(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

Rapid progress in the development of metamaterials and metaphotonics allowed bulky optical assemblies to be replaced with thin nanostructured films, often called metasurfaces, opening a broad range of novel and superior applications of flat optics to the generation, manipulation and detection of classical light. Recently, these developments started making headway in quantum photonics, where novel opportunities arose for the control of non-classical nature of light, including photon statistics, quantum state superposition, quantum entanglement and single-photon detection. In this Perspective, we review recent progress in the emerging field of quantum-photonics applications of metasurfaces, focusing on innovative and promising approaches to create, manipulate and detect non-classical light. Progress in the field of quantum-photonics applications of metasurfaces is reviewed. Cutting-edge research, including the development of optical chips supporting high-dimensional quantum entanglement and advanced quantum tomography, is summarized.

Metasurfaces for quantum photonics

Rapid progress in the development of metasurfaces allowed to replace bulky optical assemblies with thin nanostructured films, often called metasurfaces, opening a broad range of novel and superior applications to the generation, manipulation, and detection of light in classical optics. Recently, these developments started making a headway in quantum photonics, where novel opportunities arose for the control of nonclassical nature of light, including photon statistics, quantum state superposition, quantum entanglement, and single-photon detection. In this Perspective, we review recent progress in the field of quantum-photonics applications of metasurfaces, focusing on innovative and promising approaches to create, manipulate, and detect nonclassical light.

Metasurfaces for Quantum Photonics

Optical nanoantennas have shown a great capacity for efficient extraction of photons from the near to the far field, enabling directional emission from nanoscale single-photon sources. However, their potential for the generation and extraction of multi-photon quantum states remains unexplored. Here we experimentally demonstrate the nanoscale generation of two-photon quantum states at telecommunication wavelengths based on spontaneous parametric down-conversion in an optical nanoantenna. The antenna is a crystalline AlGaAs nanocylinder, possessing Mie-type resonances at both the pump and the bi-photon wavelengths, and when excited by a pump beam it generates photon pairs with a rate of 35 Hz. Normalized to the pump energy stored by the nanoantenna, this rate corresponds to 1.4 GHz/Wm, being 1 order of magnitude higher than conventional on-chip or bulk photon-pair sources. Our experiments open the way for multiplexing several antennas for coherent generation of multi-photon quantum states with complex spatial-mode entanglement and applications in free-space quantum communications and sensing.

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Spontaneous photon-pair generation from a dielectric nanoantenna

The full manipulation of intrinsic properties of electromagnetic waves has become the central target in various modern optical technologies. Optical metasurfaces have been suggested for a complete control of light-matter interaction with subwavelength structures, and they have been explored widely in the past decade for creating next-generation multifunctional flat-optics devices. The current studies of metasurfaces have reached a mature stage where common materials, basic optical physics, and conventional engineering tools have been explored extensively for various applications such as light bending, metalenses, metaholograms, and many others. A natural question is where the future research on metasurfaces will be going: Quo vadis, metasurfaces? In this Mini Review, we provide perspectives on the future developments of optical metasurfaces. Specifically, we highlight recent progresses on hybrid metasurfaces employing low-dimensional materials and discuss biomedical, computational, and quantum applications of metasurfaces, followed by discussions of challenges and foreseeing the future of metasurface physics and engineering.

Quo Vadis, Metasurfaces?

All-dielectric optical metasurfaces are a workhorse in nano-optics, because of both their ability to manipulate light in different degrees of freedom and their excellent performance at light frequency conversion. Here, we demonstrate first-time generation of photon pairs via spontaneous parametric-down conversion in lithium niobate quantum optical metasurfaces with electric and magnetic Mie-like resonances at various wavelengths. By engineering the quantum optical metasurface, we tailor the photon-pair spectrum in a controlled way. Within a narrow bandwidth around the resonance, the rate of pair production is enhanced up to 2 orders of magnitude, compared to an unpatterned film of the same thickness and material. These results enable flat-optics sources of entangled photons-a new promising platform for quantum optics experiments.

Photon Pairs from Resonant Metasurfaces.

