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arXiv:1604.03712 · Entropy

A Langevin Canonical Approach to the Study of Quantum Stochastic Resonance in Chiral Molecules

Germán A. Rojas-Lorenzo, H.C. Peñate-Rodríguez, Anaís Dorta-Urra, Pedro Bargueño, Salvador Miret-Artés

Published Sep 29, 2016• 1 citation
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Abstract

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.

DOI
10.3390/e18100354

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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.
In order to explore parity violating effects in chiral molecules, of interest in some models of evolution towards homochirality, quantum stochastic resonance (QSR) is studied for the population difference between the two enantiomers of a chiral molecule (hence for the optical activity of the sample), under low viscous friction and in the deep quantum regime. The molecule is described by a two-state model in an asymmetric double well potential where the asymmetry is given by the known predicted parity violating energy difference (PVED) between enantiomers. In the linear response to an external driving field that lowers and rises alternatively each one of the minima of the well, a signal of QSR is predicted only in the case that the PVED is different from zero, the resonance condition being independent on tunneling between the two enantiomers. It is shown that, at resonance, the fluctuations of the first order contribution to the internal energy are zero. Due to the small value of the PVED, the resonance would occur in the ultracold regime. Some proposals concerning the external driving field are suggested.
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 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.
Within a generalized Langevin framework for open quantum systems, the cyclic evolution of a two-level system is analyzed in terms of the geometric phase extended to dissipative systems for Ohmic friction. This proposal is applied to the dynamics of chiral molecules where the tunneling and parity violating effects are competing. The effect of different system-bath coupling functions in the dissipated energy is shown to be crucial to understand the behavior of the geometric phase as well as the decoherence displayed by the corresponding interference patterns.
A study of (aperiodic) quantum stochastic resonance (QSR) in parallel is put forward. By doing so, a generally stochastic input signal is fed into an array of parallel dissipative quantum two-level systems (TLS) and its integral response is studied against increasing temperature. The response is quantified by means of an information-theoretic measure provided by the rate of mutual information per element and, in addition, by the cross-correlation between the information-carrying input signal and the output response. For ohmic-like quantum dissipation, both measures exhibit QSR for biased two-level systems. Our prime focus here, however, is on the case with zero asymmetry between the two localized stable states. We then find that the mutual information measure exhibits QSR only for sufficiently strong dissipation ( > 3/2), as measured by the dimensionless ohmic friction strength . Moreover, the mutual information measure relates QSR within quantum linear response theory to the signal-to-noise-ratio (SNR) , being independent of the input driving frequencies in this limit. In contrast, the cross correlation measure connects QSR to a genuine synchronization phenomenon . For a single symmetric TLS, aperiodic QSR is exhibited in the cross-correlation measure for a Gaussian exponentially correlated input signal for > 1 already. Upon feeding the aperiodic input signal into a parallel array of unbiased TLS's, QSR successively emerges above the critical ohmic dissipation strength > 1/2 with increasing number n of parallel units. Thus, QSR can occur in parallel despite the fact that it does not occur in each individual, unbiased, TLS for < 1. This paradoxical phenomenon - which can be tested with an array of bistable superconducting quantum interference devices - constitutes a true quantum effect: it is due to the power-law dependence on temperature of the tunnelling rate and the stochastic linearization of quantum fluctuations with increasing number of parallel units.
Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of Egs+kBTE_{gs}+k_BTEgs​+kB​T. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.