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arXiv:2303.10998

The maximum refractive index of an atomic crystal $\unicode{x2013}$ from quantum optics to quantum chemistry

Andreoli, Francesco, Bennet Windt, Stefano Grava, Gian Marcello Andolina, Michael J. Gullans, Alexander A. High, Darrick E. Chang

Published Mar 20, 2023• 0 citations
arXivHTMLPDF

Abstract

All known optical materials have an index of refraction of order unity. Despite the tremendous implications that an ultrahigh index could have for optical technologies, little research has been done on why the refractive index of materials is universally small, and whether this observation is fundamental. Here, we investigate the index of an ordered arrangement of atoms, as a function of atomic density. At dilute densities, this problem falls into the realm of quantum optics, where atoms do not interact with one another except via the scattering of light. On the other hand, when the lattice constant becomes comparable to the Bohr radius, the electronic orbitals begin to overlap, giving rise to quantum chemistry. We present a minimal model that allows for a unifying theory of index spanning these two regimes. A key aspect is the treatment of multiple light scattering, which can be highly non-perturbative over a large density range, and which is the reason that conventional theories of the index break down. In the quantum optics regime, we show that ideal light-matter interactions can have a single-mode nature, allowing for a purely real refractive index that grows with density as $(N/V)^{1/3}$. At the onset of quantum chemistry, we show how two physical mechanisms (excited electron tunneling dynamics and the buildup of electronic density-density correlations) can open up inelastic or spatial multi-mode light scattering processes, which ultimately reduce the index back to order unity while introducing absorption. Around the onset of chemistry, our theory predicts that ultrahigh index ($n\sim 30$), low-loss materials could in principle be allowed by the laws of nature.

