There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

In recent years, notions drawn from non-Hermitian physics and parity–time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss can lead to entirely new and unexpected features has initiated an intense research effort to explore non-Hermitian systems both theoretically and experimentally. Here we review recent progress in this emerging field, and provide an outlook to future directions and developments. This Review Article outlines the exploration of the interplay between parity–time symmetry and non-Hermitian physics in optics, plasmonics and optomechanics.

Non-Hermitian physics and PT symmetry

Tables of integrals

Hybrid quantum circuits combine two or more physical systems, with the goal of harnessing the advantages and strengths of the different systems in order to better explore new phenomena and potentially bring about novel quantum technologies. This article presents a brief overview of the progress achieved so far in the field of hybrid circuits involving atoms, spins, and solid-state devices (including superconducting and nanomechanical systems). How these circuits combine elements from atomic physics, quantum optics, condensed matter physics, and nanoscience is discussed, and different possible approaches for integrating various systems into a single circuit are presented. In particular, hybrid quantum circuits can be fabricated on a chip, facilitating their future scalability, which is crucial for building future quantum technologies, including quantum detectors, simulators, and computers.

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Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems

Sensors play an important part in many aspects of daily life such as infrared sensors in home security systems, particle sensors for environmental monitoring and motion sensors in mobile phones. High-quality optical microcavities are prime candidates for sensing applications because of their ability to enhance light-matter interactions in a very confined volume. Examples of such devices include mechanical transducers, magnetometers, single-particle absorption spectrometers, and microcavity sensors for sizing single particles and detecting nanometre-scale objects such as single nanoparticles and atomic ions. Traditionally, a very small perturbation near an optical microcavity introduces either a change in the linewidth or a frequency shift or splitting of a resonance that is proportional to the strength of the perturbation. Here we demonstrate an alternative sensing scheme, by which the sensitivity of microcavities can be enhanced when operated at non-Hermitian spectral degeneracies known as exceptional points. In our experiments, we use two nanoscale scatterers to tune a whispering-gallery-mode micro-toroid cavity, in which light propagates along a concave surface by continuous total internal reflection, in a precise and controlled manner to exceptional points. A target nanoscale object that subsequently enters the evanescent field of the cavity perturbs the system from its exceptional point, leading to frequency splitting. Owing to the complex-square-root topology near an exceptional point, this frequency splitting scales as the square root of the perturbation strength and is therefore larger (for sufficiently small perturbations) than the splitting observed in traditional non-exceptional-point sensing schemes. Our demonstration of exceptional-point-enhanced sensitivity paves the way for sensors with unprecedented sensitivity.

Exceptional points enhance sensing in an optical microcavity

So-called exceptional points, degenerate quantum states, allow higher energy splitting under the same perturbation conditions, greatly improving the detection sensitivity of sensors.

Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection

/pdf/dielectric-microcavities-model-systems-for-wave-chaos-and-103vgugf73.pdf

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

Controlling the emission and the flow of light in micro- and nanostructures is crucial for on-chip information processing. Here we show how to impose a strong chirality and a switchable direction of light propagation in an optical system by steering it to an exceptional point (EP)—a degeneracy universally occurring in all open physical systems when two eigenvalues and the corresponding eigenstates coalesce. In our experiments with a fiber-coupled whispering-gallery-mode (WGM) resonator, we dynamically control the chirality of resonator modes and the emission direction of a WGM microlaser in the vicinity of an EP: Away from the EPs, the resonator modes are nonchiral and laser emission is bidirectional. As the system approaches an EP, the modes become chiral and allow unidirectional emission such that by transiting from one EP to another one the direction of emission can be completely reversed. Our results exemplify a very counterintuitive feature of non-Hermitian physics that paves the way to chiral photonics on a chip.

https://www.pnas.org/content/pnas/113/25/6845.full.pdf

Chiral modes and directional lasing at exceptional points.

A general theory of sensors based on the detection of splittings of resonant frequencies or energy levels operating at so-called exceptional points is presented. Exploiting the complex-square-root topology near such non-Hermitian degeneracies has a great potential for enhanced sensitivity. Passive and active systems are discussed. The theory is specified for whispering-gallery microcavity sensors for particle detection. As example, a microdisk with two holes is studied numerically. The theory and numerical simulations demonstrate a sevenfold enhancement of the sensitivity.

Jan Wiersig

Papers

Exceptional points enhance sensing in an optical microcavity

Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

Chiral modes and directional lasing at exceptional points.

Sensors operating at exceptional points: General theory