A long-standing problem of the low-energy dynamics of a disordered XY spin chain is reexamined. The case of a rigid chain is studied, where the quantum effects can be treated quasiclassically. It is shown that, as the frequency decreases, the relevant excitations change from localized spin waves to two-level systems to soliton-antisoliton pairs. The linear-response correlation functions are calculated. The results apply to other periodic glassy systems such as pinned density waves, planar vortex lattices, stripes, and disordered Luttinger liquids.
For the Davydov Hamiltonian several special cases are known which can be solved analytically. Starting from these cases we show that the initial state for a simulation using Davydov's \ensuremath{\Vert}
D1〉 approximation has to be constructed from a given set of initial lattice displacements and momenta in the form of a coherent state with its amplitudes independent of the lattices site, corresponding to Davydov's \ensuremath{\Vert}
D2〉 approximation. The site dependences in the \ensuremath{\Vert}
D1〉 ansatz evolve from this initial state exclusively via the equations of motion. Starting the \ensuremath{\Vert}
D1〉 simulation from an ansatz with site-dependent coherent-state amplitudes leads to an evolution which is different from the analytical solutions for the special cases. Thus also in applications of the \ensuremath{\Vert}
D1〉 ansatz to polyacetylene \ensuremath{\Vert}
D2〉-type initial states always have to be used in contrast to our previous suggestion [W. F\"orner, J. Phys. Condens. Matter 6, 9105 (1994)]. Further we expand the known exact solutions in Taylor series in time and compare expectation values in different orders with the exact results.We find that for an approximation up to third order in time (for the wave function) norm, and total energy, as well as displacements and momenta are reasonably correct for a time up to \ensuremath{\approxeq}0.12--0.14 ps, depending somewhat on the coupling strengh for the transportless case. We performed long-time simulations using the \ensuremath{\Vert}
D1〉 approximation where we computed expectation values of the relevant operators with the state (H^/J)\ensuremath{\Vert}
D1〉 and the deviation \ensuremath{\Vert}\ensuremath{\delta}〉 from the exact solution over long times, namely 10 ns. We found that in the very long-time scale the \ensuremath{\Vert}
D1〉 ansatz is very close to an exact solution. Further we report results from an investigation of the very short-time behavior of the \ensuremath{\Vert}
D1〉 state compared with that of an expansion of the exact solution in powers of time t. Within a time of roughly 0.10--0.15 ps the second- and third-order corrections turned out to be not very important. This is due to the fact that our first-order state contains already some terms of the expansion, summed up to infinite order. We found good agreement of the results obtained with our expansion and those from the corresponding \ensuremath{\Vert}
D1〉 simulations within the time of about 0.10 ps. Altogether we have shown that the \ensuremath{\Vert}
D1〉 state, although of approximative nature, is very close to an exact solution of the Davydov model on time scales from some fs up to ns. \textcopyright{} 1996 The American Physical Society.
Davydov trial states, in combination with the Green's function method, we study the dynamics of the Tavis-Cummings model and the Holstein-Tavis-Cummings model in the presence of diagonal disorder and cavity-qubit coupling disorder. For the Tavis-Cummings model, time evolution of the photon population, the optical absorption spectra, and the hetero-entanglement between the qubits and the cavity mode are calculated by using the Green's function method to corroborate numerically exact results of Davydov's Ansätze. For the Holstein-Tavis-Cummings model, only the latter is utilized to simulate effects of disorder on the photon population dynamics and the absorption spectra. We have demonstrated that the complementary employment of analytical and numerical methods permits uncovering a fairly comprehensive picture of a variety of complex behaviors in disordered multidimensional polaritonic cavity quantum electrodynamics systems.
Quantum electrodynamics of a single two-level atom placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied. Despite a nonlocal effective polariton-polariton coupling, the system is proven to be integrable and is diagonalized exactly by means of the Bethe-ansatz technique. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits some unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons propagate within the gap, while a soliton with an odd number of polaritons is pinned to the atom and forms a many-polariton-atom bound state, in which the radiation and the medium polarization are localized in the vicinity of the atom. The model under consideration has a quite general structure and can also be applied to the case of photonic-band-gap materials. \textcopyright{} 1996 The American Physical Society.
A system of polaritons interacting with a two-level atom placed within a frequency dispersive medium is proven to be integrable and diagonalized exactly by the Bethe ansatz method, despite a nonlocal effective polariton-polariton coupling. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons (``even'' solitons) propagate within the gap, while an ``odd'' soliton is pinned to the atom and forms a many-polariton--atom bound state.
We study the real-time evolution of solitary excitations in one-dimensional quantum spin chains using exact diagonalization and the density-matrix renormalization group. The underlying question of this work is the correspondence between classical solitons and solitons in quantum mechanics. While classical solitons as eigensolutions of nonlinear wave equations are localized and have a sharp momentum, this is not possible in the corresponding quantum case due to the linearity of the Schr\"odinger equation or, seen in a more pictorial way, because of the uncertainty relation. For the case of the XXZ model it is shown that the real-time evolution of quantum wave packets accompanied by spreading is in qualitative accordance with that predicted by classical solitons.