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Ji L, He Y, Cai Q, Fang Z, Wang Y, Qiu L, Zhou L, Wu S, Grava S, Chang DE. Superradiant Detection of Microscopic Optical Dipolar Interactions. PHYSICAL REVIEW LETTERS 2023; 131:253602. [PMID: 38181370 DOI: 10.1103/physrevlett.131.253602] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2021] [Revised: 08/16/2023] [Accepted: 10/31/2023] [Indexed: 01/07/2024]
Abstract
The interaction between light and cold atoms is a complex phenomenon potentially featuring many-body resonant dipole interactions. A major obstacle toward exploring these quantum resources of the system is macroscopic light propagation effects, which not only limit the available time for the microscopic correlations to locally build up, but also create a directional, superradiant emission background whose variations can overwhelm the microscopic effects. In this Letter, we demonstrate a method to perform "background-free" detection of the microscopic optical dynamics in a laser-cooled atomic ensemble. This is made possible by transiently suppressing the macroscopic optical propagation over a substantial time, before a recall of superradiance that imprints the effect of the accumulated microscopic dynamics onto an efficiently detectable outgoing field. We apply this technique to unveil and precisely characterize a density-dependent, microscopic dipolar dephasing effect that generally limits the lifetime of optical spin-wave order in ensemble-based atom-light interfaces.
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Affiliation(s)
- Lingjing Ji
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Yizun He
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Qingnan Cai
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Zhening Fang
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Yuzhuo Wang
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Liyang Qiu
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Lei Zhou
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Saijun Wu
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Stefano Grava
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain and ICREA-Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain
| | - Darrick E Chang
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain and ICREA-Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain
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2
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No Cooperative Lamb Shift in Response of Thin Slab to cw Beam of Resonant Light. ATOMS 2022. [DOI: 10.3390/atoms10020049] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023] Open
Abstract
We review the definition of cooperative Lamb shift originally introduced by ourselves and S. R. Hartmann in 1973. We point out that the definition specified the preparation of a sample of identical two-level atoms prepared with partial excitation by a short pulse. We spell out in some detail the reasoning behind our assertion that the CLS does not enter into the dielectric constant, which determines the transmission of cw radiation through a sample. We give a prescription, using the transfer matrix formalism, for determining the transmission coefficient through a slab, given the thickness in wavelengths and the dielectric constant. We explore the possibility of achieving a true measurement of the CLS in a gaseous cold-atom cloud, with the help of a large foreign gas broadening of the resonant line.
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3
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Yoo SM. Optical cooperative effects of multiemitters in a one-dimensional (1D) dense array. OPTICS EXPRESS 2021; 29:35314-35326. [PMID: 34808968 DOI: 10.1364/oe.440558] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2021] [Accepted: 09/30/2021] [Indexed: 06/13/2023]
Abstract
We theoretically explore cooperative effects of equally spaced multiemitters in a 1D dense array driven by a low-intensity probe field propagating through a 1D waveguide by modeling the emitters as point-like coupled electric dipoles. We calculate the collective optical spectra of a number of 1D emitter arrays with any radiation-retention coefficient η using both exact classical-electrodynamics and mean-field-theory formalisms. We illustrate cooperative effects of lossless 1D emitter arrays with η = 1 at the emitter spacings, which are displayed by steep edges accompanied by a deep minimum and Fano resonances in the plots of transmissivities as a function of the detuning of the incident light from the emitter resonance. Numerical simulation of the full width of such optical bandgaps reveals that cooperativity between emitters is greater in a small array of size N ≤ 8 than in a larger one of size N > 8. For a lossy 1D emitter array in which the radiation retention coefficient is equal to or less than 0.1 the transmissivity obtained by exact-electrodynamics scheme exhibits no bandgap structures, being in good agreement with the mean-field-theory result. We propose that a 1D multiemitter array may work as a nanoscale filter blocking transmission of light with a frequency in the range of optical bandgaps.
