Saturation Absorption Spectroscopy of Rubidium Atom Jayash Panigrahi August 17, 2013 Abstract Saturated absorption spectroscopy has various application in laser cooling which have many relevant uses in atomic physics and quantum optics. The spectroscopic studies in this project includes the measurement of the Dopplerbroadened absorption profiles of the D2 transitions of Rubidium at 780 nm and then use the technique of saturated absorption spectroscopy to improvement the resolution beyond the Doppler limit and measurement of the nuclear hyperfine splittings. 1
Contents 1 Introduction 3 2 Theoretical background 3 3 Doppler free spectroscopy 6 4 Doppler free Saturation Absorption Spectroscopy 7 5 Conclusion 11 6 Acknowledgment 11 7 References 11 2
1 Introduction Rb has been one of the model atomic systems studied for a long time and has been extensively studied in quantum optics especially in laser cooling and atom traps. There are 2 stable isotopes of Rb, viz. 87 Rb and 85 Rb with natural abundance of 30 % and 70 % respectively. The energy level diagram is shown in Fig. 1. The fine structure of Rb arising from the spin-orbit coupling gives rise to two strong transitions viz. the D1 and the D2 lines. D2 line is the transition from the 5 2 P 3/2 excited state to the 5S 1/2 ground state having a wavelength of 780.027 nm. D1 line is the transition from the 5P 1/2 excited state to the 5S 1/2 ground state. with a wavelength of 794.760 nm. The excited state lifetime of the D1 line is 29.4 ns and 27.02 ns for the D2 line. The linewidth of D2 line from the transition probability is 5.9 MHz. 2 Theoretical background Laser interactions with atoms The difference E = E 1 E 0 between the excited state energy E 1 and the ground state energy E 0 is used with Planck s law to deter- mine the photon frequency v 0 associated with transitions between the two states: E = hv 0 (1) This energy-frequency proportionality is why energies are often given directly in frequency units. For example, MHz is a common energy unit for high precision laser experiments. There are three transition processes involving atoms and laser fields: Stimulated absorption in which the atom starts in the ground state, absorbs a photon from the laser field, and then ends up in the excited state. Stimulated emission in which the atom starts in the excited state, emits a photon with the same direction, frequency, and polarization as those in the laser field, and then ends up in the ground state. Spontaneous emission in which the atom starts in the excited state, emits a photon in an arbitrary direction unrelated to the laser photons, and then ends up in the ground state. Stimulated emission and absorption are associated with external electromagnetic fields such as from a laser or thermal radiation. We consider spontaneous emission first as a process characterized by a transition rate or probability per unit time for an atom in the excited state to decay to the ground state. In the absence of an external field, any initial population of excited state atoms would decay exponentially to the ground state with a mean lifetime t = 1 1 γ 8ns. In the rest frame of the atom, spontaneous photons are emitted in all directions with an energy spectrum having a mean E = hv 0 and a full width at half maximum (FWHM) E given by the Heisenberg uncertainty principle E t = h or E = γh. Expressed in frequency units, the FWHM is called the natural linewidth and given the symbol Γ. Thus Γ = γ 2π 3 (2)
For our rubidium levels, E 2.5 10 8 ev or Γ 6MHz.[2] Structure of Rubidium atom The Rubidium atom (Rb) has atomic number 37. In its ground state configuration it has one electron outside an inert gas core and is described with the notation 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 5s 1. The integers 1 through 5 above specify principal quan- tum numbers n. The letters s, p, and d specify orbital angular momentum quantum numbers l. as 0, 1, and 2, respectively. The superscripts indicate the number of electrons with those values of n and l. The Rb ground state configuration is said to have filled shells to the 4p orbitals and a single valence (or optical) electron in a 5s orbital. The next higher energy configuration has the 5s valence electron promoted to a 5p orbital with no change to the description of the remaining 36 electrons. Electron spins & Fine Stuctures It was observed that magnetic field split the energy levels into other than the three lines that we have dealt with, accompanied by unequal spacing. In some cases, energy level splittings similar to that of Zeeman splitting were observed, even when there was no magnetic field applied. The spectra of hydrogen, when observed in high-resolution, showed lines consisting of sets of closely spaced lines called multiplets. Electron Spin When a neutral bean was passed through a non-uniform magnetic field, atoms were found to deflect according to their magnetic moments with respect to their fields. It was also observed that some atomic beams were split into an even number of components. If only orbital angular momentum was involved, an odd number of deflections corresponding to 2j + 1 components would have been observed. The obtained results inferred a half-integer angular momentum j = 1 2, 3 2, 5 2.... This inference led to a proposal that the electron might have other additional motion (i.e.), the electron might possess a spinning motion instead of behaving like a particle. This would make the electron acquire spin angular momentum and magnetic moment. In a magnetic field, the spin magnetic moment would have an additional interaction energy, besides the one from the orbital magnetic moment. The quantization of this quantity was necessary to the aforementioned anomalous splitting. Spin Quantum Number It was found through precise spectroscopic experiments and analysis that the spin angular momentum of an electron was quantized as well (i.e.), the value of the components of the spin angular momentum had only two possible values, S z = ± 1 2 (3) Comparing with the orbital angular momentum relation given by Eqn: (1.6) and replacing with the spin quantum number s = 1 2, we get 1 3 S = 2 (1 2 + 1) h = (4) 4 h 4
