The shape of the saturated fluorescence decay line is sensitive to the distortion of the laser beam in the case of the atomic beam. The remaining Doppler broadening is caused by the thermal motion of the atoms in the vapor cell. 20 3.2 Comparison of probe absorption obtained using TOC and the cladding state approach for a five-level atomic system.

## Introduction

Introduction

### Motivation

Hyperfine measurement of the splitting of the 6P states has been performed using saturation absorption [21] for both the 6P1/2 and 6P3/2 states, or fluorescence spectroscopy [54] for the 6P3/2 state in the 5S→6P transition, double resonance at a Doppler misconfigurations for 6P1/2 [22] and using RF transition with electrical discharge [55]. The range of beating frequencies is determined by the inverse of the population relaxation times of the upper levels. The effect of partial Doppler broadening is removed for multiphoton resonance by subtraction of the broad transparency or absorption background.

### Thesis overview

The complete overlap of the solid red curve (AT peaks plus interference) and the dashed green curve (AT peaks only) in Fig. The imaginary part of the zeroth harmonic, ρ(0)12 corresponds to the IR pump absorption, while the imaginary part of the first harmonic, ρ(+1)12 is proportional to IR probe absorption to first order and all others are for wave mixing [25]. Therefore, optical pumping [108,110] gives rise to enhanced absorption (EA) Doppler-free peaks of the 6P3/2 hyperfine planes.

## Theoretical background of light-atom interaction

### The Hamiltonian

*Rotating wave approximation**Corotating frame*

The Hamiltonian H of the atomic system is given as H =H0 +HI, where H0 is the Hamiltonian of the atom without interaction and HI is the interaction Hamiltonian between the atom and the driving light fields. Note that the dipole matrix element hi|ε.dˆ |ii = 0, because the dipole operator ε.dˆ is of odd parity and does not connect states of the same parity. While the slowly oscillating exponential terms with imaginary arguments of the form ±i(ωlij−(ωj−ωi))t are preserved.

Density matrix formalism

### The fine and hyperfine structures

*Fine structure**Hyperfine structure*

5.2: (a) Steady-state population for atoms with velocity vx = 0, (b) population oscillations for atoms with velocityvx = Γ/2k, (c) Fourier transform of the oscillating population, and (d) absolute value of the Fourier transform of the oscillating population. The transparency spectrum of the hyperfine levels 6P3/2 (F = 1,2,3) (when all three gratings are closed) is shown by the red trace in Fig. The transparency spectrum of 6P3/2 (F = 0, 1,2) hyperfine levels (when all three shutters are closed) is shown by the red trace of Fig.

## Nature of interference between excitation paths associated with AT peaks in

### Introduction

In the three-level system, the excitation paths connected to the two AT peaks interfere in pairs, almost similar to the two-slit interference. In the four-level system, the interference between the excitation paths connected to the three AT peaks is also pairwise analogous to the three-slit interference, but has a slightly more complicated nature of the interference. However, in the four-level system the nature of the interference between the excitation paths connected to the two extreme AT peaks can be tuned from constructive to destructive by tuning the power of the control laser.

### The theoretical model

*Dressed state approach**Bare state TOC approach*

3.6) The coherent decay terms κij give rise to interference between the excitation paths of the dressed states. In this case, there will be no interference between the excitation paths of the dressed states. The validity of the dressed state approach is tested by comparing the probe absorption in the two approaches.

### The nature of interference in the multi-level systems

*Three-level system**Four-level system**Chain configurations with all the control lasers at res-**Branching configurations with all control lasers at res-**Four-level loopy system*

The configuration of the VIPO at IR transition corresponds to the energy scheme shown in Fig. The energy level scheme and configuration of the optical pumping system is shown in Fig. The EA spectrum of the optical pumping system is shown by the red trace in Fig. .

From the equations of motion (Eq. E.3), the density matrix element ˙ρ21 is oscillating at the frequency of the probe laser field (Ω∗pe−ωpt).

## Hyperfine measurement of 6P 1/2 state in 87 Rb using double resonance on blue

### Introduction

This chapter describes spectroscopy of the 6P1/2 state i87Rb using the double resonance technique at 780 nm and 421 nm. The double resonance technique is implemented using electromagnetically induced transparency (EIT) and optical pumping methods. Using these spectroscopy methods, the hyperfine splitting of the 6P1/2 state is measured with a precision of < 400 kHz, and the hyperfine constant of the magnetic dipole is also calculated.

The chapter begins with a detailed theoretical description of the double resonance technique using the EIT effect in a V-type system (or coherent control scheme) and the optical pumping scheme.

### Measurement Schemes

*Coherent Control Scheme**Optical Pumping Scheme*

The Rabi frequency for the fields is ΩL = −dijELeiφL/~ where, dij = hi|ˆd|ji is the dipole matrix element, ˆd is the atomic dipole operator, φL is the phase of the fields and subscript L=p, c represents the fields (i.e. p is the probe of the 780 nm laser and c is the pump of the 421 nm laser). The dynamics of the atomic field interactions are described by Eq. 2.15 yields the following set of equations of motion: The absorption of the probe field is obtained by dividing the steady-state solution of Eq.

4.3) The probe absorption is proportional to the imaginary part of ρ12 and the Doppler-enhanced spectrum of Eq. 4.3, is calculated by the thermal average over all the velocities of the atoms in the vapor cell. H The Hamiltonian of the open V-type system is the same as that of the closed V-type system given in Eq.

