Atom Microscopy via Dual Resonant Superposition

Commun. Theor. Phys. 64 (2015) 741 746 Vol. 64, No. 6, December 1, 2015 Atom Microscopy via Dual Resonant Superposition M.S. Abdul Jabar, Bakht Amin Bacha, M. Jalaluddin, and Iftikhar Ahmad Department of Physics, University of Malakand, at Chakdara Dir(L), Pakistan (Received June 16, 2015; revised manuscript received September 11, 2015) Abstract An M-type Rb 87 atomic system is proposed for one-dimensional atom microscopy under the condition of Electromagnetically Induced Transparency. Super-localization of the atom in the absorption spectrum while its delocalization in the dispersion spectrum is observed due to the dual superposition effect of the resonant fields. The observed minimum uncertainty peaks will find important applications in Laser cooling, creating focused atom beams, atom nanolithography, and in measurement of the center-of-mass wave function of moving atoms. PACS numbers: 42.65.-k, 42.50.Gy Key words: atom microscopy, resonant fields, susceptibility 1 Introduction The precise measurement of atom localization has become an active research topic from the theoretical as well as experimental point of view, because of its potential applications in various quantum optical effects such as laser cooling and trapping of neutral atoms, atom nanolithography, Bose Einstein condensation 1] and measurement of the center-of-mass wave function of moving atoms, 2] etc. The concept of accurate position measurement of an atom came from the start of quantum mechanics discussed by Heisenberg. 3] Quantum coherence and interference based, several localization schemes have been proposed during the last three decades. For example, localization in a three and four level atomic systems were discussed by Zubairy and his coworkers. 4 5] Paspalakis and Knight 6] achieved localization to higher degree of precision by measuring the upper-state population of the atom, whereas Agarwal and Kapale 7] presented their localization scheme on the basis of coherent population trapping. Kapale and Zubairy 8] discussed the phase of driving field for sub-wavelength atom localization. The effect of quantum interference arising from spontaneously generated coherence has also been introduced on the subhalf wavelength atom localization. 9] Macovei et al. 10] proposed atom localization via superfluorescence. Mompart and his coworkers 11] proposed the subwavelength localization via adiabatic passage technique, to coherently achieve state selective patterning of matter waves well beyond the diffraction limit. Sub wavelength localization via the probe absorbtion was also reported in a Y-type four level atomic system, 12] where the position probability of the atom was controlled by intensities and detunings of the optical fields. Atom localization has also been demonstrated in a proof of principle experiment, using the electromagnetically induced transparency (EIT) technique. 13] In other schemes, 14 15] the 2D atom localization based on the probe absorbtion measurements, in a micro-wave driven and a radio-frequency driven four level atomic systems were analyzed. In both of these schemes, the maximum probability of finding the atom in one period of the standing wave-field reached unity by properly adjusting the system parameters. Wang et al. 16] investigated the one- and two-dimensional atom localization behaviors via spontaneous emission in a coherently driven five level atomic system by means of a radio-frequency field driving a hyperfine transition. El-Nabi 17] presented her localization scheme of the atom and found that the precision of localization is dependent on the dephasing rates of atomic coherence. Yu and coworkers 18] claimed 100% detecting probability of the atom in the subwavelength domain in a scheme of one-dimensional atom localization via measurement of upper state population or the probe absorption in a four-level atomic system. 100% localization probability has also been attained in the sub-half wavelength domain in a scheme of 2D atomic localization, through measuring the population in excited states of a four level atomic system. 19] Some other relative schemes 20 24] for realizing atom localization in four and five level atomic systems are also studied. Motivated by these achievements, in this article we propose a one-dimensional (1D) localization scheme, based on the superposition of resonant fields in an experimental five level rubidium or cesium atomic system. The atom localizes by probe absorption, while the uncertainty in localization peaks decreases by increasing strength of the space independent rabi frequencies of the superimposing resonant fields. Our localization scheme may be helpful in the development of controllable, focused atom beams, Laser cooling, atom nanolithography, and Bose Einstein condensation. 2 Model and Its Dynamics Here we consider an experimental M-type five level atomic system, e.g. rubidium or cesium. 25] This M-type E-mail: a jabar80@yahoo.com; aminoptics@gmail.com c 2015 Chinese Physical Society and IOP Publishing Ltd http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn

