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arXiv:quant-ph/0110084 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

From Davydov solitons to decoherence-free subspaces: Self-consistent propagation of coherent-product states

S. Gheorghiu-Svirschevski

Published Oct 23, 2001• 17 citations
arXivHTMLPDF

Abstract

The self-consistent propagation of generalized D1 [coherent-product] states and of a class of Gaussian density-matrix generalizations is examined, at both zero and finite temperature, for arbitrary interactions between the localized lattice (electronic or vibronic) excitations and the phonon modes. It is shown that in all legitimate cases, the evolution of D1 states reduces to the disentangled evolution of the component D2 states. The self-consistency conditions for the latter amount to conditions for decoherence-free propagation, which complement the D2 Davydov soliton equations in such a way as to lift the nonlinearity of the evolution for the on-site degrees of freedom. Although it cannot support Davydov solitons, the coherent-product ansatz does provide a wide class of exact density-matrix solutions for the joint evolution of the lattice and phonon bath in compatible systems. Included are solutions for initial states given as a product of a [largely arbitrary] lattice state and a thermal equilibrium state of the phonons. It is also shown that external pumping can produce self-consistent Frohlich-like effects. A few sample cases of coherent, albeit not solitonic, propagation are briefly discussed.

DOI
10.1103/physreve.64.051907

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Journal of Physics Condensed Matter•Oct 24, 1994•10 citations
The self-consistent propagation of generalized D1 [coherent-product] states and of a class of Gaussian density-matrix generalizations is examined, at both zero and finite temperature, for arbitrary interactions between the localized lattice (electronic or vibronic) excitations and the phonon modes. It is shown that in all legitimate cases, the evolution of D1 states reduces to the disentangled evolution of the component D2 states. The self-consistency conditions for the latter amount to conditions for decoherence-free propagation, which complement the D2 Davydov soliton equations in such a way as to lift the nonlinearity of the evolution for the on-site degrees of freedom. Although it cannot support Davydov solitons, the coherent-product ansatz does provide a wide class of exact density-matrix solutions for the joint evolution of the lattice and phonon bath in compatible systems. Included are solutions for initial states given as a product of a [largely arbitrary] lattice state and a thermal equilibrium state of the phonons. It is also shown that external pumping can produce self-consistent Frohlich-like effects. A few sample cases of coherent, albeit not solitonic, propagation are briefly discussed.
The theoretical description of materials' properties driven out of equilibrium has important consequences in various fields such as semiconductor spintronics, nonlinear optics, and continuous and discrete quantum information science and technology. The coupling of a quantum many-body system to an external bath can dramatically modify its dynamics compared to that of closed systems; new phenomena like relaxation and decoherence appear as a consequence of the non-unitary evolution of the quantum system. In addition, electron-electron correlations must be properly accounted for in order to go beyond a simple one-electron or mean-field description of the electronic system. Here, we discuss a first-principles methodology based on the evolution of the electronic density matrix capable of treating electron-environment interactions and electron-electron correlations at the same level of description. The effect of the environment is separated into a coherent contribution, like the coupling to applied external electromagnetic fields, and an incoherent contribution, like the interaction with lattice vibrations or the thermal background of radiation. Electron-electron interactions are included using the nonequilibrium Green's function plus the generalized Kadanoff-Baym ansatz. The obtained non-Markovian coupled set of equations reduces to the ordinary Lindblad quantum master equation form in the Markovian limit.
Coherent-state functional-integral methods are used to develop approximation schemes for the description of coupled electron-phonon systems far from equilibrium. We consider an interacting-electron system which is coupled to a system of phonons and subjected to an external---not necessarily weak---perturbation. For systems where the electron-phonon interaction is linear in the phonon variables, the phonon degrees of freedom can be accounted for exactly. From a formally exact representation of a suitable generating function for the expectation value of electronic observables, approximation schemes based on mean-field solutions can be derived. Within the lowest order of approximation, the coupled electron-phonon system may be treated as a self-consistent system of indepenent electrons. Higher-order corrections are shown to give a random-phase approximation based on self-consistent mean fields. This approach is applied to electronic two-level systems coupled to phonons under a large external bias. We find that the coupling to acoustical-phonon modes leads to a damping of the quantum beats and drives the system into a new equilibrium state. The coupling to LO phonons is shown to account correctly for resonance effects which arise when the level splitting is close to the LO-phonon energy.
