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Physical Review A

Spectroscopic measurement of an atomic wave function

Kishore T. Kapale, Shahid Qamar, Muhammad Suhail Zubairy

Published Feb 10, 2003• 91 citations
Publisher

Abstract

We present a simple spectroscopic method based on Autler-Townes spectroscopy to determine the center-of-mass atomic wave function. The detection of spontaneously emitted photons from a three-level atom, in which two upper levels are driven by a classical standing light, yields information about the position and momentum distribution of the atom [A. M. Herkommer, W. P. Schleich, and M. S. Zubairy, J. Mod. Opt. $44,$ 2507 (1997)]. In this paper, we show that both the amplitude and phase information of the center-of-mass atomic wave function can be obtained from these distributions after a series of conditional measurements on the atom and the emitted photon.

DOI
10.1103/physreva.67.023805

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We present a simple spectroscopic method based on Autler-Townes spectroscopy to determine the center-of-mass atomic wave function. The detection of spontaneously emitted photons from a three-level atom, in which two upper levels are driven by a classical standing light, yields information about the position and momentum distribution of the atom [A. M. Herkommer, W. P. Schleich, and M. S. Zubairy, J. Mod. Opt. 44,44,44, 2507 (1997)]. In this paper, we show that both the amplitude and phase information of the center-of-mass atomic wave function can be obtained from these distributions after a series of conditional measurements on the atom and the emitted photon.
Entanglement of the atom and field leads towards the measurement of the state of the atom or field depending on the kind of measurement made. Taking advantage of this fact, we propose spectroscopic methods for the single atom localization within the subwavelength domain of the optical standing wave field. \n \nWhen a two-level atom interacts with some intense radiation field the spectrum of the scattered light signal exhibits a three-peak Mollow spectrum. We propose a scheme of atom localization based on the resonance fluorescence from a two-level atom interacting with a standing wave field. The Rabi frequency is considered to be position dependent and therefore, the spontaneously emitted photon carries the information about the center-of-mass motion of the atom. This leads to the localization of the single atom by measuring the frequency of the spontaneously emitted photon even during the passage of the atom through the standing wave field. We predict a spatial resolution in position information of up to λ/60 where λ is the optical wavelength. \n \nDetuning of the coherent field with the atomic transition plays an important role in the characterization of the peaks present in the Autler- Townes spectrum of a three-level atom. Strong line narrowing is observed in one of the peaks of the Autler- Townes doublet depending on the choice of the detuning. We investigate how this detuning effects the position information of the atom inside the standing wave field in upper-level and lower-level coupling schemes of Autler- Townes microscopy. We show an enhancement in precision position measurement by factor of at least 4 in upper-level coupling scheme due to the quantum interference phenomenon. This enhancement is not observed in lower-level coupling scheme where the quantum interference is not taking place. \n \nWe also consider a system where a three-level atom is interacting with a monochromatic radiation field in such a way that the decay takes place from upper level to the intermediate level with a rate T and from intermediate level to lower level at a rate γ. This type of system exhibits a non stationary scattered light signal. We report the results of the spontaneous emission spectrum for non stationary scattered light signal by using the definition of time-dependent physical spectrum. This is a rare example of problems where time-dependent physical spectrum can be calculated exactly. We study the characteristics of the spontaneous spectrum for a resonant and off-resonant case and show that this physical spectrum recovers the results of earlier work quite successfully. We also show that the decay rate r disturbs the atomic coherence and finally eliminates the dark line from the center of the spontaneous spectrum. Here we only consider upper-level coupling case for the time-dependent spontaneous spectrum because in lower-level coupling case the system automatically evolved a pumping mechanism due to the coupling of ground level to the upper level via radiation field. Thus a steady-state limit exists in lower-level coupling scheme. \n \nWe investigate the phenomenon of atom localization by using the non stationary Autler-Townes microscopy. The decay rate T reduces the precision in position measurement which is consistent with the results of the non stationary spontaneous emission spectrum. This is due to the fact that for the upper-level coupling case, the standing wave field is coupled to atomic transition and it induces a coherence in the upper-level atomic transition but the spontaneous decay from the upper level to the intermediate level disturbs this coherence. When the effect of the spontaneous decay dominates the coherence effect the quantum interference phenomenon vanishes and thus the precision in localization reduces.
We propose a three-level atomic system where quantum coherence is generated by a classical standing-wave field coupled to the two upper excited decaying levels. Quantum coherence results in cancellation of the spontaneously emitted spectral lines depending on the choice of the phase of the standing wave. We exploit this phenomenon for precision measurement of the atomic position in the standing wave. Measurement of the conditional position probability distribution shows one to eight peaks per unit wavelength of the standing wave. Only one controllable parameter, that is, the phase of the driving standing wave, is enough to control these atomic positions. Along with the other results, the result of obtaining a single peak is remarkable as it enhances the efficiency of our system by a factor of 8.
We present a scheme to reconstruct a two-mode entangled state in a high-QQQ cavity from the spontaneous emission spectrum in a driven four-level atomic system. The two modes of the cavity field drive the upper three levels of the atom and thus the spontaneous emission spectrum contains the information about the photon statistics of the cavity field. Wigner function of the driving field is recovered from the photon statistics in a straightforward manner.