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The Journal of Chemical Physics

Implications of parity conservation for superpositions of chiral states

Christopher S. Maierle, Robert A. Harris

Published Sep 8, 1998• 21 citations
Publisher

Abstract

In this paper, we consider how parity conservation limits the type of superpositions of chiral states that can be created and detected. We also show how spontaneous emission can cause the left and right hands of the superposition to become coupled to different states of the radiation field. This coupling, aside from being an interesting effect in its own right, helps to enforce the restrictions that parity conservation places upon superpositions of chiral states. Finally, we outline an analogy that exists between aspects of this work and the Einstein, Podolsky, Rosen paradox.

DOI
10.1063/1.476972

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In this paper, we consider how parity conservation limits the type of superpositions of chiral states that can be created and detected. We also show how spontaneous emission can cause the left and right hands of the superposition to become coupled to different states of the radiation field. This coupling, aside from being an interesting effect in its own right, helps to enforce the restrictions that parity conservation places upon superpositions of chiral states. Finally, we outline an analogy that exists between aspects of this work and the Einstein, Podolsky, Rosen paradox.
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.
Enantio-conversion with the help of electromagnetic fields is an essential issue due to the chirality dependence of many chemical, biological, and pharmaceutical processes. Here, we propose a method for this issue based on a five-level double-$\mathrm{\ensuremath{\Delta}}$ model of chiral molecules. By utilizing the breaking of left-right symmetry in two $\mathrm{\ensuremath{\Delta}}$-type substructures, we can establish chiral-state-selective excitations with one chiral ground state being excited to an achiral excited state and the other one being undisturbed. In the meanwhile, the achiral excited state will relax to two chiral ground states. The two effects simultaneously acting on chiral mixtures can convert molecules of different chiralities to ones of the same chirality, i.e., enantio-conversion via optical pumping. We numerically show that highly efficient enantio-conversion can be achieved. Our method works in the appearance of decoherences and without the precise control of pulse durations (pulse areas) or pulse shapes. These advantages offer it promising features in promoting the future exploration of enantio-conversion.
Spontaneous emission may be regarded as a result of the radiation-reaction fields emitted by an atom acting back on the atom itself. It is commonly (and often implicitly) assumed that the frequencies of different transitions in the atom are sufficiently different that the field radiated by one dipole is not in near resonance with any other transition. The “secular assumption” is then made that the field couples back only on the same dipole that originally sent it. In this work we study spontaneous emission without making the secular approximation.
This thesis considers measurements that could be used to verify the existence of a coherent superposition, using the specific example of a superposition of a Bose-Einstein condensate in a double-well. There is significant interest in superposition states in atomic condensates, due to the possibility of large numbers of atoms being involved in the superposition, and while various experimental proposals exist for generating such states none have yet been successfully verified in this system. We use a number-state description of the condensate in a double well, which is a number-conserving approach with basis states specifying the number of atoms in each well. This allows us to consider 'NOON`-type states, i.e. a superposition of the 'N,0' state of all atoms localized in the left well and the '0,N' state where all N atoms are localized in the right hand well. MATLAB is used to solve for the ground state, dynamics, and measurement expectation values for the double-well condensate in this basis. We consider measurements that would distinguish this coherent superposition from a statistical mixture of the same states. Such a suitable measurement is the parity of the atoms in one well, after a Ramsey-type interference experiment has been performed. We have found that unambiguously distinguishing even a perfect 'NOON' state from a statistical mixture using standard detection methods will present experimental difficulties. We find that interactions between condensate atoms degrade the visibility of the parity oscillations, and we also consider the effects of detector inefficiency on these parity measurements. We suggest that a 'NOON'-type superposition state in the double-well system could be generated in a reasonable time frame by making use of a Feshbach resonance to tune the inter-atomic interactions from repulsive to negative. We find that even with imperfect superpositions, parity measurements retain their frequency dependence, which is the signature of NOON-type coherence.
This paper is a straightforward generalization of Maierle-Harris proposal regarding parity implications on the superpositions of chiral states of a molecule. It is shown that the inclusion of electric quadrupole and magnetic dipole interactions removes several of restrictions on the preparation of superpositions of mid R:L and mid R:R states of a chiral molecule. It is also found that the dephasing of mid R:L and mid R:R superpositions, due to the spontaneous emission from the chiral molecule, has opposing contributions from electric quadrupole-magnetic dipole and electric dipole interactions.