time-independent, excitation-preserving spin Hamiltonians realising perfect state transfer from a localised two-excitation state to the symmetric two-excitation Dicke state, for every $N\ge4$. The Hamiltonian has the physical form $H=\sum_{i<j}J_{ij}(\sigma_i^+\sigma_j^-+\sigma_j^+\sigma_i^-)+\sum_i\epsilon_i n_i$ with real couplings, and satisfies $e^{-iHt}|110\cdots0\rangle=e^{i\phi}|D_N^{(2)}\rangle$ at a finite time. An $S_{N-2}$ permutation symmetry on the unoccupied spins reduces the dynamics to a four-dimensional invariant subspace. Requiring $(|\psi_0\rangle+|D_N^{(2)}\rangle)/2$ to be a zero eigenvector fixes the on-site energies in closed form and leaves three coupling parameters free. The inverse spectral problem then becomes two polynomial equations in two coupling ratios; eliminating one gives a degree-six reciprocal polynomial, which $z=x+x^{-1}$ converts to a cubic. For the spectral family $(-n,-1,1)$ with odd $n$, factorising the cubic's leading coefficient and evaluating it at $z=-2$ shows that some odd $n$ always produces a real root below $-2$, which a subresultant lifts back to the original system. The existence argument is symbolic and uses no numerical optimisation. Since that coefficient contains no odd powers of $n$, the required $n$ comes with an explicit threshold, not an asymptotic guarantee. I also cost the construction: couplings grow as $N^{1/2}$ and the on-site range as $N^{3/2}$, the transfer time stays within a factor 2.3-3.0 of the Mandelstam-Tamm limit at every size, and the fidelity is sensitive to systematic drift of the spectator-spectator coupling class but tolerant of independent bond disorder, which self-averages. This is a constrained analogue of perfect state transfer: for general real states an unconstrained real symmetric matrix suffices, whereas here the Hamiltonian must have excitation-preserving spin-network form.