DOI
10.48550/arxiv.2303.10998

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All known optical materials have an index of refraction of order unity. Despite the tremendous implications that an ultrahigh index could have for optical technologies, little research has been done on why the refractive index of materials is universally small, and whether this observation is fundamental. Here, we investigate the index of an ordered arrangement of atoms, as a function of atomic density. At dilute densities, this problem falls into the realm of quantum optics, where atoms do not interact with one another except via the scattering of light. On the other hand, when the lattice constant becomes comparable to the Bohr radius, the electronic orbitals begin to overlap, giving rise to quantum chemistry. We present a minimal model that allows for a unifying theory of index spanning these two regimes. A key aspect is the treatment of multiple light scattering, which can be highly non-perturbative over a large density range, and which is the reason that conventional theories of the index break down. In the quantum optics regime, we show that ideal light-matter interactions can have a single-mode nature, allowing for a purely real refractive index that grows with density as (N/V)1/3(N/V)^{1/3}(N/V)1/3. At the onset of quantum chemistry, we show how two physical mechanisms (excited electron tunneling dynamics and the buildup of electronic density-density correlations) can open up inelastic or spatial multi-mode light scattering processes, which ultimately reduce the index back to order unity while introducing absorption. Around the onset of chemistry, our theory predicts that ultrahigh index (n∼30n\sim 30n∼30), low-loss materials could in principle be allowed by the laws of nature.
It is interesting to observe that all optical materials with a positive refractive index have a value of index that is of order unity.Surprisingly, though, a deep understanding of the mechanisms that lead to this universal behavior seems to be lacking.Moreover, this observation is difficult to reconcile with the fact that a single isolated atom is known to have a giant optical response, as characterized by a resonant scattering cross section that far exceeds its physical size.Here, we theoretically and numerically investigate the evolution of the optical properties of an ensemble of ideal atoms as a function of density, starting from the dilute gas limit, including the effects of multiple scattering and near-field interactions.Interestingly, despite the giant response of an isolated atom, we find that the maximum index does not indefinitely grow with increasing density but rather reaches a limiting value of n ≈ 1.7.This limit arises purely from electrodynamics, as it occurs at densities far below those where chemical processes become important.We propose an explanation based upon strong-disorder renormalization group theory, in which the nearfield interaction combined with random atomic positions results in an inhomogeneous broadening of atomic resonance frequencies.This mechanism ensures that, regardless of the physical atomic density, light at any given frequency only interacts with at most a few near-resonant atoms per cubic wavelength, thus limiting the maximum index attainable.Our work is a promising first step to understand the limits of the refractive index from a bottom-up, atomic physics perspective, and it also introduces the renormalization group as a powerful tool to understand the generally complex problem of multiple scattering of light overall.
It is interesting to observe that all optical materials with a positive refractive index have a value of index that is of order unity. Surprisingly, though, a deep understanding of the mechanisms that lead to this universal behavior seems to be lacking. Moreover, this observation is difficult to reconcile with the fact that a single isolated atom is known to have a giant optical response, as characterized by a resonant scattering cross section that far exceeds its physical size. Here, we theoretically and numerically investigate the evolution of the optical properties of an ensemble of ideal atoms as a function of density, starting from the dilute gas limit, including the effects of multiple scattering and near-field interactions. Interestingly, despite the giant response of an isolated atom, we find that the maximum index does not indefinitely grow with increasing density but rather reaches a limiting value of n ≈ 1.7. This limit arises purely from electrodynamics, as it occurs at densities far below those where chemical processes become important. We propose an explanation based upon strong-disorder renormalization group theory, in which the near-field interaction combined with random atomic positions results in an inhomogeneous broadening of atomic resonance frequencies. This mechanism ensures that, regardless of the physical atomic density, light at any given frequency only interacts with at most a few near-resonant atoms per cubic wavelength, thus limiting the maximum index attainable. Our work is a promising first step to understand the limits of the refractive index from a bottom-up, atomic physics perspective, and it also introduces the renormalization group as a powerful tool to understand the generally complex problem of multiple scattering of light overall.
(English) Our ability to confine, guide, and bend light has led to astonishing technological achievements, playing a fundamental role in diverse fields like microscopy, photochemistry, telecommunications or material design. The key property of materials that allows to control light is the refractive index. Notably, regardless of very different microscopic structures, all natural materials exhibit a modest, near-unity index of refraction, n ~ 1. This universality suggests the existence of some simple, ubiquitous origin, whose complete characterization from microscopic considerations, surprisingly, is still missing. Moreover, one can wonder which principles might allow to synthesize a material with an ultra-high index, to boost the performance of photonic devices. In this thesis, we address these questions from an atomic-physics standpoint, exploring if the macroscopic optical properties can be related to simple, electrodynamical processes occurring between well-separated atoms, which only interact via light scattering. Standard theories neglect that light can be scattered multiple times, and lead to unphysical predictions when strong interference occurs between the coherent atomic emission, such as in dense atomic ensembles or ordered lattices. We here develop new techniques to address the physics of multiple light scattering, with the ultimate goal of understanding the fundamental limits to the refractive index, as well as proposing unexpected photonic applications. Our results are divided in three parts. First, we investigate an ensemble of ideal atoms with increasing atomic density, starting from the dilute gas limit, up to dense regimes where a non-perturbative treatment of multiple scattering and near-field interactions is required. In this situation, we find that these effects limit the index to a maximum value of n ~ 1.7, in contrast with standard theories. We propose an explanation based upon strong-disorder renormalization group theory, in which the near-field interactions combined with random atomic positions result in an inhomogeneous broadening of the