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Atom localization in cascade type system

A three-level cascade type system is subjected to a standing wave (SW) field acting between the ground energy level and the intermediate energy level of the system and the probe field scans the uppermost energy level from the intermediate energy level. Optical Bloch equations (OBE) for this three-level system are derived from the Liouville equation (Master equation) where the decay terms are added phenomenologically. These OBEs are solved analytically under steady state condition by using weak probe approximation. Under doppler free condition precession of localization was controlled by tuning the SW rabi frequency and relative orientation of the applied fields.

acting between the ground energy level and the intermediate energy level of the system and the probe field scans the uppermost energy level from the intermediate energy level. Optical Bloch equations (OBE) for this three-level system are derived from the Liouville equation (Master equation) where the decay terms are added phenomenologically. These OBEs are solved analytically under steady state condition by using weak probe approximation. Under doppler free condition precession of localization was controlled by tuning the SW rabi frequency and relative orientation of the applied fields.

Introduction:
Atom localization [1] is a process in which atoms get confined within a very narrow spatial region. Precession measurement of a single atom has potential application in nanolithography [2], Bose Einstein Condensation [3] and laser cooling [4]. The strong localization of atoms in cold atomic system also modifies the optical properties of the medium and can be used in fabricating optical logic gates, storage of light etc. There are several reports on different techniques to localize atoms within a narrow spatial region. Thomas and his co-workers demonstrated that sub optical wavelength localization could be achieved via a light-shift gradient for atom imaging [5].Later atom localization was achieved by atoms interacting with a standing wave and this was confirmed by using the phase shift measurement of the optical field [6], homodyne detection [7] and quantum trajectories [8]. Later phase shift of atomic dipole-moment [9] and entanglement between the atomic position and its internal state were used to localize the atom without directly affecting the spatial wave function of the particle [10]. Detection of spontaneously emitted photon due to its interaction with a classical standing wave field and the reservoir modes [11] has also been suggested by several groups but it is not easy to control spontaneous emission experimentally. To overcome this difficulty measurement of upper level population [12], probe absorption [13] and coherent population trapping [14] were used for atom localization study. All these mentioned phenomena [12][13][14] have experimental realization in pump-probe experiment. B.K. Dutta et al discussed the electromagnetically induced grating [15] phenomenon by using a three level Ξ type system interacting with one dimensional (1D) standing wave field. Ivanov and Rozhdestvensky have proposed a two-dimensional (2D) atom localization scheme using a four-level tripod system via measurement of the population in the upper state or in any ground state [16]. Atom localization via spontaneous emission in a five-level M-type atomic system  [17] and probe absorption in a microwave-driven [18] four-level atomic system were also studied. Knight et al [12] observed 1D subwavelength localization of a moving atom in a three-level Ʌ type system by measuring the upper level population and recently Rahamatullah et al extended this study [19] for 2D atom localization by probe absorption measurement. In this article a three level Ξ type atomic system interacting with 1D strong standing wave and a weak travelling field has been studied for measurement of atom localization using population of different levels as well as probe absorption. This study is done in Doppler free environment and the parameters of 87 Rb 5S 1/2 → 5P 3/2 → 5D 3/2 transitions are used in simulation. The position and precession of atom localization are controlled by varying Rabi frequency of the standing wave field and relative orientation of the applied fields.

Theory:
In the theoretical model we have considered a three-level Ξ type system. The strong standing wave field (control or pump field) acts between the ground level |1> and the intermediate level |2> whereas level |2> and level |3> are coupled by a weak probe field. The model is shown in figure 1. The standing wave is formed along the x-direction due to counter-propagating components of pump fields. The probe field which is assumed to be spatially uniform along the x-direction can pass through the standing wave regime along the orthogonal z-direction. The relative orientation (not shown in figure 1.) of propagation vectors of the probe field ‫ܭ(‬ ሬ ሬ⃗ ) and SW field ‫ܭ(‬ ሬ ሬ⃗ ) is denoted by the angle ϕ. The components of ‫ܭ‬ ሬ ሬ⃗ along z-axis and x-axis are ‫ܭ‬ ௭ and ‫ܭ‬ ௫ respectively and ‫ܭ‬ is the magnitude of ‫ܭ‬ ሬ ሬ⃗ . The propagation vectors of the probe field and SW field are given below. The Hamiltonian for the three-level Ξ type system is given by The time evolution of the density matrix operator ߩ of the system is governed by the Liouville equation (or the Master equation) with the phenomenological decay terms added.
‫ܪ‬ is the Hamiltonian of the bare atomic system and ‫ܪ‬ ூ is interaction Hamiltonian of the system concerned. Ω p and Ω c stand for the Rabi frequencies of the probe field and control or pump field respectively. ߱ and ߱ represent the probe and control frequencies in angular scale. Ʌ denotes the separation between two consecutive nodes and antinodes of the standing wave. γ 32 and γ 21 are the spontaneous decay rates from the level |3> to level |2> and from level |2> to level |1> respectively and these are shown by dashed lines in Fig 1. A set of nine Optical Bloch equations (OBEs) for the threelevel system are derived under rotating wave approximation. ) to determine the population (ߩ ) of different levels and probe coherence term (ߩ ଷଶ ) . These solutions are given below: Here ‫ܮ‬ = The expressions of the probe absorption (imaginary part of ߩ ଷଶ ) and ߩ are used to study atom localization.

Results:
The population of levels |1> and |2> vs different position of the atoms is plotted for different ߗ ( figure 2). At low ߗ almost entire population is trapped in level |1> and shows little undulation along x direction. The strength of control field is position dependent due to the standing wave formation in x direction and this variation becomes prominent for higher values of ߗ resulting in redistribution of population between the energy levels. At ߗ = 30 MHz a periodic array of sharp spikes are observed in the anti-node positions of the standing wave indicating strong atom localization at these positions within sub-wavelength region .The degree of localization increases if level |2> be metastable state but for experimental realization of this theoretical study we have used the parameters of 87 Rb 5S 1/2 → 5P 3/2 → 5D 3/2 where 5P 3/2 is not a metastable state. At node positions the effective strength of the control field is maximum and the population is distributed almost equally between the two states |1> and |2> (50% each). The separation between two consecutive node and antinode (Ʌ) depends on the wavelength of the control field as well as the relative orientation of the applied fields. This causes alteration in position of atom localization.

Conclusion:
In this work the 1D atom localization in a three-level Ξ type system, where unlike the conventional pump-probe study we have applied a strong standing wave field between the ground level and intermediate level, has been studied using population measurement of different energy levels as well as probe absorption. The degree of atom localization can be controlled by Rabi frequency of control field. Under this atom localization almost 100% reduction in probe absorption has been made possible by tuning the relative orientation of the applied fields. Atom localization within a narrow region can be used in nanolithography and periodic transparencies in probe absorption profile can have potential application in optical devices fabrication.