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4
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He Y, Ji L, Wang Y, Qiu L, Zhao J, Ma Y, Huang X, Wu S, Chang DE. Geometric Control of Collective Spontaneous Emission. PHYSICAL REVIEW LETTERS 2020; 125:213602. [PMID: 33275003 DOI: 10.1103/physrevlett.125.213602] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2019] [Accepted: 10/05/2020] [Indexed: 06/12/2023]
Abstract
Dipole spin-wave states of atomic ensembles with wave vector k(ω) mismatched from the dispersion relation of light are difficult to access by far-field excitation but may support rich phenomena beyond the traditional phase-matched scenario in quantum optics. We propose and demonstrate an optical technique to efficiently access these states. In particular, subnanosecond laser pulses shaped by a home-developed wideband modulation method are applied to shift the spin wave in k space with state-dependent geometric phase patterning, in an error-resilient fashion and on timescales much faster than spontaneous emission. We verify this control through the redirection, switch off, and recall of collectively enhanced emission from a ^{87}Rb gas with ∼75% single-step efficiency. Our work represents a first step toward efficient control of electric dipole spin waves for studying many-body dissipative dynamics of excited gases, as well as for numerous quantum optical applications.
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Affiliation(s)
- Yizun He
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Lingjing Ji
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Yuzhuo Wang
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Liyang Qiu
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Jian Zhao
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Yudi Ma
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Xing Huang
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Saijun Wu
- Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China
| | - Darrick E Chang
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain and ICREA-Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain
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5
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Sinha K, Venkatesh BP, Meystre P. Collective Effects in Casimir-Polder Forces. PHYSICAL REVIEW LETTERS 2018; 121:183605. [PMID: 30444396 DOI: 10.1103/physrevlett.121.183605] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2018] [Indexed: 06/09/2023]
Abstract
We study cooperative phenomena in the fluctuation-induced forces between a surface and a system of neutral two-level quantum emitters prepared in a coherent collective state, showing that the total Casimir-Polder force on the emitters can be modified via their mutual correlations. Particularly, we find that a one-dimensional chain of emitters prepared in a super- or subradiant state experiences an enhanced or suppressed collective vacuum-induced force, respectively. The collective nature of dispersion forces can be understood as resulting from the interference between the different processes contributing to the surface-modified resonant dipole-dipole interaction. Such cooperative fluctuation forces depend singularly on the surface response at the resonance frequency of the emitters, thus being easily maneuverable. Our results demonstrate the potential of collective phenomena as a new tool to selectively tailor vacuum forces.
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Affiliation(s)
- Kanupriya Sinha
- US Army Research Laboratory, Adelphi, Maryland 20783, USA; Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA; and Max Planck Institute for the Physics of Complex Systems, Dresden 01187, Germany
| | - B Prasanna Venkatesh
- Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria, and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
| | - Pierre Meystre
- Department of Physics and College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
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Peyrot T, Sortais YRP, Browaeys A, Sargsyan A, Sarkisyan D, Keaveney J, Hughes IG, Adams CS. Collective Lamb Shift of a Nanoscale Atomic Vapor Layer within a Sapphire Cavity. PHYSICAL REVIEW LETTERS 2018; 120:243401. [PMID: 29956978 DOI: 10.1103/physrevlett.120.243401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2017] [Revised: 04/17/2018] [Indexed: 06/08/2023]
Abstract
We measure the near-resonant transmission of light through a dense medium of potassium vapor confined in a cell with nanometer thickness in order to investigate the origin and validity of the collective Lamb shift. A complete model including the multiple reflections in the nanocell reproduces accurately the observed line shape. It allows the extraction of a density-dependent shift and width of the bulk atomic medium resonance, deconvolved from the cavity effect. We observe an additional, unexpected dependence of the shift with the thickness of the medium. This extra dependence demands further experimental and theoretical investigations.