742 Communications in Theoretical Physics Vol. 64 configuration is convertible to all experimental configurations, e.g., Λ type, V type, and N type atomic configurations. The energy-level diagram of the atom field interaction is presented in Fig. 1. The figure demonstrates three degenerate ground states 1, 2, and 3, while two degenerate excited states 4 and 5. The upper excited level 4 is coupled with the ground levels 1 and 2 by superposition of two resonant fields having space independent rabi frequencies, Ω 1,2 while a probe field of Ω p having detuning p. The excited level 5 is coupled with levels 2 and 3 by coupling field Ω c having detuning c and superposition of resonant fields, Ω 3,4. The coupling field Ω c interconnects the interference of the resonant suprimposing fields Ω 1,2 and Ω 3,4. The system is complicated but it is most like duplicated Wang 26] three level system having more advantages than others. The atom is initially prepared in the ground state 2. Here we assume that the center of mass position of the atom along the directions of the standing-wave fields is nearly constant and neglect the kinetic part of the atom in the Hamiltonian by applying the Raman Nath approximation. 1] Fig. 1 Energy level diagram of five level atomic system. The Hamiltonian in the interaction picture for this system is written as: H i = 2 Ω 1 sinη 1 kx + Ω 2 sin(η 2 kx + ϕ 1 )] 1 4 2 Ω 3 sinη 3 kx + Ω 4 sin(η 4 kx + ϕ 2 )] 3 5 2 Ω p e i pt 2 4 2 Ω c e i ct 2 5 + H.c. (1) H.c. means the Hermitian conjugate. ϕ 1 and ϕ 2 are phase difference between Ω 1,2 and Ω 3,4, while η 1,2,3,4 indicates propagation directions of resonant fields relative to the probe field direction and k is the resultant wave number of the superimposing resonant fields. The general form of density matrix equation is written as: ρ = i H i, ρ] 1 2 γ ij (σ σρ + ρσ σ 2σρσ ), (2) where σ and σ are raising/lowering operators. γ ij represent the decay rates from excited to ground states. Using the master Eq. (2) for the dynamics of the system, the most important six coupling rates equations are obtained and presented in Appendix A. The atoms are initially prepared in the ground state 2. The population of atoms primarily in other states are assumed to be zero. Therefore, ρ (0) 22 = 1 while ρ(0) 44 = ρ(0) 41 = 0, ρ(0) 52 = ρ(0) 32 = 0. The susceptibility is a complex response function to the applied electric field. The real and imaginary parts of the complex susceptibility are associated with absorption and dispersion spectrum of the probe field respectively. To find out the susceptibility we define the electric polarization of the medium as P = ǫ 0 χe and due to the coherence of the probe field the polarization is P = 24 2 ρ 24, where 24 is the dipole matrix element. The complex susceptibility is written for the atomic system, if the two polarizations are compared. The complex susceptibility for this system in the first order is written as: χ = 2N 2 24 ǫ 0 Ω p ρ 24, (3) where N represents the atomic density. ρ 24 is given in Appendix A. Both the real and imaginary parts of χ describe the position distribution function for the atom localization. 3 Results and Discussion We will present results for several cases using the numerical procedure described in the previous section. We supposed a scaling parameter γ to be 1 MHz and scaled all other parameters to this γ. The constants and ǫ 0 are supposed to be unit. The detunings p = c are taken 0γ, for all our results. Ω is written for Ω 1,2,3,4. The plots are traced for atom localization verses kx. The imaginary and real parts of the susceptibility are traced for the accurate positional information within the domain π < kx < π. In Fig. 2 for η 1 = 0.5, η 2 = 0.7, η (3,4) = 1 and Ω c = 1γ, a single localization peak is observed in the half wavelength domain of the standing wave field as a result of the resultant superposition between two superpositions, Ω 1,2 and Ω 3,4. The localization probability is 1 or 100%. Uncertainty in the peak decreases with the strength of space independent rabi frequency, while probability remains the same. At Ω = 1000γ, uncertainty becomes negligible and the atom localizes to a single point. Further the peak shifts from the position π < kx < 0, Fig. 2(a) to the position 0 < kx < π, Fig. 2(b) by changing ϕ 1,2 from π/2 to 3π/2. Figure 3 are traced for η (1,2,3,4) = 1 and Ω c = 2γ. Four localization peaks and two dark lines are observed in one wavelength domain. Dark lines are due to the destructive interference of the atomic waves. The dark lines disappear and each two peaks merge into a single peak, with the increase of space independent rabi frequency i.e. Ω = 30γ, 50γ, 1000γ. Four peaks change into two peaks with negligible uncertainty and appear in the same one wave length domain π < kx < π. The localization probability at each peak is about 50%. The two peaks are because of the two superpositions i.e. Ω 1,2 and Ω 3,4.

