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Physical Review A

Bichromatic atomic lens

M. K. Olsen, Thomas G. Wong, S. M. Tan, Daniel F. Walls

Published May 1, 1996• 13 citations
Publisher

Abstract

We investigate the focusing of three-level atoms with a bichromatic standing wave laser field, using both classical and quantum treatments of the problem. We find that, for the appropriate ratio of detunings to Rabi frequencies, the atoms will experience a periodic potential which is close to harmonic across half an optical wavelength. The field thus becomes equivalent to a periodic array of microlenses, which could be utilized to deposit lines of atoms upon a substrate. We consider and compare two regimes, differentiated by the interaction time of the atoms in the optical field. The first case considered, the Raman-Nath regime, is analogous to the thin lens regime in classical optics. The second case treats the transverse atomic motion within the light field, and investigates the distribution of atoms upon a substrate placed within the field. We investigate the extent to which this case can be modeled classically. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.3358

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The focusing of atoms after interacting with both far-detuned and resonant standing wave fields in the thin-lens regime is considered. Exact quantum expressions for the Fourier components of the density (that include all spherical aberration) are used to study the focusing numerically. The following lens parameters and density profiles are calculated as functions of the pulsed field area $\ensuremath{\theta}:$ the position of the focal plane, peak atomic density, atomic density pattern at the focus, focal spot size, depth of focus, and background density. The lens parameters are compared to asymptotic, analytical results derived from a scalar diffraction theory for which spherical aberration is small but non-negligible $(\ensuremath{\theta}\ensuremath{\gg}1).$ Within the diffraction theory analytical expressions show that the focused atoms in the far-detuned case have an approximately constant background density $0.5(1\ensuremath{-}0.635{\ensuremath{\theta}}^{\ensuremath{-}1/2})$ while the peak density behaves as $3.83{\ensuremath{\theta}}^{1/2},$ the focal distance as ${\ensuremath{\theta}}^{\ensuremath{-}1}(1+1.27{\ensuremath{\theta}}^{\ensuremath{-}1/2})$ ${L}_{T}/2\ensuremath{\pi},$ the depth of focus as ${0.304L}_{T}{\ensuremath{\theta}}^{\ensuremath{-}3/2},$ and the focal spot size $0.0592\ensuremath{\lambda}{\ensuremath{\theta}}^{\ensuremath{-}3/4},$ where ${L}_{T}$ is the Talbot distance and \ensuremath{\lambda} is the wavelength of the light. Focusing by the resonant standing wave field leads to a new effect, a Rabi-like oscillation of the atom density. For the far-detuned lens, chromatic aberration is studied quantitatively with the exact Fourier results. Similarly, the degradation of the focus that results from angular divergence in beams or thermal velocity distributions in traps is studied quantitatively with the exact Fourier method and understood analytically using the asymptotic results. Overall, we show that strong thin-lens focusing is possible with modest laser powers and with currently achievable atomic beam characteristics.

