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arXiv:physics/0012002 · Physical Review A

Recoil-induced resonances in nonlinear, ground-state, pump-probe spectroscopy

Christopher P. Search, P. R. Berman

Published May 16, 2001• 1 citation
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Abstract

A theory of pump-probe spectroscopy is developed in which optical fields drive two-photon Raman transitions between ground states of an ensemble of three-level atoms. Effects related to the recoil the atoms undergo as a result of their interactions with the fields are fully accounted for in this theory. The linear absorption coefficient of a weak probe field in the presence of two pump fields of arbitrary strength is calculated. For subrecoil cooled atoms, the spectrum consists of eight absorption lines and eight emission lines. In the limit that ${\ensuremath{\chi}}_{1}\ensuremath{\ll}{\ensuremath{\chi}}_{2},$ where ${\ensuremath{\chi}}_{1}$ and ${\ensuremath{\chi}}_{2}$ are the Rabi frequencies of the two pump fields, one recovers the absorption spectrum for a probe field interacting with an effective two-level atom in the presence of a single pump field. However, when ${\ensuremath{\chi}}_{1}\ensuremath{\gtrsim}{\ensuremath{\chi}}_{2},$ new interference effects arise that allow one to selectively turn on and off some of these recoil-induced resonances.

DOI
10.1103/physreva.63.063411

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A theory of pump-probe spectroscopy is developed in which optical fields drive two-photon Raman transitions between ground states of an ensemble of three-level atoms. Effects related to the recoil the atoms undergo as a result of their interactions with the fields are fully accounted for in this theory. The linear absorption coefficient of a weak probe field in the presence of two pump fields of arbitrary strength is calculated. For subrecoil cooled atoms, the spectrum consists of eight absorption lines and eight emission lines. In the limit that \ensuremathχ1\ensuremath≪\ensuremathχ2,{\ensuremath{\chi}}_{1}\ensuremath{\ll}{\ensuremath{\chi}}_{2},\ensuremathχ1​\ensuremath≪\ensuremathχ2​, where \ensuremathχ1{\ensuremath{\chi}}_{1}\ensuremathχ1​ and \ensuremathχ2{\ensuremath{\chi}}_{2}\ensuremathχ2​ are the Rabi frequencies of the two pump fields, one recovers the absorption spectrum for a probe field interacting with an effective two-level atom in the presence of a single pump field. However, when \ensuremathχ1\ensuremath≳\ensuremathχ2,{\ensuremath{\chi}}_{1}\ensuremath{\gtrsim}{\ensuremath{\chi}}_{2},\ensuremathχ1​\ensuremath≳\ensuremathχ2​, new interference effects arise that allow one to selectively turn on and off some of these recoil-induced resonances.
When a pump field and probe field simultaneously drive an electronic-state atomic transition, the probe field absorption spectrum can consist of an absorption peak centered near \ensuremath{\Delta}'=0 and amplification peak centered near \ensuremath{\Delta}'=2\ensuremath{\Delta} (\ensuremath{\Delta} and \ensuremath{\Delta}' are the pump and probe field detunings from the atomic transition frequency, respectively). This type of spectrum is seen in the limit \ensuremath{\Vert}\ensuremath{\Delta}\ensuremath{\Vert}\ensuremath{\gg}\ensuremath{\chi}\ensuremath{\gg}\ensuremath{\Gamma}, where \ensuremath{\chi} is the Rabi frequency associated with the pump field and 2\ensuremath{\Gamma} is the homogeneous width associated with the atomic transition. For atoms cooled below the recoil limit of laser cooling, the qualitative nature of the probe absorption spectrum can undergo a dramatic change. Provided that the recoil splitting ${\mathrm{\ensuremath{\omega}}}_{\mathit{k}}$ is larger than the homogeneous decay rate (as might occur in the case of a forbidden transition), the absorption and amplification features each split into an absorption-amplification doublet. In addition, structure is found in the probe absorption spectrum near \ensuremath{\Delta}'=\ensuremath{\Delta}; this structure consists of two absorption-amplification doublets. Both doublets can be resolved if ${\mathrm{\ensuremath{\omega}}}_{\mathit{k}}$>\ensuremath{\Gamma}. If ${\mathrm{\ensuremath{\omega}}}_{\mathit{k}}$, one of the doublets can be resolved provided that ${\mathrm{\ensuremath{\omega}}}_{\mathit{k}}$>${\mathrm{\ensuremath{\Gamma}}}_{\mathit{A}}$, where ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{A}}$ is some effective atomic ground state width in the problem. The positions, widths, and relative weights of all the components are readily predicted using a dressed-atom theory in which quantization of the center-of-mass momentum is included. An analytical expression for the probe field spectrum is obtained for a simple case in which spontaneous decay to the lower level is neglected. Validity criteria for the results are discussed.
