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arXiv:1310.3155 · Chirality

A Langevin Canonical Approach to the Dynamics of Chiral Systems: Thermal Averages and Heat Capacity

Helen C. Peñate‐Rodríguez, Anais Dorta‐Urra, Pedro Bargueño, Germán A. Rojas-Lorenzo, Salvador Miret-Artés

Published May 2, 2014• 6 citations
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Abstract

A Langevin canonical framework for a chiral two-level system coupled to a bath of harmonic oscillators was developed within a coupling scheme different from the well-known spin-boson model. Thermal equilibrium values were reached at asymptotic times by solving the corresponding set of nonlinear coupled equations in a Markovian regime. In particular, phase difference thermal values (or, equivalently, the so-called coherence factor) and heat capacity through energy fluctuations were obtained and are discussed in terms of tunneling rates and asymmetries.

DOI
10.1002/chir.22326

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A Langevin canonical framework for a chiral two-level system coupled to a bath of harmonic oscillators was developed within a coupling scheme different from the well-known spin-boson model. Thermal equilibrium values were reached at asymptotic times by solving the corresponding set of nonlinear coupled equations in a Markovian regime. In particular, phase difference thermal values (or, equivalently, the so-called coherence factor) and heat capacity through energy fluctuations were obtained and are discussed in terms of tunneling rates and asymmetries.
A canonical framework for chiral two-level systems coupled to a bath of harmonic oscillators is developed to extract, from a stochastic dynamics, the thermodynamic equilibrium values of both the population difference and coherences. The incoherent and coherent tunneling regimes are analyzed for an Ohmic environment in terms of a critical temperature defined by the maximum of the heat capacity. The corresponding numerical results issued from solving a non-linear coupled system of equations are fitted to approximate path-integral analytical expressions beyond the so-called non-interacting blip approximation in order to determine the different time scales governing both regimes.
A Langevin canonical framework for a chiral two-level system coupled to a bath of harmonic oscillators is used within a coupling scheme different from the well-known spin-boson model to study the quantum stochastic resonance for chiral molecules. This process refers to the amplification of the response to an external periodic signal at a certain value of the noise strength, being a cooperative effect of friction, noise, and periodic driving occurring in a bistable system. Furthermore, from this stochastic dynamics within the Markovian regime and Ohmic friction, the competing process between tunneling and the parity violating energy difference present in this type of chiral systems plays a fundamental role. This mechanism is finally proposed to observe the so-far elusive parity-violating energy difference in chiral molecules.
Thermostated time evolutions are on a firm ground and widely used in classical molecular dynamics (MD) simulations. Hamilton´s equations of motion are supplemented by time-dependent pseudofriction terms that convert the microcanonical isoenergetic time evolution into a canonical isothermal time evolution, thus permitting the calculation of canonical ensemble averages by time averaging. However, similar methods for quantum MD schemes are still lacking. Given the rich dynamical behavior of ultracold trapped quantum gases depending on the value of the s-wave scattering length, it is timely to investigate how classical thermostating methods can be combined with powerful approximate quantum dynamics schemes to deal with interacting quantum systems at finite temperature. In this work, the popular method of Nose and Hoover to create canonically distributed positions and momenta in classical MD simulations is generalized to a genuine quantum system of infinite dimensionality. We show that for the quantum harmonic oscillator, the equations of motion in terms of coherent states may be modified in a Nose-Hoover manner to mimic the coupling of the system to a thermal bath and create a quantum canonical ensemble. The method is developed initially for a single particle and then generalized to the case of an arbitrary number of identical quantum particles, involving entangled distribution functions. The resulting isothermal equations of motion for bosons and fermions contain additional terms leading to Bose-attraction and Pauli-blocking, respectively. Questions of ergodicity are discussed for different coupling schemes. In the many-particle case, the superiority of the Nose-Hoover technique to a Langevin approach is demonstrated. In addition, the work contains an investigation of the Grilli-Tosatti thermostating method applied to the harmonic oscillator, and calculations for quantum wavefunctions moving with a time-invariant shape in a harmonic potential.
We investigate the dynamics of a two-qubit open quantum system, in particular the two-qubit spin-boson model in the strong coupling regime, coupled to two thermal bosonic baths under non-Markovian and non-equilibrium conditions. Two complementary approaches, the Hierarchical Equations of Motion (HEOM) and Reaction Coordinate Mapping (RCM), are employed to examine various coupling regimes between the qubits and their respective baths. The dynamical features of the model and the impact of the tunneling amplitude on quantum coherence of the system are probed using the l1l_1l1​-norm of coherence. The model is further shown to have non-Markovian evolution. The nontrivial task of calculating entropy production in the strong-coupling regime is performed using auxiliary density operators in HEOM. Motivated by the realization of a quantum thermal device in the strong-coupling regime, the non-equilibrium steady-state behavior of the system is investigated. Furthermore, the relationship between the heat and spin currents and the tunneling amplitude is probed.
We derive a semi-classical Langevin equation that, in conjuction with a quantum fluctuation–dissipation theorem, can be used to run canonical dynamics and to determine thermal averages in the second-order quantized Hamilton dynamics (QHD-2). In contrast to our earlier work [J. Chem. Phys. 122 (2005) 234109], the bath is no longer treated classically, and the friction is represented at the second-order. As a result, the quantum fluctuation variables are kept close to their optimal values, similarly to the recently developed phase-space canonical averaging for QHD-2 [J. Chem. Phys. 126 (2007) 204108]. The theory is exact for the quantum harmonic oscillator and the free particle, and gives good agreement with the quantum answer for a variety of quartic potentials.
A method is proposed to describe Fermi or Bose systems coupled to one or several heat baths composed of fermions and/or bosons. The method, called the coupled equations of motion method, properly includes non-Markovian effects. The approach is exact in the full-coupling approximation when only bosonic particles are present in the system and baths. The approach provides an approximate treatment when fermions are present either in the system and/or in one or several environments. Our approach has the advantage of properly respecting the Pauli exclusion principle for fermions during the evolution. We illustrate the approach for the single fermionic or bosonic oscillator coupled to one or two heat baths assuming different types of quantum statistics (fermion or boson) for them. The cases of a Fermi system coupled to fermion or boson heat baths or a mixture of both are analyzed in detail. With the future goal of treating Fermi systems formed of an increasing number of two-level systems (qubits), we discuss possible simplifications that could be made in the equations of motion and their limits of validity in terms of the system-bath coupling or of the initial heat bath temperatures.