Fourth Order Algorithms for Solving the Multivariable Langevin Equation and the Kramers Equation

We develop a fourth order simulation algorithm for solving the stochastic Langevin equation. The method consists of identifying solvable operators in the Fokker-Planck equation, factorizing the evolution operator for small time steps to fourth order and implementing the factorization process numerically. A key contribution of this work is to show how certain double commutators in the factorization process can be simulated in practice. The method is general, applicable to the multivariable case, and systematic, with known procedures for doing fourth order factorizations. The fourth order convergence of the resulting algorithm allowed very large time steps to be used. In simulating the Brownian dynamics of 121 Yukawa particles in two dimensions, the converged result of a first order algorithm can be obtained by using time steps 50 times as large. To further demostrate the versatility of our method, we derive two new classes of fourth order algorithms for solving the simpler Kramers equation without requiring the derivative of the force. The convergence of many fourth order algorithms for solving this equation are compared.Comment: 19 pages, 2 figure

Coherence and Decoherence in Biological Systems: Principles of Noise Assisted Transport and the Origin of Long-lived Coherences

The quantum dynamics of transport networks in the presence of noisy environments have recently received renewed attention with the discovery of long-lived coherences in different photosynthetic complexes. This experimental evidence has raised two fundamental questions: Firstly, what are the mechanisms supporting long-lived coherences and secondly, how can we assess the possible functional role that the interplay of noise and quantum coherence might play in the seemingly optimal operation of biological systems under natural conditions? Here we review recent results, illuminate them at the hand of two paradigmatic systems, the Fenna-Matthew-Olson (FMO) complex and the light harvesting complex LHII, and present new progress on both questions. In particular we introduce the concept of the phonon antennae and discuss the possible microscopic origin or long-lived electronic coherences.Comment: Paper delivered at the Royal Society Discussion Meeting "Quantum-coherent energy transfer: implications for biology and new energy technologies", 27 - 28 April 2011 at The Kavli Royal Society International Centre, Buckinghamshire, UK. Accepted for publication in Philosophical Transactions of the Royal Society

Any order imaginary time propagation method for solving the Schrodinger equation

The eigenvalue-function pair of the 3D Schr\"odinger equation can be efficiently computed by use of high order, imaginary time propagators. Due to the diffusion character of the kinetic energy operator in imaginary time, algorithms developed so far are at most fourth-order. In this work, we show that for a grid based algorithm, imaginary time propagation of any even order can be devised on the basis of multi-product splitting. The effectiveness of these algorithms, up to the 12$^{\rm th}$ order, is demonstrated by computing all 120 eigenstates of a model C$_{60}$ molecule to very high precisions. The algorithms are particularly useful when implemented on parallel computer architectures.Comment: 8 pages, 3 figure

Observation of the Pairing Gap in a Strongly Interacting Fermi Gas

We study fermionic pairing in an ultracold two-component gas of $^6$Li atoms by observing an energy gap in the radio-frequency excitation spectra. With control of the two-body interactions via a Feshbach resonance we demonstrate the dependence of the pairing gap on coupling strength, temperature, and Fermi energy. The appearance of an energy gap with moderate evaporative cooling suggests that our full evaporation brings the strongly interacting system deep into a superfluid state.Comment: 18 pages, 3 figure

Isospin mode splitting and mixing in asymmetric nuclear matter

We estimate exclusive density and asymmetry parameter dependent dispersion relations of various charged states of pions in asymmetric nuclear matter. The possibility of matter induced mixing of $\pi^0$ with $\eta$ is clearly exposed with the further mass modification of $\pi^0$ meson due to mixing. Asymmetry driven mass splitting and mixing amplitude are of the same order as the corresponding values in vacuum. Closed form analytic results for the mass shifts and dispersion relations with and without mixing are presented. Furthermore, we discuss the sensitivity of our results on the scalar mean field within the framework of Quantum Hadrodynamics.Comment: 8 pages, 4 Figure