With the knowledge of the fourth quantum number, called the spin quantum number, m s = ± 1 2, one can describe the state of a hydrogen atom completely. The spin angular momentum vector S can have only one of the two configurations : spin up or spin down corresponding to + 2 h or 2 h 1 1 respectively. Spin-Orbit Coupling It was stated in the previous section that in the absence of an external magnetic field, the spin magnetic dipole moment gives rise to splitting of energy levels. This can be explained by the following reasoning. The interaction energy U can be written in terms of the scalar product of the two angular momentum vectors L and S. This particular effect is called spin-orbit coupling. Fine Structure & Hyperfine Structure The combination of spin and angular momenta can be brought about in different manners. The total angular momentum is given by the vector sum of the two quantum numbers, J = L + S (5) Since the two angular momentum vectors L and S are quantized separately, their sum J is also quantized. The possible values which this total angular momentum vector can take, are given by J = j(j + 1) h (6) Now, we can see that it is possible to have states in which j = l ± 1 2. The l + 1 2 states correspond to the ones where L and S have parallel z-components, and similarly l 1 2 states correspond to the ones where L and S have anti-parallel z-components. To illustrate this, let us take the case where l = 1. Now, j can be either 1 2 or 3 2. In common spectroscopic notation, these two states are represented as 2 P 1 and 2 P 3 respectively. The letter denotes the l = 1 state, 2 2 the superscript the number of possible spin orientations, and the subscript the value of j. These splitting of lines resulting from the magnetic interactions are altogether known as fine structure. In addition to the fine structure, there are smaller splitting arising from the fact that the nucleus of the atom itself has a magnetic dipole moment, orders of magnitude smaller than that of the electrons. These smaller splittings are called as hyperfine splittings. The hyperfine splitting can be observed in hydrogen atom. Its ground level is split into two energy levels separated by an interval of just 5.9 10 6 ev. During this transition, a photon of wavelength 21cm is emitted. However, the measurement of hyperfine structure becomes difficult owing to the Doppler broadening of the atomic spectrum, which has an appreciable effect on the hyperfine energy splittings. 5
3 Doppler free spectroscopy Random thermal motion of atoms or molecules creates a Doppler shift in the emitted or absorbed radiation. The spectral lines of such atoms or molecules are said to be Doppler broadened since the frequency of the radiation emitted or absorbed depends on the atomic velocities. Individual spectral lines may not be resolved due to Doppler broadening, and, hence, subtle details in the atomic or molecular structure are not revealed. Doppler Broadening Suppose the lab reference frame is at rest and an atom is moving at a velocity v with respect to this frame parallel to the x-axis, as shown in the diagram. Now suppose an EM radiation of angular frequency ω falls on the atom. The angular frequency of the radiation as seen from the reference point of the atom will be: where ω = ω kv; (7) k = 2π/λ = ω/c (8) Note that it is the velocity component that is parallel (or antiparallel) to the direction of propagation of the radiation that is relevant and not the total velocity. The speed distribution can be obtained from the Maxwell-Boltzmann Speed Distribution, which is: [ ] m mv 2 f(v i ) = 2πkT exp i (9) 2kT Define: δ = ω ω 0 = kv v = ω ω 0 k Substituting this in equation (3), we get: [ m m(ω g(w i ) = 2πkT exp ω0 ) 2 ] 2k 3 T (10) (11) Observe that the maximum occurs at ω - ω 0 = δ 1/2, and the function falls to half its maximum at ω - ω 0 = δ 1/2 where δ 1/2 = uω 0 ln(2) (12) c The Doppler-broadened line has a full width at half maximum (FWHM) given by: 2δ 1/2 = 2 uω 0 ln(2) (13) c Some simple calculations using Equation(7) show that even for heavy elements, which have a considerably smaller value of u, the value of the FWHM comes out close to 1 GHz. As one can see, Doppler-Broadening gives a significant line 6
width and hence limits the resolution of the Spectroscopic techniques. However, very often we need to go beyond this resolution to observe some of the properties and hence it becomes important to get rid of this broadening. There are several techniques such as Saturation Absorption Spectroscopy, Crossed-beam method, Two-photon experiment to eliminate the effects of Doppler broadening. 4 Doppler free Saturation Absorption Spectroscopy The experiment The setup for the Doppler-free saturated absorption spectroscopy of rubidium is shown in Figure 2. The output beam from the laser is split into three beams, two less intense probe beams and a more intense pump beam, at the thick beamsplitter. The two probe beams pass through the rubidium cell from top to bottom, and they are separately detected by two photodiodes. The two photodiodes form a balanced photodetector. After being reflected twice by mirrors, the more intense pump beam passes through the rubidium cell from bottom to top. Inside the rubidium cell there is a region of space where the pump and a probe beam overlap and, hence, interact with the same atoms. This overlapping probe beam will be referred to as the first probe beam and the other one the second probe beam. Figure 1: Experimental setup for DFSAS 7
Spectrum The spectrum obtained for various transitions of rubidium atom at different combinations of pump and probe beam polarization configurations, which we changed using wave-plates is shown in fig. Figure 2: Spectrum showing transition at different polarization 8
9 Figure 3: Spectrum showing transition at different polarization of pump & probe beam
Figure 4: Spectrum showing transition at different polarization 10
5 Conclusion As a rule of thumb, the ratio of the intensities of the pump and the probe beam is kept to about 3. The best linewidth that we achieved was about 17 MHz in 85 Rb5 2 S 1/2, F = 25 2 P 3/2, F transition. We also found that the linewidth is also dependent on the electronic time-constant of the photodiode amplifier, which limits the scan rate. 6 Acknowledgment I would like thank Prof. Vasant Natarajan for providing me the bestowed opportunity to work in his lab and his fellow PhD scholars at Atomic and Optical physics lab at IISc for supporting and backing me during various technical glitches. 7 References The above work and results are original and true to best of my knowledge. For theoretical background I followed the Experiment SAS lab manual of Advanced Physics Laboratory at University of Florida and PhD theses of Dr. Umakant pursued at IISc. 11