No. 6 Communications in Theoretical Physics 743 Fig. 2 Atomic localized peaks vs. kx such that γ = 1 MHz, γ 41,42,43,51,52,53 = 1γ, Ω = 30γ, 50γ, 1000γ, Ω c = 1γ, ϕ 1,2 = π/2 (solid line), 3π/2 (dashed line), η 1 = 0.5, η 2 = 0.7, η (3,4) = 1. Fig. 3 Atomic localized peaks vs. kx such that γ = 1 MHz, γ 41,42,43,51,52,53 = 1γ, Ω = 30γ, 50γ, 1000γ, Ω c = 2γ ϕ 1,2 = π/2 (solid line), 3π/2 (dashed line), η (1,2,3,4) = 1.

744 Communications in Theoretical Physics Vol. 64 In Fig. 4 a single localization peak is also observed in a half wavelength domain by setting η (1,2,3,4) = 0.5 and Ω c = 2γ. Localization probability in this case is 68%. The reason for the single peak is the resultant superposition of the two superpositions Ω 1,2 and Ω 3,4. In this case the destructive interference between the two superpositions Ω 1,2 and Ω 3,4 disappear at Ω = 1000γ, the uncertainty in peak vanishes and the atom localizes to a single point. probe, which is much easier to realize in experiment than the spontaneous emission measurement. 14] Second, uncertainty in the localization peaks decreases only with the rabi frequencies of the superimposing resonant fields without requiring significant changes in other parameters. Third, the atom is initially prepared at the ground state, so the experiment can be easily conducted in laboratory with this scheme. Fourth, the system is a generalized one. By setting the strength of either one or more of the five fields to zero, except the probe, one may find interesting features of any other type of atomic configuration and may search for the best results. Fifth, the probe absorption at certain frequencies is position dependent, such position dependent probe absorption can be reflected by standard spectroscopic methods or the heterodyne measurement of fluorescence images. 15] In view of the above technical aspects of our proposed scheme, this scheme of localization is experimental feasible and will contribute to the researchers for converging their attention towards this new technique of atom microscopy in labortary. Fig. 4 Atomic localized peaks vs. kx such that γ = 1 MHz, γ 41,42,43,51,52,53 = 1γ, Ω = 30γ, 50γ,1000γ, Ω c = 2γ, ϕ 1,2 = π/2, 3π/2, η (1,2,3,4) = 0.5. Figure 5 shows three localization peaks in one wavelength domain of the standing wave field at η (1,2) = 1, η 3 = 0.1, η 4 = 0.8, and Ω c = 1γ. Large uncertainty is observed at Ω = 8γ, 10γ. But for Ω = 200γ the uncertainty becomes negligible while the three peaks change into two sharp peaks with 100% localization probability at each of the two superpositions. In Fig. 6, atom localization in the dispersion spectrum is investigated in this plot at η (1,2) = 1, η 3 = 0.1, η 4 = 0.8, and Ω c = 0.5γ. Single localization peak in the half wavelength domain of the standing wave field is observed for each value of space independent rabi frequency. Localization peak is investigated for Ω = 4γ, 6γ, 8γ, 10γ and is observed that the probability amplitude and also the uncertainity of the peak decreases with increase in the strength of space independent rabi frequency. At sufficient high strength of the rabi frequency the probability amplitude of localization disappear and the atom delocalizes. For all the plots peaks shift can be observed in Fig. 6(b) with change of phase ϕ 1,2 from π/2 to 3π/2. Our proposed scheme of localization is a new one having various advantages. First, the atom is localized to a single point in the absorption spectrum of the Fig. 5 Atomic localized peaks vs. kx such that γ = 1 MHz, γ 41,42,43,51,52,53 = 1γ, Ω = 8γ, 10γ, 200γ, Ω c = 1γ, ϕ 1,2 = π/2, 3π/2, η (1,2) = 1, η 3 = 0.1, η 4 = 0.8. The only apparent disadvantage in our proposed model is its complexity due to the presence of six laser fields which are necessary for the super-localization of our scheme. However this complexity is naturally the beauty of this scheme. In nature we are not only limited to the domain of simple atomic systems. 27] In fact there are a lot of complex structures (atoms) in nature for which the

