Perfect State Transfer from a Localised Two-Excitation State to a Dicke State via Static Spin-Network Hamiltonians
Abstract
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≥4. The Hamiltonian has the physical form H=∑i<jJij(σi+σj-+σj+σi-)+∑iεi ni with real couplings, and satisfies e−iHt|110⋯0⟩=eiϕ|DN(2)⟩ at a finite time. An SN−2 permutation symmetry on the unoccupied spins reduces the dynamics to a four-dimensional invariant subspace. Requiring (|ψ0⟩+|DN(2)⟩)/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 N1/2 and the on-site range as N3/2, 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.
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BibTeX
@misc{dutta2026perfect,
title = {{Perfect State Transfer from a Localised Two-Excitation State to a Dicke State via Static Spin-Network Hamiltonians}},
author = {Soumyojyoti Dutta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09654},
eprint = {2609.09654},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09654},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Perfect State Transfer from a Localised Two-Excitation State to a Dicke State via Static Spin-Network Hamiltonians
AU - Soumyojyoti Dutta
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.09654
UR - https://arxiv.org/abs/2609.09654
AB - 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≥4. The Hamiltonian has the physical form H=∑_i<jJ_ij(σ_i^+σ_j^-+σ_j^+σ_i^-)+∑_iε_i n_i with real couplings, and satisfies e^−iHt|110⋯0⟩=e^iϕ|D_N^(2)⟩ at a finite time. An S_N−2 permutation symmetry on the unoccupied spins reduces the dynamics to a four-dimensional invariant subspace. Requiring (|ψ_0⟩+|D_N^(2)⟩)/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.
ER -