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Affiliation(s)
- T Peyrot
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, F-91127 Palaiseau Cedex, France
| | - Y R P Sortais
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, F-91127 Palaiseau Cedex, France
| | - A Browaeys
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, F-91127 Palaiseau Cedex, France
| | - A Sargsyan
- Institute for Physical Research, National Academy of Sciences, Ashtarak 2, 0203, Armenia
| | - D Sarkisyan
- Institute for Physical Research, National Academy of Sciences, Ashtarak 2, 0203, Armenia
| | - J Keaveney
- Department of Physics, Rochester Building, Durham University, South Road, Durham DH1 3LE, United Kingdom
| | - I G Hughes
- Department of Physics, Rochester Building, Durham University, South Road, Durham DH1 3LE, United Kingdom
| | - C S Adams
- Department of Physics, Rochester Building, Durham University, South Road, Durham DH1 3LE, United Kingdom
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7
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Wang BX, Zhao CY, Kan YH, Huang TC. Design of metasurface polarizers based on two-dimensional cold atomic arrays. OPTICS EXPRESS 2017; 25:18760-18773. [PMID: 29041070 DOI: 10.1364/oe.25.018760] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2017] [Accepted: 07/21/2017] [Indexed: 06/07/2023]
Abstract
Engineering light-matter interaction using cold atomic arrays is one of the central topics in modern optics. Here we have demonstrated the capability of two-dimensional asymmetric cold atomic arrays as microscopic metasurfaces for controlling polarization states of light. The designed linear polarizer can lead to an extinction ratio over 20dB as well as a high transmittance over 0.8 for the permitted polarization at zero detuning. For detuned driving light, changing lattice constants can also achieve high performance linear polarizers. We have also accomplished a circular polarizer by manipulating the phases of transmitted light. A theoretical analysis based on Bloch theorem shows the underlying mechanism for this performance is actually attributed to cooperative effects in periodic lattices. Finally, we discuss in detail the effects of system size, lattice imperfection and nonzero driving light linewidth in practical implementation. The present study paves a way to design extremely miniaturized metasurfaces using cold atoms and other two-level systems, showing great potential in quantum information and quantum metrology sciences as well as the fundamental physics of light-matter interaction.
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8
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Jenkins SD, Ruostekoski J, Papasimakis N, Savo S, Zheludev NI. Many-Body Subradiant Excitations in Metamaterial Arrays: Experiment and Theory. PHYSICAL REVIEW LETTERS 2017; 119:053901. [PMID: 28949761 DOI: 10.1103/physrevlett.119.053901] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2016] [Indexed: 06/07/2023]
Abstract
Subradiant excitations, originally predicted by Dicke, have posed a long-standing challenge in physics owing to their weak radiative coupling to environment. Here we engineer massive coherently driven classical subradiance in planar metamaterial arrays as a spatially extended eigenmode comprising over 1000 metamolecules. By comparing the near- and far-field response in large-scale numerical simulations with those in experimental observations we identify strong evidence for classically correlated multimetamolecule subradiant states that dominate the total excitation energy. We show that similar spatially extended many-body subradiance can also exist in plasmonic metamaterial arrays at optical frequencies.
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Affiliation(s)
- Stewart D Jenkins
- Mathematical Sciences and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - Janne Ruostekoski
- Mathematical Sciences and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - Nikitas Papasimakis
- Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - Salvatore Savo
- Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom
- TetraScience Inc., 114 Western Ave, Boston, Massachusetts 02134, USA
| | - Nikolay I Zheludev
- Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom
- Centre for Disruptive Photonic Technologies, School of Physical and Mathematical Sciences and The Photonics Institute, Nanyang Technological University, Singapore 637378, Singapore
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9
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Ruostekoski J, Javanainen J. Emergence of correlated optics in one-dimensional waveguides for classical and quantum atomic gases. PHYSICAL REVIEW LETTERS 2016; 117:143602. [PMID: 27740794 DOI: 10.1103/physrevlett.117.143602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2016] [Indexed: 06/06/2023]
Abstract
We analyze the emergence of correlated optical phenomena in the transmission of light through a waveguide that confines classical or ultracold quantum degenerate atomic ensembles. The conditions of the correlated collective response are identified in terms of atom density, thermal broadening, and photon losses by using stochastic Monte Carlo simulations and transfer matrix methods of transport theory. We also calculate the "cooperative Lamb shift" for the waveguide transmission resonance, and discuss line shifts that are specific to effectively one-dimensional waveguide systems.
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Affiliation(s)
- Janne Ruostekoski
- Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - Juha Javanainen
- Department of Physics, University of Connecticut, Storrs, Connecticut 06269-3046
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10
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Araújo MO, Krešić I, Kaiser R, Guerin W. Superradiance in a Large and Dilute Cloud of Cold Atoms in the Linear-Optics Regime. PHYSICAL REVIEW LETTERS 2016; 117:073002. [PMID: 27563957 DOI: 10.1103/physrevlett.117.073002] [Citation(s) in RCA: 37] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2016] [Indexed: 05/05/2023]
Abstract
Superradiance has been extensively studied in the 1970s and 1980s in the regime of superfluorescence, where a large number of atoms are initially excited. Cooperative scattering in the linear-optics regime, or "single-photon superradiance," has been investigated much more recently, and superradiant decay has also been predicted, even for a spherical sample of large extent and low density, where the distance between atoms is much larger than the wavelength. Here, we demonstrate this effect experimentally by directly measuring the decay rate of the off-axis fluorescence of a large and dilute cloud of cold rubidium atoms after the sudden switch off of a low-intensity laser driving the atomic transition. We show that, at large detuning, the decay rate increases with the on-resonance optical depth. In contrast to forward scattering, the superradiant decay of off-axis fluorescence is suppressed near resonance due to attenuation and multiple-scattering effects.
