Maximum Refractive Index of an Atomic Medium

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 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.

arXiv:2006.01680Physical Review XFeb 9, 202152 citations

Multiple light scattering in atomic media : from metasurfaces to the ultimate refractive index

Francesco Andreoli

(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. Aquí, desarrollamos nuevas técnicas para tratar la dispersión múltiple de la luz, con el objetivo final de comprender los límites fundamentales del índice de refracción, así como proponer aplicaciones fotónicas innovadoras. Nuestros resultados se dividen en tres partes. Primero, investigamos un conjunto desordenado de átomos con densidad creciente, hasta regímenes donde se requiere un tratamiento completo de la dispersión múltiple y de las interacciones de campo cercano. En esta situación, encontramos que estos efectos limitan el índice de refracción a un valor máximo de n ~ 1,7. Proponemos una explicación basada en la teoría del grupo de renormalización, en la que las interacciones de campo cercano, combinadas con posiciones atómicas aleatorias, desarrollan una ampliación no homogénea de las frecuencias atómicas de resonancia. 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. Desarrollamos un modelo para incluir el inicio de este régimen en nuestro análisis, argumentando que la aparición del magnetismo cuántico, las correlaciones de densidad y la dinámica de efecto túnel de los electrones suprimen efectivamente la respuesta monomodo, bajando nuevamente el índice a la unidad. No obstante, justo antes del inicio de los procesos químicos, nuestra teoría predice la posibilidad teórica de sintetizar un material con un índice de refracción sorprendentemente alto (n ~ 30) y pérdidas bajas. Por último, inspirándonos en la impresionante respuesta óptica de las redes atómicas, proponemos su uso para imitar un dispositivo óptico complejo, a saber, una lente delgada. El componente básico de esta "metalente atómica" está compuesto por tres redes atómicas bidimensionales consecutivas, cuyas distancias y constantes de red se eligen adecuadamente para garantizar una alta transmisión de la luz, así como un cambio de fase arbitrario. 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.(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. Aquí, desarrollamos nuevas técnicas para tratar la dispersión múltiple de la luz, con el objetivo final de comprender los límites fundamentales del índice de refracción, así como proponer aplicaciones fotónicas innovadoras. Nuestros resultados se dividen en tres partes. Primero, investigamos un conjunto desordenado de átomos con densidad creciente, hasta regímenes donde se requiere un tratamiento completo de la dispersión múltiple y de las interacciones de campo cercano. En esta situación, encontramos que estos efectos limitan el índice de refracción a un valor máximo de n ~ 1,7. Proponemos una explicación basada en la teoría del grupo de renormalización, en la que las interacciones de campo cercano, combinadas con posiciones atómicas aleatorias, desarrollan una ampliación no homogénea de las frecuencias atómicas de resonancia. 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. Desarrollamos un modelo para incluir el inicio de este régimen en nuestro análisis, argumentando que la aparición del magnetismo cuántico, las correlaciones de densidad y la dinámica de efecto túnel de los electrones suprimen efectivamente la respuesta monomodo, bajando nuevamente el índice a la unidad. No obstante, justo antes del inicio de los procesos químicos, nuestra teoría predice la posibilidad teórica de sintetizar un material con un índice de refracción sorprendentemente alto (n ~ 30) y pérdidas bajas. Por último, inspirándonos en la impresionante respuesta óptica de las redes atómicas, proponemos su uso para imitar un dispositivo óptico complejo, a saber, una lente delgada. El componente básico de esta "metalente atómica" está compuesto por tres redes atómicas bidimensionales consecutivas, cuyas distancias y constantes de red se eligen adecuadamente para garantizar una alta transmisión de la luz, así como un cambio de fase arbitrario. 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.

Mar 23, 20230 citations

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

Andreoli, Francesco, Bennet Windt, Stefano Grava +4 more

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.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.

arXiv:2303.10998Mar 20, 20230 citations

Renormalization group analysis of near-field induced dephasing of optical spin waves in an atomic medium

Stefano Grava, Yizun He, Saijun Wu +1 more

Abstract While typical theories of atom–light interactions treat the atomic medium as being smooth, it is well-known that microscopic optical effects driven by atomic granularity, dipole–dipole interactions, and multiple scattering can lead to important effects. Recently, for example, it was experimentally observed that these ingredients can lead to a fundamental, density-dependent dephasing of optical spin waves in a disordered atomic medium. Here, we go beyond the short-time and dilute limits considered previously, to develop a comprehensive theory of dephasing dynamics for arbitrary times and atomic densities. In particular, we develop a novel, non-perturbative theory based on strong disorder renormalization group (RG), in order to quantitatively predict the dominant role that near-field optical interactions between nearby neighbors has in driving the dephasing process. This theory also enables one to capture the key features of the many-atom dephasing dynamics in terms of an effective single-atom model. These results should shed light on the limits imposed by near-field interactions on quantum optical phenomena in dense atomic media, and illustrate the promise of strong disorder RG as a method of dealing with complex microscopic optical phenomena in such systems.Abstract While typical theories of atom–light interactions treat the atomic medium as being smooth, it is well-known that microscopic optical effects driven by atomic granularity, dipole–dipole interactions, and multiple scattering can lead to important effects. Recently, for example, it was experimentally observed that these ingredients can lead to a fundamental, density-dependent dephasing of optical spin waves in a disordered atomic medium. Here, we go beyond the short-time and dilute limits considered previously, to develop a comprehensive theory of dephasing dynamics for arbitrary times and atomic densities. In particular, we develop a novel, non-perturbative theory based on strong disorder renormalization group (RG), in order to quantitatively predict the dominant role that near-field optical interactions between nearby neighbors has in driving the dephasing process. This theory also enables one to capture the key features of the many-atom dephasing dynamics in terms of an effective single-atom model. These results should shed light on the limits imposed by near-field interactions on quantum optical phenomena in dense atomic media, and illustrate the promise of strong disorder RG as a method of dealing with complex microscopic optical phenomena in such systems.

arXiv:2108.09268New Journal of PhysicsDec 24, 20217 citations

Refractive index of a dilute Bose gas

O. Morice, Yvan Castin, Jean Dalibard

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 ${)}^{3}$ . 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 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${)}^{3}$. 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.

Physical Review AMay 1, 1995238 citations

Emergence of an Epsilon-Near-Zero Medium from Microscopic Atomic Principles

Lewis Ruks, Janne Ruostekoski

We demonstrate that an effective near-zero refractive index can emerge from collective light scattering in a discrete atomic lattice, using essentially exact microscopic simulations. In a 25-layer array, cooperative response leads to over a thirtyfold increase in the effective optical wavelength within the medium, almost eliminating optical phase accumulation, with potential applications in spectroscopy and optical manipulation of quantum emitters. Crucially, the near-zero refractive index arises from first-principles microscopic theory, rather than being imposed through continuous phenomenological effective-medium models-providing conceptually important insight into the emergence of macroscopic electromagnetism from atomic-scale interactions.We demonstrate that an effective near-zero refractive index can emerge from collective light scattering in a discrete atomic lattice, using essentially exact microscopic simulations. In a 25-layer array, cooperative response leads to over a thirtyfold increase in the effective optical wavelength within the medium, almost eliminating optical phase accumulation, with potential applications in spectroscopy and optical manipulation of quantum emitters. Crucially, the near-zero refractive index arises from first-principles microscopic theory, rather than being imposed through continuous phenomenological effective-medium models-providing conceptually important insight into the emergence of macroscopic electromagnetism from atomic-scale interactions.

arXiv:2506.13294Physical Review LettersNov 17, 20253 citations

(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.

{)}^{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.