Evolution in time of an N-atom system. II. Calculation of the eigenstates (9 pages)

U. Fano, John L. Bohn

Terry Rudolph, Itay Yavin, Helen S. Freedhoff

We calculate the energy eigenvalues and eigenstates corresponding to coherent single and multiple excitations of a number of different arrays of N identical two-level atoms (TLA's) or qubits, including polygons, ``diamond'' structures, polygon multilayers, icosahedra, and dodecahedra. We assume only that the coupling occurs via an exchange interaction which depends on the separation between the atoms. We include the interactions between all pairs of atoms, and our results are valid for arbitrary separations relative to the radiation wavelength.

Emmanuel Fromager

Lev Vidmar, Lucas Hackl, Eugenio Bianchi A +1 more

In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is ${S}_{\text{ave}}\ensuremath{\simeq}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(1/2){\mathcal{D}}_{\mathrm{A}}^{2}/\mathcal{D}$ for $1\ensuremath{\ll}{\mathcal{D}}_{\mathrm{A}}\ensuremath{\le}\sqrt{\mathcal{D}}$, where ${\mathcal{D}}_{\mathrm{A}}$ and $\mathcal{D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy $⟨S⟩$ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}\ensuremath{\le}⟨S⟩\ensuremath{\le}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}[1/(2\mathrm{ln}2)](\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}$. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then $⟨S⟩<\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}$ in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

T. Wolfram, Şinasi S. Ellialtioğlu

Osvaldo Civitarese, M. Reboiro, Diego Tielas

A cavity quantum electrodynamical Hamiltonian is solved to study spin observables, such as the orientation and fluctuations of the total spin and entropy, in a system of N two-level atoms placed in a resonant cavity. The eigenstates of the Hamiltonian are calculated, both exactly and approximately. We calculate the time evolution of spin observables such as spin and entropy squeezing. Conclusions are drawn about the validity of the approximations in the limit of a large number of atoms.