We report, for the first time, the observation of spontaneous parametric down-conversion (SPDC) free of phase matching (momentum conservation). We alleviate the need to conserve momentum by exploiting the position-momentum uncertainty relation and using a planar geometry source, a $6\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ thick layer of lithium niobate. Nonphase-matched SPDC opens up a new platform on which to investigate fundamental quantum effects but it also has practical applications. The ultrasmall thickness leads to a frequency spectrum an order of magnitude broader than that of phase-matched SPDC. The strong two-photon correlations are still preserved due to energy conservation. This results in ultrashort temporal correlation widths and huge frequency entanglement. The studies we make here can be considered as the initial steps into the emerging field of nonlinear quantum optics on the microscale and nanoscale.

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Microscale Generation of Entangled Photons without Momentum Conservation

Quantum state engineering, the cornerstone of quantum photonic technologies, mainly relies on spontaneous parametric downconversion and four-wave mixing, where one or two pump photons spontaneously decay into a photon pair. Both of these nonlinear effects require momentum conservation for the participating photons, which strongly limits the versatility of the resulting quantum states. Nonlinear metasurfaces have subwavelength thickness and allow the relaxation of this constraint; when combined with resonances, they greatly expand the possibilities of quantum state engineering. Here, we generated entangled photons via spontaneous parametric downconversion in semiconductor metasurfaces with high–quality factor, quasi-bound state in the continuum resonances. By enhancing the quantum vacuum field, our metasurfaces boost the emission of nondegenerate entangled photons within multiple narrow resonance bands and over a wide spectral range. A single resonance or several resonances in the same sample, pumped at multiple wavelengths, can generate multifrequency quantum states, including cluster states. These features reveal metasurfaces as versatile sources of complex states for quantum information. Description Another twist Metasurfaces are specially designed arrays of dielectric components that transform the function of bulk optical components into thin films. Exploiting the physics of bulk states in the continuum for the highly efficient trapping of light, Santiago-Cruz et al. demonstrate metasurfaces that operate as sources of quantum and chiral light, respectively. Patterned in gallium arsenide, the quantum source can provide entangled pairs of photons across a broad range of wavelengths, allowing for the formation of complex quantum states. The authors also used a dielectric metasurface doped with emitting molecules to produce chiral light and lasing. Both approaches will be useful for the development of integrated optical and quantum optical devices. —ISO Resonant metasurfaces generate photon pairs at multiple selected wavelengths, forming complex quantum states.

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Resonant metasurfaces for generating complex quantum states

Miniaturized entangled photon sources, in particular based on subwavelength metasurfaces, are highly demanded for the development of integrated quantum photonics. Here, as a first step towards the development of quantum optical metasurfaces (QOMs), we demonstrate generation of entangled photons via spontaneous parametric down-conversion (SPDC) from subwavelength films. We achieve photon pair generation with a high coincidence-to-accidental ratio in lithium niobate and gallium phosphide nanofilms. By implementing the fiber spectroscopy of SPDC in nanofilms, we measure a spectrum with a bandwidth of 500 nm, limited only by the overall detection efficiency. The spectrum reveals vacuum field enhancement due to a Fabry-Perot resonance inside the nonlinear films. It also suggests a strategy for observing SPDC from QOM. Our experiments lay the groundwork for future development of flat SPDC sources, including QOM.

Entangled photons from subwavelength nonlinear films.

The concept of "flat optics" is quickly conquering different fields of photonics, but its implementation in quantum optics is still in its infancy. In particular, polarization entanglement, strongly required in quantum photonics, is so far not realized on "flat" platforms. Meanwhile, relaxed phase matching of "flat" nonlinear optical sources enables enormous freedom in tailoring their polarization properties. Here we use this freedom to generate photon pairs with tunable polarization entanglement via spontaneous parametric downconversion (SPDC) in a 400-nm GaP film. By changing the pump polarization, we tune the polarization state of photon pairs from maximally entangled to almost disentangled, which is impossible in a single bulk SPDC source. Polarization entanglement, together with the broadband frequency spectrum, results in an ultranarrow (12 fs) Hong-Ou-Mandel effect and promises extensions to hyperentanglement.

Tomas Santiago-Cruz

Papers

Photon Pairs from Resonant Metasurfaces.

Microscale Generation of Entangled Photons without Momentum Conservation

Resonant metasurfaces for generating complex quantum states

Entangled photons from subwavelength nonlinear films.

Flat-optics generation of broadband photon pairs with tunable polarization entanglement.