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arXiv:physics/9909044•Physical Review A•Feb 15, 2000•6 citations
We investigate the focusing of three-level atoms with a bichromatic standing wave laser field, using both classical and quantum treatments of the problem. We find that, for the appropriate ratio of detunings to Rabi frequencies, the atoms will experience a periodic potential which is close to harmonic across half an optical wavelength. The field thus becomes equivalent to a periodic array of microlenses, which could be utilized to deposit lines of atoms upon a substrate. We consider and compare two regimes, differentiated by the interaction time of the atoms in the optical field. The first case considered, the Raman-Nath regime, is analogous to the thin lens regime in classical optics. The second case treats the transverse atomic motion within the light field, and investigates the distribution of atoms upon a substrate placed within the field. We investigate the extent to which this case can be modeled classically. \textcopyright{} 1996 The American Physical Society.
The focusing of atoms after interacting with both far-detuned and resonant standing wave fields in the thin-lens regime is considered. Exact quantum expressions for the Fourier components of the density (that include all spherical aberration) are used to study the focusing numerically. The following lens parameters and density profiles are calculated as functions of the pulsed field area $\ensuremath{\theta}:$ the position of the focal plane, peak atomic density, atomic density pattern at the focus, focal spot size, depth of focus, and background density. The lens parameters are compared to asymptotic, analytical results derived from a scalar diffraction theory for which spherical aberration is small but non-negligible $(\ensuremath{\theta}\ensuremath{\gg}1).$ Within the diffraction theory analytical expressions show that the focused atoms in the far-detuned case have an approximately constant background density $0.5(1\ensuremath{-}0.635{\ensuremath{\theta}}^{\ensuremath{-}1/2})$ while the peak density behaves as $3.83{\ensuremath{\theta}}^{1/2},$ the focal distance as ${\ensuremath{\theta}}^{\ensuremath{-}1}(1+1.27{\ensuremath{\theta}}^{\ensuremath{-}1/2})$ ${L}_{T}/2\ensuremath{\pi},$ the depth of focus as ${0.304L}_{T}{\ensuremath{\theta}}^{\ensuremath{-}3/2},$ and the focal spot size $0.0592\ensuremath{\lambda}{\ensuremath{\theta}}^{\ensuremath{-}3/4},$ where LT{L}_{T}LT​ is the Talbot distance and \ensuremath{\lambda} is the wavelength of the light. Focusing by the resonant standing wave field leads to a new effect, a Rabi-like oscillation of the atom density. For the far-detuned lens, chromatic aberration is studied quantitatively with the exact Fourier results. Similarly, the degradation of the focus that results from angular divergence in beams or thermal velocity distributions in traps is studied quantitatively with the exact Fourier method and understood analytically using the asymptotic results. Overall, we show that strong thin-lens focusing is possible with modest laser powers and with currently achievable atomic beam characteristics.
We study the optimal focusing of two-level atoms with a near-resonant standing wave light, using both classical and quantum treatments of the problem in the thin- and thick-lens regimes. It is found that the near-resonant standing wave focuses the atoms with a reduced background in comparison with far-detuned light fields. For some parameters, the quantum atomic distribution shows even better localization than the classical one. Spontaneous emission effects are included via the technique of quantum Monte Carlo wave function simulations. We investigate the extent to which nonadiabatic and spontaneous emission effects limit the achievable minimal size of the deposited structures.
This dissertation explores the spatial patterns that evolve dynamically in atomic clouds and beams after they interact with one or more standing wave fields, typically formed by laser beams. The main systems under study are ensembles of free atoms, but a model Bose-Einstein condensate and the harmonic oscillator are considered as well. Focused atomic patterns with features much smaller than the wave length of the light fields are possible owing to the nonlinear atom-field interaction and are emphasized. The quantitative results combine numerical simulations and analytical expressions. Computer algorithms were developed from known methods in partial differential equations to perform the numerical simulations. The analytical results relied on standard and newly contrived techniques in quantum physics. The analytical expressions were evaluated by computer in order to extract valuable quantitative information and to compare different analytical methods with one another and with the numerical simulations. The thesis is written as a self-contained piece, and as such, develops specific examples of atom-field configurations and atomic patterns of interest from a general theory of the atoms interacting with a spatially varying light field that influences the subsequent center-of-mass motion. Beyond those already published by our group and expanded upon in the thesis, significant advances are made in: the link between transient interactions in the time- and spatial-domains; an understanding of the Raman-Nath approximation, its range of validity, and its corrections; atomic focusing by pulsed adiabatic interactions between an atom and a single standing wave light field; the differences between classical and quantum free motion after periodic interactions; the distinctions between free motion after amplitude and phase gratings; harmonic oscillator ground states kicked by spatially varying pulses; and quasiperiodic echo effects. The general theoretical structure that was detailed can be applied effectively, as intended from the outset, to other problems in atom optics.
We investigate schemes for the clean splitting of beams of three-level atoms using two standing-wave laser fields within an optical cavity. The proposed beam splitter is shown to work for atoms in the \ensuremath{\Lambda} ladder, and ssV configurations. For appropriate values of Rabi frequencies and detunings, we obtain a triangular type of potential for the atomic states of interest. As well as modeling the coherent evolution of the systems, we have used quantum Monte Carlo wave-function methods to model the effects of spontaneous emission on the resulting diffraction pattern, finding significant differences between the three configurations. We also investigate the limits of the Raman-Nath approximation for our systems, using the symmetric split-operator technique to include the effects of the kinetic term in the Hamiltonian. We also present the results of calculations in which the split output beams are recombined, demonstrating the expected interference for differently prepared input beams. In comparison with two-level beam splitters using a single standing wave, we obtain a superior splitting, while, in comparison with magneto-optical beam splitters, our system possesses the worthwhile practical advantages of experimental simplicity.
This dissertation covers the field of atom optics and is divided into four main chapters: In Chapter 2 we investigate the effects of light forces on the center-of-mass motion of two-level atoms. This will lead to the discussion of two regimes: the "ray optic" and the "wave optic" regime. In the first case, an atom is well localized in the field which allows a comparison to be made with classical ray optics. In the second case, the atom is strongly delocalized which leads to a wave treatment and allows a comparison with diffractive optics. We finish this chapter with an example in each regime: Doppler cooling for ray optics and an atomic Fabry-Perot for wave optics. In Chapter 3 we extend the results of the previous chapter to the diffraction of atoms by a standing light field. We cover three regimes in the near resonant Kapitza-Dirac effect: the Raman-Nath, the Bragg and the optical Stern-Gerlach regime. In the Raman-Nath and Bragg regimes, the wave-packet is strongly delocalized compared to the period of the standing wave. In contrast, the Stern-Gerlach regime has a small spatial extent. The Raman-Nath and Bragg regimes are differentiated in their treatment of the kinetic energy. Initially we only discuss coherent interactions. In the later half of this chapter we introduce spontaneous emission and show how its presence affects the diffraction pattern in each of these regimes. In Chapter 4 we cover various atomic cooling schemes: strong field Sisyphus cooling, adiabatic cooling, evaporative cooling, polarization gradient cooling and velocity selective coherent population trapping. We begin with a brief discussion of atomic temperature. We then cover two cooling schemes for two-level atoms. We eventually move to multi-level atoms and end this chapter with a two-atom multi-level system. In Chapter 5 we conclude with a brief discussion of practical uses and devices that may arise from atom optics such as lenses, mirrors, gravitational interferometry, lithography and atomic clocks.
We propose a scheme to create high-contrast, periodic atom density distributions having period $\ensuremath{\lambda}/2n$ using the Talbot effect, where $\ensuremath{\lambda}$ is the wavelength of the optical fields that scatter the atoms and n is a positive integer. This filtered Talbot lens is comprised of two standing-wave optical fields. An atomic beam propagates perpendicular to the fields. The first field, which is far detuned from the atomic transition frequency, acts as an array of lenses that focuses the atoms. The second field, positioned at the atom optical focus of the first, is resonant with the atomic transition frequency and acts as an amplitude mask, leaving unperturbed only those atoms that pass through its nodes. At distances following the interaction with the second field that are equal to an integral fraction of the Talbot length, atomic density gratings having period $\ensuremath{\lambda}/2n$ are formed.