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Physical Review A

Light scattering by an ultracold trapped atom

Mariusz Gajda, Piotr Krekora, Jan Mostowski

Published Jul 1, 1996• 3 citations
Publisher

Abstract

We study the scattering of light by a single ultracold trapped atom initially in the ground state of the trapping harmonic potential. We have found interesting features of scattering in the regime where the atomic recoil energy is much larger than the separation between oscillatory trap levels. Although we present the quantum-mechanical expression for the scattering cross section, special attention is paid to the semiclassical analysis of the process. We show that the major characteristics of the scattering might be deduced from two conservation laws: conservation of energy and momentum in absorption and emission of photon process separately. These conservation laws impose a strong correlation between scattered light characteristics and the position of the trapped atom at the moment of photon emission. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.928

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We study the scattering of light by a single ultracold trapped atom initially in the ground state of the trapping harmonic potential. We have found interesting features of scattering in the regime where the atomic recoil energy is much larger than the separation between oscillatory trap levels. Although we present the quantum-mechanical expression for the scattering cross section, special attention is paid to the semiclassical analysis of the process. We show that the major characteristics of the scattering might be deduced from two conservation laws: conservation of energy and momentum in absorption and emission of photon process separately. These conservation laws impose a strong correlation between scattered light characteristics and the position of the trapped atom at the moment of photon emission. \textcopyright{} 1996 The American Physical Society.
The recoil momentum and energy of ultracold atoms is studied for situations where collective scattering or emission of photons is important. Effects due to the dipole-dipole interaction are emphasized. For two and many atoms confined in optical traps, calculations of the energy and momentum with respect to atomic separation are performed. When the atomic separation is comparable to or larger than the transition wavelength, the ultracold atoms' recoil energy depends mainly on collective spontaneous decay rather than forces from the cooperative frequency shift. Calculations for a laser pulse reflecting from an atomic array suggest that the recoil energy of atoms can be substantially larger than might be expected from an independent atom picture.
Both far off-resonance and resonant scattering of light from single atoms trapped by 3D harmonic potentials has thoroughly been studied. Novel effects are predicted for different physical regimes. We have shown that dynamics of the atomic center-of-mass strongly influences the scattering cross section. Possibility of using spectrum of the scattered light in far-off-resonance regime to nondestructively measure the temperature of ultracold atoms is advocated: off-resonance scattering can be used as an ‘optical thermometer’. The realistic Compton-like regime in resonant scattering has been investigated in detail. Another interesting quantum effect in resonant regime, which has not been discussed here due to the lack of space, is the time resolved scattering, showing up when the atom can remain in the excited state long enough to make many trips back and forth in the trap before emitting a photon. The possibility of the experimental observation of the predicted effects is now being scrutinized.
We investigate theoretically light scattering of photons by ultracold atoms in an optical lattice in the linear regime. A full quantum theory for the atom-photon interactions is developed as a function of the atomic state in the lattice along the Mott-insulator--superfluid phase transition, and the photonic-scattering cross section is evaluated as a function of the energy and of the direction of emission. The predictions of this theory are compared with the theoretical results of a recent work on Bragg scattering in time-of-flight measurements [A.M. Rey et al., Phys. Rev. A 72, 023407 (2005)]. We show that, when performing Bragg spectroscopy with light scattering, the photon recoil gives rise to an additional atomic site-to-site hopping, which can interfere with ordinary tunneling of matter waves and can significantly affect the photonic-scattering cross section.
In the present thesis we study several schemes where the detection of the emitted light from an atomic ensemble gives information about the state of the atoms. In chapter 1 we review the basic theory for the light matter interaction in second quantized form. In chapter 2 we show how conservative potentials for the atoms, such as dispersive optical lattices, can be realized experimentally due to the dispersive interaction with a far detuned laser We then review some basic properties of the physics of a particle in a periodic potential. We show that the ground state and the lowest lying excitations of a Bose-Einstein condensate in a dispersive optical lattice can be described by the Bose-Hubbard Hamiltonian. In chapter 3 we study the photonic bandstructure of a chain of fixed pointlike atoms in a biperiodic configuration. We calculate the photonic spectra and the probe transmission if the atoms are in free space and inside a standing wave optical resonator. In chapter 4 we consider light scattering from ultracold atoms in an optical lattice where we calculate the photonic scattering cross section as a function of energy and direction of emission along the Mott-insulator-superfluid phase transition. We take into account the finite tunneling rate for the atoms, when evaluating the scattering cross section of the photons and the effect of photon recoil onto the atoms. In chapter 5 we consider a gas of ultracold atoms trapped at two spatially separated regions in space. Using a two Raman lasers to couple atoms out of the systems we show that it is possible to measure the mean value of the atomic field operator of the two systems by measuring the scattered light intensity of one of the Raman lasers. We show how this setup might be used to measure the temperature of a Bose-Einstein condensate and to monitor the superfluid to Mott-insulator phase transition for a gas of ultracold atoms in an optical lattice.
Ultracold atoms confined in a dipole trap are submitted to a potential whose depth is proportional to the real part of their dynamic dipole polarizability. The atoms also experience photon scattering whose rate is proportional to the imaginary part of their dynamic dipole polarizability. In this article we calculate the complex dynamic dipole polarizability of ground-state erbium, a rare-earth atom that was recently Bose condensed. The polarizability is calculated with the sum-over-state formula inherent to second-order perturbation theory. The summation is performed on transition energies and transition dipole moments from ground-state erbium, which are computed using the Racah-Slater least-squares fitting procedure provided by the cowan codes. This allows us to predict nine unobserved odd-parity energy levels of total angular momentum J=5J=5J=5, 6, and 7, in the range 25 000--31 000 cm${}^{\ensuremath{-}1}$ above the ground state. Regarding the trapping potential, we find that ground-state erbium essentially behaves like a spherically symmetric atom, in spite of its large electronic angular momentum. We also find a mostly isotropic van der Waals interaction between two ground-state erbium atoms, characterized by a coefficient C6iso=1760{C}_{6}^{\mathrm{iso}}=1760C6iso​=1760 a.u. To the contrary, the photon-scattering rate shows a pronounced anisotropy since it strongly depends on the polarization of the trapping light.
In ultracold atoms settings, inelastic light scattering is a preeminent technique to reveal static and dynamic properties at nonzero momentum. In this work, we investigate an array of one-dimensional trapped Bose gases, by measuring both the energy and the momentum imparted to the system via light scattering experiments. The measurements are performed in the weak perturbation regime, where these two quantities—the energy and momentum transferred—are expected to be related to the dynamic structure factor of the system. We discuss this relation, with special attention to the role of in-trap dynamics on the transferred momentum.