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Long-lived dark states, in which an experimentally accessible qubit is not in thermal equilibrium with a surrounding spin bath, are pervasive in solid-state systems. We explain the ubiquity of dark states in a large class of inhomogeneous central spin models using the proximity to integrable lines with exact dark eigenstates. At numerically accessible sizes, dark states persist as eigenstates at large deviations from integrability, and the qubit retains memory of its initial polarization at long times. Although the eigenstates of the system are chaotic, exhibiting exponential sensitivity to small perturbations, they do not satisfy the eigenstate thermalization hypothesis. Rather, we predict long relaxation times that increase exponentially with system size. We propose that this intermediatechaotic but non-ergodicregime characterizes mesoscopic quantum dot and diamond defect systems, as we see no numerical tendency towards conventional thermalization with a finite relaxation time.

 

We discuss how the language of wave functions (state vectors) andassociated non-commuting Hermitian operators naturally emerges fromclassical mechanics by applying the inverse Wigner-Weyl transform to thephase space probability distribution and observables. In this language,the Schr"odinger equation follows from the Liouville equation, with \hbar ℏ now a free parameter. Classical stationary distributions can berepresented as sums over stationary states with discrete (quantized)energies, where these states directly correspond to quantum eigenstates.Interestingly, it is now classical mechanics which allows for apparentnegative probabilities to occupy eigenstates, dual to the negativeprobabilities in Wigner’s quasiprobability distribution. These negativeprobabilities are shown to disappear when allowing sufficientuncertainty in the classical distributions. We show that thiscorrespondence is particularly pronounced for canonical Gibbs ensembles,where classical eigenstates satisfy an integral eigenvalue equation thatreduces to the Schr"odinger equation in a saddle-pointapproximation controlled by the inverse temperature. We illustrate thiscorrespondence by showing that some paradigmatic examples such astunneling, band structures, Berry phases, Landau levels, levelstatistics and quantum eigenstates in chaotic potentials can bereproduced to a surprising precision from a classical Gibbs ensemble,without any reference to quantum mechanics and with all parameters(including \hbar ℏ )on the order of unity.