arXiv subjects

Nikesh S. Dattani

Publications and source records attributed to Nikesh S. Dattani.

27 records · Page 2Linked to original sources

arXiv2012

Optimal representation of the bath response function & fast calculation of influence functional coefficients in open quantum systems with BATHFIT 1

Today's most popular techniques for accurately calculating the dynamics of the reduced density operator in an open quantum system, either require, or gain great computational benefits, from representing the bath response function a(t) in the form a(t)=Σ_k^K p_k e^{O_k t} . For some of these techniques, the number of terms in the series K plays the lead role in the computational cost of the calculation, and is therefore often a limiting factor in simulating open quantum system dynamics. We present an open source MATLAB program called BATHFIT 1, whose input is any spectral distribution functions J(w) or bath response function, and whose output attempts to be the set of parameters {p_k,w_k}_k=1^K such that for a given value of K, the series Σ_k^k p_k e^{O_k t} is as close as possible to a(t). This should allow the user to represent a(t) as accurately as possible with as few parameters as possible. The program executes non-linear least squares fitting, and for a very wide variety of spectral distribution functions, competent starting parameters are used for these fits. For most forms of J(w), these starting parameters, and the exact a(t) corresponding to the given J(w), are calculated using the recent Pade decomposition technique - therefore this program can also be used to merely implement the Pade decomposition for these spectral distribution functions; and it can also be used just to efficiently and accurately calculate a(t) for any given J(w) . The program also gives the J(w) corresponding to a given a(t), which may allow one to assess the quality (in the w-domain) of a representation of a(t) being used. Finally, the program can calculate the discretized influence functional coefficients for any J(w), and this is computed very efficiently for most forms of J(w) by implementing the recent technique published in [Quantum Physics Letters (2012) 1 (1) pg. 35].

quant-ph↗

arXiv2012

Analytic influence functionals for numerical Feynman integrals in most open quantum systems

Fully analytic formulas, which do not involve any numerical integration, are derived for the discretized influence functionals of a very extensive assortment of spectral distributions. For Feynman integrals derived using the Trotter splitting and Strang splitting, we present general formulas for the discretized influence functionals in terms of proper integrals of the bath response function. When an analytic expression exists for the bath response function, these integrals can almost always be evaluated analytically. In cases where these proper integrals cannot be integrated analytically, numerically computing them is much faster and less error-prone than calculating the discretized influence functionals in the traditional way, which involves numerically calculating integrals whose bounds are both infinite. As an example, we present the analytic discretized influence functional for a bath response function of the form α(t)=\sum_{j}^{K}p_{j}e^{Ω_{j}t}, which is a natural form for many spectral distribution functions (including the very popular Lorentz-Drude/Debye function), and for other spectral distribution functions it is a form that is easily obtainable by a least-squares fit . Evaluating our analytic formulas for this example case is much faster and easier to implement than numerically calculating the discretized influence funcitonal in the traditional way. In the appendix we provide analytic expressions for p_{j} and Ω_{j} for a variety of spectral distribution forms, and as a second example we provide the analytic bath response function and analytic influence functionals for spectral distributions of the form J(ω)\proptoω^{s}e^{-(ω/ω_{c})^{q}}. The value of the analytic expression for this bath response function extends beyond its use for calculating Feynman integrals.

quant-ph↗

arXiv2011

A general approach to quantum dynamics using a variational master equation: Application to phonon-damped Rabi rotations in quantum dots

We develop a versatile master equation approach to describe the non-equilibrium dynamics of a two-level system in contact with a bosonic environment, which allows for the exploration of a wide range of parameter regimes within a single formalism. As an experimentally relevant example, we apply this technique to the study of excitonic Rabi rotations in a driven quantum dot, and compare its predictions to the numerical Feynman integral approach. We find excellent agreement between the two methods across a generally difficult range of parameters. In particular, the variational master equation technique captures effects usually considered to be non-perturbative, such as multi-phonon processes and bath-induced driving renormalisation, and can give reliable results even in regimes in which previous master equation approaches fail.

cond-mat.mes-hall↗

arXiv2011

A DPF analysis yields accurate analytic potentials for Li$_2(a\,^3Σ^+_u)$ and Li$_2(1\,^3 Σ^+_g)$ that incorporate 3-state mixing near the $1\,^3 Σ^+_g$-state asymptote

