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Physical Review A

Pump-probe spectroscopy approach to Bragg scattering

P. R. Berman, B. M. Bian

Published Jun 1, 1997• 13 citations
Publisher

Abstract

A theory of Bragg scattering of atoms is presented in which atoms are scattered by counterpropagating optical fields having different frequencies. It is shown that, for this geometry, the system reduces to that encountered in the pump-probe spectroscopy of two or multilevel systems. As a result, one can tune the Bragg resonances by varying the frequency difference between the optical fields. The possiblity of using these techniques to construct an atom-beam splitter, whose output arms are separated in momentum space by a hundred photon momenta, is discussed.

DOI
10.1103/physreva.55.4382

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arXiv:physics/0012002•Physical Review A•May 16, 2001•1 citation
A theory of Bragg scattering of atoms is presented in which atoms are scattered by counterpropagating optical fields having different frequencies. It is shown that, for this geometry, the system reduces to that encountered in the pump-probe spectroscopy of two or multilevel systems. As a result, one can tune the Bragg resonances by varying the frequency difference between the optical fields. The possiblity of using these techniques to construct an atom-beam splitter, whose output arms are separated in momentum space by a hundred photon momenta, is discussed.
Effective atomic beam splitters providing large scattering angles are central to atom optics and atom interferometry. Their most common optical realization relies on first-order Bragg scattering, which generates a splitting of the atomic wave function in momentum space. For typical atomic beam velocities of the order of 100 m/s, 2ħk produces very small scattering angles. Higher-order Bragg scattering is of questionable practicality, because of the need for atom–field interaction times that scale exponentially with the scattering order. Attempts at solving this problem include the introduction of multiple laser beams, the use of three-level transition schemes, and the application of velocity-tuned, or Doppleron, resonances. We present a novel scheme that uses a Doppleron resonance as a catalyst to speed up a high-order Bragg resonance by orders of magnitude. This effect occurs when the atom–field frequency detuning is such that the Bragg resonance and the Doppleron resonance become degenerate. Alternatively, in the band-theoretical description of the near-resonant Kapitza–Dirac effect, the Doppleron resonance is required to occur either at the center or at the boundary of the first Brillouin zone.
Atomic Kapitza-Dirac Bragg regime scattering is a multiphoton process in which a neutral atom undergoes a change of momentum through an interaction with a coherent light source. When the Bragg conditions are met, the outgoing atom beams are spatially quantized. Counterpropagating lasers act as pump and probe scattering from far-off-resonant excited intermediate electronic states, leaving the atoms in the electronic ground state with quantized transverse momentum. Each nontrivial scattering event imparts transverse velocity and therefore kinetic energy to the deflected atoms through recoil. In the Bragg regime, the loss of energy in the light fields is equal to the gain of kinetic energy in the atom. Energy nonconserving intermediate states, which are described by nonadiabatic contributions, are not accounted for by first-order off-resonant states. By comparing the solutions of increasing orders of off-resonant intermediate states, it becomes clear that calculating the correct Pendell\"osung frequencies and phases requires including an ever increasing number of off-resonant states in proportion to the square root of the field strength.
Sub-Doppler resonances observed in miniature cells (V≪1cm3V \ll 1 cm^3V≪1cm3) with Rb or Cs atoms in the field of counterpropagating bichromatic laser beams are of interest for the creation of a miniature quantum frequency standard (QFS) of the optical range. However, a theory of these resonances that takes into full account the real structure of energy levels in an atom has not yet been constructed, which would make it possible to obtain explicit analytical expressions for the absorption coefficient of the medium. This paper addresses this gap using the density matrix formalism and the algebra of irreducible tensor operators. Namely, we obtained an analytical expression for the absorption coefficient of the probe beam in the presence of a more intense counterpropagating beam (pump-probe regime). Moreover, the calculations were carried out for arbitrary parameters of laser beam polarization, arbitrary angular momenta of optical dipole transitions in the atom and various relaxation processes. The constructed theory allows us to correctly take into account the simultaneous action of various nonlinear optical effects, in contrast to the simplified three-level model of the atom, and also to investigate the partial contribution of each of them to the overall observed resonant circuit. The analysis performed, in particular, made it possible to predict an increase in the amplitude of the sub-Doppler resonance when using counterpropagating light beams with opposite circular polarizations in relation to the configuration of counterpropagating beams with orthogonal linear polarizations, which is often used in practice.
A novel phase-modulation technique which permits subkilohertz-laser stability and new levels of precision in laser spectroscopy was reported recently. For spectroscopy, the basic arrangement consists of a combination of an optical pump and a probe field which is phase modulated. The pump prepares the atomic sample by burning a narrow hole within the atom's inhomogeneous line shape, and the probe beam samples the prepared hole when its modulation sidebands are swept into resonance. Off resonance, the probe is balanced as pairs of sidebands produce heterodyne beat signals of opposite phase which just cancel. On resonance, the balance is upset and yields a nonvanishing beat signal with a Lorentzian absorption or dispersion line shape and with residual noise approaching the shot noise limit. Here we investigate the theory of phase-modulation spectroscopy. We treat the nonlinear response of an atomic two-level quantum system subject to an intense pump and a weak copropagating or counterpropagating phase-modulated probe beam. The density-matrix equations of motion are solved by a Laplace-transform method and by the novel use of a translation operator which allows the infinite hierarchy of coupled equations to close. A solution equivalent to the rate-equation result is developed and coherence corrections are found which predict new resonances that have just been detected in this laboratory. The delayed pump-probe technique encountered in solid-state laser spectroscopy is analyzed in this context for two- and three-level quantum systems. The response of a Fabry-Perot cavity to a phase-modulated light wave is examined also and reveals an unexpected absorption feature.
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.