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Affiliation(s)
- Michelle O Araújo
- Université Côte d'Azur, CNRS, INLN, 06560 Valbonne, France
- CAPES Foundation, Ministry of Education of Brazil, Brasília, DF 70040-020, Brazil
| | - Ivor Krešić
- SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, United Kingdom
| | - Robin Kaiser
- Université Côte d'Azur, CNRS, INLN, 06560 Valbonne, France
| | - William Guerin
- Université Côte d'Azur, CNRS, INLN, 06560 Valbonne, France
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11
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Jennewein S, Besbes M, Schilder NJ, Jenkins SD, Sauvan C, Ruostekoski J, Greffet JJ, Sortais YRP, Browaeys A. Coherent Scattering of Near-Resonant Light by a Dense Microscopic Cold Atomic Cloud. PHYSICAL REVIEW LETTERS 2016; 116:233601. [PMID: 27341230 DOI: 10.1103/physrevlett.116.233601] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2015] [Indexed: 06/06/2023]
Abstract
We measure the coherent scattering of light by a cloud of laser-cooled atoms with a size comparable to the wavelength of light. By interfering a laser beam tuned near an atomic resonance with the field scattered by the atoms, we observe a resonance with a redshift, a broadening, and a saturation of the extinction for increasing atom numbers. We attribute these features to enhanced light-induced dipole-dipole interactions in a cold, dense atomic ensemble that result in a failure of standard predictions such as the "cooperative Lamb shift". The description of the atomic cloud by a mean-field model based on the Lorentz-Lorenz formula that ignores scattering events where light is scattered recurrently by the same atom and by a microscopic discrete dipole model that incorporates these effects lead to progressively closer agreement with the observations, despite remaining differences.
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Affiliation(s)
- S Jennewein
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, 91127 Palaiseau cedex, France
| | - M Besbes
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, 91127 Palaiseau cedex, France
| | - N J Schilder
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, 91127 Palaiseau cedex, France
| | - S D Jenkins
- Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - C Sauvan
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, 91127 Palaiseau cedex, France
| | - J Ruostekoski
- Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - J-J Greffet
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, 91127 Palaiseau cedex, France
| | - Y R P Sortais
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, 91127 Palaiseau cedex, France
| | - A Browaeys
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Université Paris-Saclay, 91127 Palaiseau cedex, France
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12
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Jenkins SD, Ruostekoski J, Javanainen J, Bourgain R, Jennewein S, Sortais YRP, Browaeys A. Optical Resonance Shifts in the Fluorescence of Thermal and Cold Atomic Gases. PHYSICAL REVIEW LETTERS 2016; 116:183601. [PMID: 27203321 DOI: 10.1103/physrevlett.116.183601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2016] [Indexed: 06/05/2023]
Abstract
We show that the resonance shifts in the fluorescence of a cold gas of rubidium atoms substantially differ from those of thermal atomic ensembles that obey the standard continuous medium electrodynamics. The analysis is based on large-scale microscopic numerical simulations and experimental measurements of the resonance shifts in a steady-state response in light propagation.
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Affiliation(s)
- S D Jenkins
- Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - J Ruostekoski
- Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - J Javanainen
- Department of Physics, University of Connecticut, Storrs, Connecticut 06269-3046, USA
| | - R Bourgain
- Laboratoire Charles Fabry, Institut d'Optique, CNRS, Univ Paris Sud, 2 Avenue Augustin Fresnel, 91127 Palaiseau cedex, France
| | - S Jennewein
- Laboratoire Charles Fabry, Institut d'Optique, CNRS, Univ Paris Sud, 2 Avenue Augustin Fresnel, 91127 Palaiseau cedex, France
| | - Y R P Sortais
- Laboratoire Charles Fabry, Institut d'Optique, CNRS, Univ Paris Sud, 2 Avenue Augustin Fresnel, 91127 Palaiseau cedex, France
| | - A Browaeys
- Laboratoire Charles Fabry, Institut d'Optique, CNRS, Univ Paris Sud, 2 Avenue Augustin Fresnel, 91127 Palaiseau cedex, France
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