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arXiv:1701.00535 · Journal of Physics B Atomic Molecular and Optical Physics

Perturbative treatment of quantum to classical transition in chiral molecules: dilute phase versus condensed phase

Farhad Taher Ghahramani, Arash Tirandaz

Published Jan 3, 2017• 2 citations
arXivHTMLPDF

Abstract

Abstract We examine the dynamics of the chiral states of chiral molecules with high tunneling rates in dilute and condensed phases in the context of time-dependent perturbation theory. The chiral molecule is effectively described by an asymmetric double-well potential, whose asymmetry is a measure of chiral interactions. The dilute and condensed phases are conjointly described by a collection of harmonic oscillators but with temperature-dependent sub-ohmic and temperature-independent ohmic spectral densities, respectively. We examine our method quantitatively by applying the dynamics to an isotopic ammonia molecule, NHDT, in an inert background gas (as the dilute phase) and in water (as the condensed phase). As the different spectral densities imply, the extension of the dynamics from the dilute phase to the condensed phase is not trivial. While the dynamics in the dilute phase leads to racemization, the chiral interactions in the condensed phase induce the quantum Zeno effect. Moreover, contrary to the condensed phase, the short-time dynamics in the dilute phase is sensitive to the initial state of the chiral molecule and to the strength of the coupling between the molecule and the environment.

DOI
10.1088/1361-6455/aa5115

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Abstract We examine the dynamics of the chiral states of chiral molecules with high tunneling rates in dilute and condensed phases in the context of time-dependent perturbation theory. The chiral molecule is effectively described by an asymmetric double-well potential, whose asymmetry is a measure of chiral interactions. The dilute and condensed phases are conjointly described by a collection of harmonic oscillators but with temperature-dependent sub-ohmic and temperature-independent ohmic spectral densities, respectively. We examine our method quantitatively by applying the dynamics to an isotopic ammonia molecule, NHDT, in an inert background gas (as the dilute phase) and in water (as the condensed phase). As the different spectral densities imply, the extension of the dynamics from the dilute phase to the condensed phase is not trivial. While the dynamics in the dilute phase leads to racemization, the chiral interactions in the condensed phase induce the quantum Zeno effect. Moreover, contrary to the condensed phase, the short-time dynamics in the dilute phase is sensitive to the initial state of the chiral molecule and to the strength of the coupling between the molecule and the environment.
We study in the framework of the Schrodinger equation the effect of intermolecular interactions on the tunneling racemization of the active molecule. The active molecule is assumed as a two-level system and the left-right isomerism is viewed in terms of a double-bottomed harmonic potential well. The active molecule is assumed to be embedded in a gas, liquid or solid, submitted to a perturbing potential U created by the molecules of the sample. In our model we take into account the difference of energy E due to the weak interactions between the left (L) and right (R) configurations. We have shown that when E is equal to zero the system cannot be optically stable: the optical activity tends asymptotically to zero in the case of dilute gases or compressed gases and liquids or oscillates periodically around zero when the molecules are isolated or submitted to a static potential. Only when E is different of zero the system can be optically stable depending on the strength parameters of the potential U and on the magnitude of the spontaneous tunneling.
The decoherence of a two-state tunneling molecule, such as a chiral molecule or ammonia, due to collisions with a buffer gas is analyzed in terms of a succession of quantum states of the molecule satisfying the conditions for a consistent family of histories. With $\ensuremath{\hbar}\ensuremath{\omega}$ the separation in energy of the levels in the isolated molecule and $\ensuremath{\gamma}$ a decoherence rate proportional to the rate of collisions, we find for $\ensuremath{\gamma}\ensuremath{\gg}\ensuremath{\omega}$ (strong decoherence) a consistent family in which the molecule flips randomly back and forth between the left- and right-handed chiral states in a stationary Markov process. For $\ensuremath{\gamma}<\ensuremath{\omega}$ there is a family in which the molecule oscillates continuously between the different chiral states, but with occasional random changes of phase, at a frequency that goes to zero at a phase transition $\ensuremath{\gamma}=\ensuremath{\omega}$. This transition is similar to the behavior of the inversion frequency of ammonia with increasing pressure, but will be difficult to observe in chiral molecules such as D2{}_{2}2​S2{}_{2}2​. There are additional consistent families both for $\ensuremath{\gamma}>\ensuremath{\omega}$ and for $\ensuremath{\gamma}<\ensuremath{\omega}$. In addition, we relate the speed with which chiral information is transferred to the environment to the rate of decrease of complementary types of information (e.g., parity information) remaining in the molecule itself.
We study the tunnel dynamics of a chiral molecule between its left (L) and right (R) conformations, under the global effect of collisional decoherence together with the effect of a mean-field generated by the environment where an energetic difference, K, between homochiral and heterochiral interactions is assumed. We show that this decoherence leads unavoidably to equal populations of the L and R chiral conformations even for a high enough value of K which tends to keep localized an initial chiral state. However, we also show that K contributes to the stabilization of an initial L or R state for times that could be many orders of magnitude larger than the tunneling time, in the case the decoherence rate is much greater than the tunneling rate. In this case, an estimation of this stabilization time and a critical tunneling time is made. Even in the case in which the tunneling rate is greater than the decoherence rate, the effect of K is to keep localized the initial chiral state for times greater than the tunneling time. A possible slight chiral asymmetry is also considered.
In this review, the concepts of quantum tunneling and parity violation are introduced in the context of chiral molecules. A particle moving in a double well potential provides a good model to study the behavior of chiral molecules, where the left well and right well represent the L and R enantiomers, respectively. If the model considers the quantum behavior of matter, the concept of quantum tunneling emerges, giving place to stereomutation dynamics between left- and right-handed chiral molecules. Parity-violating interactions, like the electroweak one, can be also considered, making possible the existence of an energy difference between the L and R enantiomers, the so-called parity-violating energy difference (PVED). Here we provide a brief account of some theoretical methods usually employed to calculate this PVED, also commenting on relevant experiments devoted to experimentally detect the aforementioned PVED in chiral molecules. Finally, we comment on some ways of solving the so-called Hund’s paradox, with emphasis on mean-field theory and decoherence.
The dynamics of tunneling of an asymmetric double well interacting with a heat bath is reexamined in the binary collision dilute gas phase limit and investigated in the harmonic bath case. (a) In both cases dynamic and static asymmetries are additive. (b) In both cases, when the tunneling amplitude is not renormalized to a value of zero, the effect of asymmetry on the approach to equilibrium due to incoherent tunneling only quantitatively differs from the symmetric double well. (c) When the renormalized tunneling amplitude is zero, the asymmetric tunneling turns dephasing into population relaxation. (d) Variational methods again give results in agreement with renormalization group and instanton calculations.
In this work, we develop a system plus environment approach for interacting chiral molecules under a quantum-classical description of the spin-spin model using a Caldeira-Leggett-like coupling. After presenting the interacting model, we show that a long-ranged parity-nonconserving interaction, encoded within a nonlinear Schr\"odinger equation, produces an energy difference between the two enantiomers of the central chiral molecule when it interacts with a chiral environment. Three examples of such interactions are considered, with particular focus on Z0{Z}^{0}Z0-photon vacuum polarization. Finally, we reveal a chirality transmission effect phenomenon, where the time-averaged population difference of the central molecule is amplified when the chiral asymmetry of the environment is considered.