atomic resonance frequencies. This basic mechanism ensures that regardless of the physical atomic density, light at any given frequency only interacts with at most a few near-resonant atoms per cubic wavelength, thus limiting the index attainable. Afterwards, we show that a radically different behavior is expected for an ideal, atomic crystal. As long as the inter-atomic interactions are only mediated by multiple scattering, each 2D array of the crystal exhibits a lossless, single-mode response, which builds up a very large and purely real refractive index. To address the limits to this picture, we extend our theoretical analysis to much higher densities, where the electronic orbitals on neighboring nuclei begin to overlap. We develop a minimal model to include the onset of this regime into our non-perturbative analysis of multiple light scattering, arguing that the emergence of quantum magnetism, density-density correlations and tunneling dynamics of the electrons effectively suppresses the single-mode response, decreasing the index back to unity. Nonetheless, right before the onset of chemistry, our theory predicts that an ultra-high-index (n ~ 30) and low-loss material could in principle be allowed by the laws of nature. Finally, inspired by the impressive optical response of atomic arrays, we propose their use as a more complex optical device, namely a thin lens. The building blocks of this "atomic metalens" are composed of three consecutive 2D arrays, whose distance and lattice constants are suitably chosen to guarantee a high transmission of light, as well as an arbitrary phase shift. To characterize its efficiency and prove its robustness against losses, we perform large-scale numerical simulations, on a number of atoms between one and two orders of magnitude larger than comparable works. (Español) Nuestra capacidad de confinar y guiar la luz nos ha llevado a logros tecnológicos asombrosos, jugando un papel fundamental en campos tan diversos como la fotoquímica, las telecomunicaciones o el diseño de materiales. La propiedad clave de un material para controlar la luz es su índice de refracción. En particular, todos los materiales naturales exhiben un índice de refracción modesto, cercano a la unidad, n ~ 1. Esta universalidad sugiere la existencia de algún origen simple y ubicuo, cuya caracterización completa a partir de consideraciones microscópicas, sorprendentemente, aún falta. Además, para aumentar el rendimiento de los dispositivos fotónicos, es crucial entender si los principios físicos permiten o prohíben la síntesis de materiales con índices más altos. En esta tesis, abordamos estas cuestiones desde un punto de vista atómico, estudiando si las propiedades ópticas macroscópicas pueden deberse a procesos electrodinámicos entre átomos bien separados, que solo interactúan a través de la dispersión de la luz. Las teorías estándar ignoran que la luz puede dispersarse varias veces y conducen a predicciones erradas en situaciones de fuerte interferencia, como en redes cristalinas o conjuntos densos de átomos. 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Este mecanismo asegura que, independientemente de la densidad atómica, la luz (para cualquier frecuencia dada) solo interactúa con unos pocos átomos resonantes por unidad cúbica de longitud de onda, limitando la respuesta óptica. Un comportamiento radicalmente diferente se manifiesta en una red cristalina de átomos. Siempre que las interacciones solo estén mediadas por dispersión múltiple, cada capa del cristal exhibe una respuesta monomodo sin pérdidas, que genera un índice de refracción muy grande y puramente real. Para abordar los límites de esta respuesta física, ampliamos nuestro análisis teórico hasta densidades tan altas que los orbitales electrónicos de los núcleos vecinos comienzan a superponerse. 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Para caracterizar su eficiencia y probar su robustez frente a pérdidas, realizamos simulaciones numéricas incluyendo un gran número de átomos, entre uno y dos órdenes de magnitud mayor que en trabajos comparables.
We derive the dispersion relation for the propagation of quasiresonant light with frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}$ in an ultracold gas of bosonic atoms in the dilute regime, i.e., for an atomic density ${\mathrm{\ensuremath{\rho}}}_{0}$\ensuremath{\ll}(${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}$/c)3{)}^{3})3. In our calculation, valid up to order 2 in density, two types of corrections to the Lorentz-Lorenz formula for the refractive index appear. The first one is due to the bosonic nature of the atoms and its contribution is related to the two-body correlation function. The second correction originates from multiple scattering of photons within pairs of close atoms, giving rise to the resonant van der Waals interaction. The temperature dependence of the refractive index gives a clear signature of quantum statistical effects, even if the degeneracy threshold for Bose-Einstein condensation is not reached.
We investigated the complex index of refraction in the x-ray regime of atoms in laser light. The laser (intensity up to 1013W∕cm2{10}^{13}\phantom{\rule{0.3em}{0ex}}\mathrm{W}∕{\mathrm{cm}}^{2}1013W∕cm2, wavelength 800nm800\phantom{\rule{0.3em}{0ex}}\mathrm{nm}800nm) modifies the atomic states but, by assumption, does not excite or ionize the atoms in their electronic ground state. Using quantum electrodynamics, we devise an ab initio theory to calculate the dynamic dipole polarizability and the photoabsorption cross section, which are subsequently used to determine the real and imaginary part, respectively, of the refractive index. The interaction with the laser is treated nonperturbatively; the x-ray interaction is described in terms of a one-photon process. We numerically solve the resolvents involved using a single-vector Lanczos algorithm. Finally, we formulate rate equations to copropagate a laser and an x-ray pulse through a gas cell. Our theory is applied to argon. We study the x-ray polarizability and absorption near the argon KKK edge over a large range of dressing-laser intensities. We find electromagnetically induced transparency (EIT) for x rays on the Ar $1s\ensuremath{\rightarrow}4p$ pre-edge resonance. We demonstrate that EIT in Ar allows one to imprint the shape of an ultrafast laser pulse on a broader x-ray pulse (duration 100ps100\phantom{\rule{0.3em}{0ex}}\mathrm{ps}100ps, photon energy 3.2keV3.2\phantom{\rule{0.3em}{0ex}}\mathrm{keV}3.2keV). Our work thus opens new opportunities for research with hard x-ray sources.
We study the index of refraction of an ultracold bosonic gas in the dilute regime. Using phase-contrast imaging with light detuned from resonance by several tens of linewidths, we image a single cloud of ultracold atoms for 100 consecutive shots, which enables the study of the scattering rate as a function of temperature and density using only a single cloud. We observe that the scattering rate is increased below the critical temperature for Bose-Einstein condensation by a factor of 3 compared to the single-atom scattering rate. We show that current atom-light interaction models to second order of the density show a similar increase, where the magnitude of the effect depends on the model that is used to calculate the pair-correlation function. This confirms that the effect of quantum statistics on the index of refraction is dominant in this regime.