However, the equations of motion will be modified to include optical pumping to the second hyperfine ground state and the ground state. The transparency spectra of the closed V-type system (Eq. 4.3) and the open V-type system (Eq. 4.5) are compared in fig. Thus, optically pumping the zero-velocity group atoms into the upper ground hyperfine level and coherently dephasing the velocity of the ground hyperfine levels increases the absorption of the probe, giving rise to Doppler-free peaks.

The EA spectrum is theoretically modeled by considering the Hamiltonian H of the four-stage optical pumping system in Fig. 1.

### Experimental Details

*Set-up and Results**Errors**Systematic Errors**Statistical Error*

The peaks of the 6P3/2(F = 2.3) hyperfine levels are well resolved, but the peaks of 6P3/2(F = 1.2) are partially resolved due to the residual Doppler broadening effect.

## Role of velocity induced population oscillation in saturated fluorescence spec-

### Introduction

The drop in fluorescence in the fluorescence spectrum is generally interpreted as a saturation effect (also known as rate-selective saturation, VSS). However, in this work we find that the non-Doppler dip in the fluorescence spectrum is due to the effects of VSS and VIPO. The fluorescence dip is further modified by Doppler averaging, and its linewidth and dip height depend on the temperature of the atomic gas and the intensity of the laser beam used.

For an atomic beam, the shift of the fluorescence dip from the line center is dependent on the average velocity of the atomic beam and the misalignment of the laser beams with respect to the atomic beam. The final section of the chapter provides a detailed discussion on the fluorescence decay in the case of an atomic beam and the effects of laser beam misalignment.

### Theoretical Formulations

*Velocity selective saturation effect on fluorescence dip**Velocity induced population oscillation effect on fluorescence dip 77**Effect of laser beams misalignment*

The Doppler-enhanced spectrum of the excited atomic gas population shown in Fig. The VIPO effect occurs only for a narrow range of beat frequencies due to the inherent inertia of the population, i.e.

## Resolving closely spaced levels for Doppler mismatched double resonance . 85

### Energy level schemes and configurations

*Transparency for a V-type open system**V-type open system**VIPO at IR transition for a V-type open system**VIPO at IR and VSS at blue transition for a V-type**Enhanced absorption for optical pumping system**Optical pumping system**VIPO at IR transition for optical pumping system . 97*

Theoretically, the VIPO at IR transition for an open V system is modeled by considering the Hamiltonian H for the configuration shown in Fig. The vertical axis of the blue curve is on the left and the red and green curve on the right. In the steady state condition (˙ρ(n)ij = 0 for all n, i and j), the absorption of the probe laser (ρ(+1)12 ) is obtained by substituting the truncated series of the Floquet expansion given in Eq. .

The vertical axis of the red and green trace is on the left and the blue and black trace on the right. The linewidth of the dip is determined by fitting a Gaussian line profile (which fits better than a Lorentzian line profile). This dip is further enhanced by the VSS effect of the two counter-propagating blue pump laser fields.

The Hamiltonian for the system under electric-dipole and rotating wave approximation and in rotating frame is obtained using Eq. The vertical axis of the red curve is on the left and the blue and green curves on the right. The imaginary part of the density matrix element ρ(+1,0)12 in the Floquet expansion is proportional to probe absorption and is expressed as follows.

The vertical axis of the red trace is on the left and the blue and green traces are on the right.

### Experimental results

*Set-up description**Resolving the 6P 3/2 hyperfine levels in 85 Rb**The V-type system**The optical pumping system**Resolving the 6P 3/2 hyperfine levels in 87 Rb**The V-type system**The optical pumping system**Effects of Power broadening on the resolution**VSS at blue transition for a V-type system*

The broad background is removed by subtraction of the absorption (or transparency) spectra of the two probes using two identical IR photodetectors (PD1 and PD2) in the differential transimpedance amplifier. The three peaks of 6P3/2(F = 2,3,4) hyperfine levels are merged to form a broad transparency spectrum due to the residual Doppler broadening effect. The effect is removed when shutter 3 is open to subtract the broad transparency profile, and the spectrum of the resolved hyperfine levels is shown by the green trace in Fig.

Additional line narrowing of the resolved peaks is achieved using the configuration shown in Fig. The broad transparency background is removed when shutter 3 is open, and the well-resolved peaks of 6P3/2(F = 2,3,4) hyperfine levels are shown by the green trace in Fig. When shutters 1 and 2 are open, the dips induced by VIPO at IR and VSS at blue transition inside the broad transparency peaks correspond to the three hyperfine levels of the 6P3/2(F = 1, 2,3) state (see the blue trace in Fig. 6.13b).

The residual Doppler broadening effect is removed when shutter 3 is open and the spectrum of the resolved hyperfine levels is shown by the green trace of figure. The contribution of the broadening effect of the IR pump power to the final result (i.e. the resolved spectrum of the 6P3/2 state), is illustrated in Figure 2. The variation of the linewidth of the resolved peak corresponding to F = 2 with the IR pump power is shown in Figure 2.

The absorption of the probe field is obtained by dividing the steady-state solution of Eq. The absorption of the probe field is obtained by substituting the truncated series of the Floquet expansion given in Eq. However, because ρ21 = ρ∗12, the imaginary part of ρ12 is proportional to the absorption of the probing field.