A theory of pump-probe spectroscopy is developed in which optical fields drive two-quantum, Raman-like transitions between ground-state sublevels. Three fields are incident on an ensemble of atoms. Two of the fields act as the pump field for the two-quantum transitions. The absorption or gain of an additional probe field is monitored as a function of its detuning from one of the fields which constitutes the pump field. Although the probe absorption spectrum displays features common to those found in pump-probe spectroscopy of single-quantum transitions, new interference effects are found to modify the spectrum. Many of these features can be explained within the context of a dressed-atom picture.
Theoretical calculations are presented for the probe absorption spectra of atoms having magnetically degenerate ground states which are subjected to weak pump fields that produce spatial polarization gradients. Apart from the Raman resonance contribution to the spectrum which has been observed experimentally, we predict that additional resonant structures should appear as a result of the atomic recoil that occurs during the absorption or emission of radiation by the atoms. The width of the atomic recoil resonances is directly related to the Doppler width associated with the driven transition. For sub-Doppler cooled atoms, the width can be much narrower than the optical pumping rate that determines the width of the Raman signal.
We present our experimental investigations of nonlinear ground-state pump-probe spectroscopy in ultracold rubidium 85 produced in a magneto-optical trap (MOT). Nonlinear ground-state pump-probe spectroscopy is a two-photon analog of traditional two-level pump-probe spectroscopy. When a two-level atom is driven with an intense, single frequency near-resonant laser field the absorption spectrum of a weak probe field exhibits absorption and gain features not seen without the pump field. These saturated absorption line shapes, obtained under different pump field conditions, are known as the Mollow spectra. Nonlinear ground-state pump-probe spectroscopy is a three-level process, in which an intense two-photon pump field drives transitions between two components of the ground state coupled through an intermediate excited state. When the frequencies of the pump fields are tuned far from the intermediate state resonance, spontaneous emission is significantly reduced and narrow spectral lines are possible. The absorption/gain spectrum of a weak probe field, tuned in the vicinity of the intermediate state virtual level, resembles the Mollow spectrum except that new interference processes arise due to competition between two-photon pathways involving the probe field and each of the two pump fields. This interference significantly affects the probe absorption spectrum, and new features appear that are not present in the two-level case. We describe our measurements of this process, including the MOT that produces the ultracold collection of atoms, and the system of injection-locked extended cavity diode lasers that produce the phase-coherent laser fields used in our measurements.
It is shown that, as a result of the atomic recoil associated with the absorption and emission of radiation, resonances can appear in nonlinear spectroscopy. The theory is illustrated by calculating the line shapes associated with nearly degenerate four-wave mixing and pump-probe spectroscopy when laser fields interact with an ensemble of two-level atoms. The recoil-induced resonances should be observable for atoms cooled below the Doppler limit of laser cooling, and the line shapes may provide a means for measuring the velocity distribution of these cooled atoms.
A theory of saturation spectroscopy is developed which accounts for effects of level degeneracy. The theory can be applied to those situations in which a pump laser field of arbitrary strength and polarization acts between two manifolds of degenerate or nearly degenerate atomic levels, while a probe laser field of arbitrary strength and polarization acts between a coupled manifold of levels. As such, this theory of "three-level" systems could, in principle, include effects arising from level degeneracy, optical pumping, atomic motion, transient laser fields, different polarizations and strengths for pump and probe fields, and atomic transit across the laser fields. The present calculation, however, is limited to a study of optical pumping and field polarization effects for pump fields of arbitrary strength and weak probe fields interacting with a single velocity subclass of atoms. Probe absorption line shapes clearly display the effects of pump-field strength (Rabi splittings), pump-field polarization (positions and number of Rabi-split resonance peaks), pump-field detuning (positions of resonance peaks), and optical pumping (relative strengths of the resonance peaks). The calculation is carried out with the use of an irreducible tensor basis for the atomic density matrix although a standard (mmm-basis) calculation is included in an appendix.