A combined-isotopologue direct-potential-fit (DPF) analysis of optical and photoassociation spectroscopy data for the $a\,^{3}Σ_u^+$ and $1\,^3Σ_g^+$ states of Li$_2$ has yielded accurate analytic potential energy functions for both states. The recommended M3LR$_{5,3}^{8.0}(3)$ potential for the $a\,^3Σ_u^+$ of $^{7,7}$Li$_2$ has a well depth of $\,{\mathfrak D}_e= 333.758(7)\,{\rm cm^{-1}}$ and equilibrium distance of $\,r_e= 4.17005(3)~$Å, and the associated scattering lengths are $\,a_{\rm SL}= -14.759(9)\,$Å for $^{7,7}$Li$_2$ and $-1906(50)\,$Å for $^{6,6}$Li$_2$. For the $1\,^3Σ_g^+$ state, in spite of a gap of $\,\sim 5200\,{\rm cm}^{-1}$ (from $\,v(1\,^3Σ_g^+)= 8-61$) for which there are no data, the DPF procedure has no difficulty determining an accurate overall potential. The $1\,^3Σ_g^+$ state of the $^{7,7}$Li$_2$ isotopologue has a well depth of $\,{\mathfrak D}_e= 7092.417(33)\,{\rm cm^{-1}}$ and equilibrium distance of $\,r_e= 3.06524(9)~$Å. The long-range tail of the recommended M3LR$_{6,3}^{3.6}(9)$ potential energy function for the $1\,^3Σ_g^+$ state is defined by the lowest eigenvalue of a $3\times 3$ long-range interstate coupling matrix to take into account the 3-state mixing near its asymptote.

physics.chem-ph↗

arXiv2010

Numerical Feynman integrals for density operator dynamics using master equation interpolants: faster convergence and significant reduction of computational cost

The Feynman integral is one of the most accurate methods for calculating density operator dynamics in open quantum systems. However, the number of time steps that can realistically be used is always limited, therefore one often obtains an approximation of the density operator at a sparse grid of points in time. Instead of relying only on \textit{ad hoc} interpolation methods such as splines to estimate the system density operator in between these points, I propose a method that uses physical information to assist with this interpolation. This method is tested on a physically significant system, on which its use allows important qualitative features of the density operator dynamics to be captured with as little as 2 time steps in the Feynman integral. This method allows for an enormous reduction in the amount of memory and CPU time required for approximating density operator dynamics within a desired accuracy. Since this method does not change the way the Feynman integral itself is calculated, the value of the density operator approximation at the points in time used to discretize the Feynamn integral will be the same whether or not this method is used, but its approximation in between these points in time is considerably improved by this method.

physics.comp-ph↗

arXiv2009

A new method for identifying vertebrates using only their mitochondrial DNA

A new method for determining whether or not a mitrochondrial DNA (mtDNA) sequence belongs to a vertebrate is described and tested. This method only needs the mtDNA sequence of the organism in question, and unlike alignment based methods, it does not require it to be compared with anything else. The method is tested on all 1877 mtDNA sequences that were on NCBI's nucleotide database on August 12, 2009, and works in 94.57% of the cases. Furthermore, all organisms on which this method failed are closely related phylogenetically in comparison to all other organisms included in the study. A list of potential extensions to this method and open problems that emerge out of this study is presented at the end.

q-bio.GN↗

arXiv2009

Modeling of neuron-semiconductor interactions in neuronal networks interfaced with silicon chips

Recent developments in the interfacing of neurons with silicon chips may pave the way for progress in constructing scalable neurocomputers. The assembly of synthetic neuronal networks with predefined synaptic connections and controlled geometric structure has been realized experimentally within the last decade. Furthermore, when such neuronal networks are interfaced with semiconductors, action potentials in neurons of the network can be elicited by capacitative stimulators, and voltage measurements can be made by transistors incorporated into the associated silicon chip. Despite the impressive progress, such preliminary devices have not yet demonstrated the performance of useful computations, and constructing larger devices can be both expensive and time-consuming. Accordingly, an appropriate modeling framework with the capability to simulate current experimental results in such devices may be used to make useful predictions regarding their potential computational power. A proposed modeling framework for functional neuronal networks interfaced with silicon chips is presented below.

q-bio.NC↗

arXiv2008

Simulating neurobiological localization of acoustic signals based on temporal and volumetric differentiations

The localization of sound sources by the human brain is computationally simulated from a neurobiological perspective. The simulation includes the neural representation of temporal differences in acoustic signals between the ipsilateral and contralateral ears for constant sound intensities (angular localization), and of volumetric differences in acoustic signals for constant azimuthal angles (radial localization). The transmission of the original acoustic signal from the environment, through each significant stage of intermediate neurons, to the primary auditory cortex, is also simulated. The errors that human brains make in attempting to localize sounds in evolutionarily uncommon environments (such as when one ear is in water and one ear is in air) are then mathematically predicted. A basic overview of the physiology behind sound localization in the brain is also provided.

q-bio.NC↗

arXiv2008

Linear Multistep Numerical Methods for Ordinary Differential Equations

A review of the most popular Linear Multistep (LM) Methods for solving Ordinary Differential Equations numerically is presented. These methods are first derived from first principles, and are discussed in terms of their order, consistency, and various types of stability. Particular varieties of stability that may not be familiar, are briefly defined first. The methods that are included are the Adams-Bashforth Methods, Adams-Moulton Methods, and Backwards Differentiation Formulas. Advantages and disadvantages of these methods are also described. Not much prior knowledge of numerical methods or ordinary differential equations is required, although knowledge of basic topics from calculus is assumed.

math.NA↗