The essential ideas of Davydov's theory of molecular solitons are used to develop a variation of the Zwanzig-Nakajima projection technique. The projection operator is time dependent and contains the feedback characteristic of the soliton dynamic. The result is a nonlinear equation of motion for a reduced density matrix. A Markoffian approximation is considered which contributes corrections to the Davydov-type evolution. The approximate equation of motion embodies the correct fluctuation-dissipation relation for this system and recovers the known exact results in the transportless limit.
We test the validity of Hamilton-equation methods for determining the time evolution of trial state vectors in quantum mechanics. Given a trial state vector, we are able to construct a differential operator under which a scalar Hamilton function must be invariant. State vectors composed of single-particle states, coherent-state products, and mixed single-particle states and coherent-state products are considered explicitly. In the latter category, we consider state vectors of the form proposed by Davydov in his treatment of the quantum soliton problem. We find that Davydov's wave vector, as determined by the Hamilton-equation method, is not a solution of the Schr\"odinger equation for the Fr\"ohlich Hamiltonian except under very restrictive circumstances. The theoretical justification for a number of conclusions about soliton transport in Fr\"ohlich-type systems is thus called into question.
The decoherence and entanglement dynamics of two interacting qubits coupled through Heisenberg XY interactions to a spin bath in thermal equilibrium are studied. The exact form of the reduced density matrix is derived for finite and infinite numbers of environmental spins. It is shown that decoherence can be minimized at low bath temperatures and strong coupling between the qubits. Some initial product states evolve into entangled ones, initially entangled states lose completely or partially their entanglement. The relation between the fidelity and the concurrence is also investigated.
We present equations of motion for the Su-Schrieffer-Heeger (SSH) Hamiltonian derived with the help of ansatz states similar to Davydov's so-called mod D 1 ) state for soliton dynamics in proteins. Such an ansatz state allows for quantum effects in the lattice and goes beyond previous calculations which mostly apply adiabatic models. In the most general case, called mod Phi 0 ), which is treated here in detail, we assume that the coherent-state amplitudes for the lattice depend on the site and the molecular orbital of the electrons. The equations of motion are derived from the Lagrangian of the system, a method which is equivalent to the time-dependent variational principle. In the resulting equations we find that, although the SSH Hamiltonian is a one-particle operator, indirect electron-electron interactions are present in the system which originate from the electron-phonon interactions. Inclusion of direct electron-electron interactions, will give insight into the interplay between electron-electron and electron-phonon interactions which can lead effectively to an attractive force between the electrons in systems other than polyacetylene, where bipolarons are known to be unstable. Further with our time-dependent wavefunction also vibrational details of absorption spectra can be computed. From the equations of motion several approximations can be derived. In a further approximation, mod Phi 2 ), the dependence of the coherent-state amplitudes on the lattice site is neglected. This mod Phi 2 ) ansatz state consists of a simple product of the electronic and the lattice wavefunctions; however, the electrons are not constrained to follow the lattice dynamics instantaneously as in the adiabatic case. Finally the classical adiabatic case is discussed on which soliton-dynamics simulations are usually based. Further we discuss how to include temperature effects in our model. Applications to soliton dynamics are discussed for the example of the mod Phi 2 ) model with emphasis on the dependence of the results on soliton width and temperature. We found that in contrast to results reported in the literature, where a similar ansatz is used, but only one electron is treated explicitly, the solitons remain stable also for small soliton widths. This indicates that the interactions of the electrons not occupying the soliton level with the lattice have a stabilizing effect on the soliton. Further our results indicate that the temperature model using random forces and dissipation terms to introduce temperature effects has to be applied with extreme care in this case due to the strong electron-lattice interactions.