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arXiv:2301.03247 · Journal of Optics

Reflection by two level system: phase singularities on the Poincaré hypersphere

Ben Lang, Edmund Harbord, Ruth Oulton

Published Oct 24, 2023• 0 citations
arXivHTMLPDF

Abstract

Abstract We consider the reflection of a photon by a two-level system in a quasi-one-dimensional waveguide. The waveguide polarisation at the location of the two-level system and the transition dipole are key determinants of the physics, controlling of the phase and amplitude of the scattered light in both directions. In most cases full control is possible by tuning only one of these two degrees of freedom. In reverse, this enables unique characterisation of the dipole from measurements of the scattered light. Phase singularities occur where the reflection coefficient is zero, with the (hyper-)spherical parameter space determining the dynamics of these singularities.

DOI
10.1088/2040-8986/ad065a

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Abstract We consider the reflection of a photon by a two-level system in a quasi-one-dimensional waveguide. The waveguide polarisation at the location of the two-level system and the transition dipole are key determinants of the physics, controlling of the phase and amplitude of the scattered light in both directions. In most cases full control is possible by tuning only one of these two degrees of freedom. In reverse, this enables unique characterisation of the dipole from measurements of the scattered light. Phase singularities occur where the reflection coefficient is zero, with the (hyper-)spherical parameter space determining the dynamics of these singularities.
We theoretically investigate the optical dipole interaction between a multi-level quantum system and a single-mode optical waveguide of any local polarisation. We investigate several paradoxical seeming situations, for example we find a situation in which there exist two non-orthogonal quantum states, each of which results in a photon flux in the opposite direction to the other. We show how, despite appearances, this does not break the unitary requirements of quantum mechanics. We also find that an isotropic quantum emitter can be either reflective or transmissive to light depending on the waveguide polarisation at the emitter location, indeed in the zero loss limit such a system changes from 100% transmission to 100% reflection due to an infinitesimal polarisation rotation. An example case for a four level system is also considered, which is found to operate as a non-destructive parity measurement of the photon number.
We consider the coherent propagation of nnn photons in a one-dimensional coupled-resonator waveguide for n=2,3,4….n=2,3,4\dots{}.n=2,3,4…. The scattering by a three-level atom, which resides in one of the resonators of the waveguide and gives rise to only two-photon transitions, results in a perfect quantum switch that allows either total reflection or total transmission. This is to be contrasted to the case of a single photon inside a one-dimensional resonant waveguide scattered by a two-level system with single-photon transitions where only total reflection can be accomplished; viz. the system behaves only as a perfect mirror but not as an ideal, transparent medium.
Summary form only given. There has recently been great interest in chiral waveguides, where the polarisation of the forwards and backwards propagating light fields are oppositely circular at certain positions in space (C-points) [1]. This results in new phenomena, such as the one way emission of light from spin-transitions in quantum dots (QDs) or atoms (simulation, fig. b). The emitted photon's direction is determined by the spin state of the emitter.To date calculations on this topic have either been restricted to certain special cases, or have relied on simplifying “rate forwards/backwards” models [2]. Using a Green's function method we have generalised to the case of any polarisable dipole at a point supporting any polarisation. Several interesting physical effects arise here that could not have been predicted using the previous models. For example in the figure we show how perfect chiral behaviour can be recovered at points of elliptical polarisation by using equally elliptical dipoles stretched along the orthogonal axis. This may be realised in QDs with a vector magnetic field.We further generalise our approach to include the population state of a multi-level system. This allows us to calculate the dynamics of complicated multi-transition “quantum dipoles”, such as a Λ-system, recovering results consistent with experiment [3].
We consider the interaction of one- and two-photon pulses in a waveguide with two two-level systems (TLSs) that are also able to interact directly either through an exchange- or a dipole-type interaction. We focus on the system's transport properties and show how the presence of a second TLS increases the control options, especially when direct interactions are also allowed. We also obtain, within a Markov (long-pulse) approximation, exact results for the nonlinear or entangled terms that arise in the two-photon case and discuss both their potential applications and ways to minimize their effects.
We unveil the inherent topological nature from a two-dimensional synthetic parameter space in a single-photon transport problem, with the simple configuration of one waveguide coupled with two four-level atoms driven by external fields. We find that, by tuning the system in the dark-state condition, the system holds two independent phase degrees of freedom from both the photon propagation and the atomic transition. As the direct output, the photon scattering characteristics show the total reflection with the frequency preservation or the complete frequency conversion at the specific choices of two phases. We uncover the underlying topological effect by analyzing the photon scattering amplitudes in the parameter space of two phases, which links to the singularity with nonzero winding numbers. Our results are verified by numerical simulations, and hence point to an alternative perspective for the research at the intersection of single-photon transport and topology, without using the topological geometric arrangement of atoms.