No. 6 Communications in Theoretical Physics 745 higher ordered spectroscopy can be realized both theoretically and experimentally. super-localization that is, the sharpest peak of the spectra have occurred in the regime of half-wavelength as well as one wavelength at Ω = 1000γ. We also have observed delocalization of the atom in dispersion spectrum with strength of space independent rabi frequency due to the dual superposition effect of the resonant fields. Our proposed theoretical scheme of localization is different from others and hopefully will have more advantages. Appendix A ρ 24 = ρ 21 = i p 1 ] 2 (γ 42 + γ 52 ) ρ 24 + i 2 R 1 ρ 21 + i 2 Ω p( ρ 22 ρ 44 ) i 2 Ω c ρ 54, (A1) i p 1 ] 2 (γ 41 + γ 42 + γ 51 + γ 52 ) ρ 21 + i 2 R 1 ρ 24 i 2 Ω p ρ 41 i 2 Ω c ρ 51 (A2) ρ 54 = i ( c p ) ρ 54 + i ] 2 R 1 ρ 51 i 2 R 2 ρ 34 ρ 51 = + i 2 Ω p ρ 52 i 2 Ω c ρ 24, (A3) i( p c ) 1 ] 2 (γ 41 + γ 51 ) ρ 51 Fig. 6 Atomic localized peaks vs. kx such that γ = 1 MHz, γ 41,42,43,51,52,53 = 1γ, Ω = 4γ,6γ, 8γ,10γ, Ω c = 0.5γ, ϕ 1,2 = π/2, 3π/2, η (1,2) = 1, η 3 = 0.1, η 4 = 0.8. 4 Conclusion A new theoretical approach, introduction of double resonant fields from a ground to an excited state, has been proposed for the physical realization of one-dimensional position microscopy of a five level rubidium 87 Rb atomic system. This five level atomic system is a generalized system and is convertible to all other experimental atomic systems. It has three ground and two excited hyperfine states. Superposition of resonant fields is applied from ground to excited states. Another coupling field and a probe field are also applied from ground to excited states. The atom is localized to a single point, in the imaginary part of the susceptibility. Localization of the atom is also observed and investigated in the real part of susceptibility. Single localization peak in half wavelength domain, while multiple peaks in one wavelength domain are observed at different directions of wave vectors, due to the superposition effects of the resonant fields. Uncertainty in localization peaks is observed to decrease with increase in the strength of space independent rabi frequency of the resonant fields. Shift of the peaks is also observed with change of phase ϕ 1,2. Various atom localization peaks can be achieved in this scheme. The most interesting result that we have observed, is the 100% localization probability for the atom in the half wavelength domain. The where ρ 34 = ρ 31 = + i 2 R 1 ρ 54 i 2 Ω c ρ 21 i 2 R 2 ρ 31, (A4) i( p c ) 1 ] 2 (γ 43 + γ 53 ) ρ 34 + i 2 R 1 ρ 31 i 2 R 2 ρ 54 + i 2 Ω p ρ 32, i( p c ) 1 ] 2 (γ 41 + γ 43 + γ 51 + γ 53 ) ρ 31 (A5) + i 2 R 1 ρ 34 i 2 R 2 ρ 51, (A6) R 1 = Ω 1 sinη 1 kx] + Ω 2 sinη 2 kx + ϕ 1 ], R 2 = Ω 3 sinη 3 kx] + Ω 4 sinη 4 kx + ϕ 2 ], and ρ ij = (ρ ji ). Equations (A1) (A6) are analytically solved by following relation. Z(t) = t e M(t t ) J dt = M 1 J, where Z(t) and J are column matrices while M is 6 6 matrix. The expression for ρ 24 is obtained. 2iA 2 T 6 + (A 6 R2 2 + A 3 T 2 )Ω 2 ρ 24 = c]ω p (4A 1 A 2 + R1 2)T 6 + 2(R2 4T 4 + T 2 T 5 )Ω 2 c + T 2 Ω 4, c where T 1 = 2A 3 A 5 + 2A 4 A 6 R 2 1, T 2 = 4A 5 A 6 + R 2 1, T 3 = 4A 3 A 4 + R 2 1, T 4 = 2A 2 A 5 + 2A 1 A 6 + R 2 1, T 5 = 2A 1 A 3 + 2A 2 A 4 R 2 1, T 6 = R 4 2 + 2R 2 2T 1 + T 2 T 3,

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