Abstract:
Coherent population trapping (CPT) and optical Ramsey interference provide new avenues for developing compact, high-performance atomic clocks. In this work, I have studied the fundamental aspects of CPT and optical Ramsey interference for Raman clock development. This thesis research is composed of two parts: theoretical and experimental studies. The theoretical component of the research was initially based on pre-existing atomic models of a three-level Λ-type system in which the phenomena of CPT and Ramsey interference are formed. This model served as a starting point for studying basic characteristics of CPT and Ramsey interference such as power dependence of CPT, effects of average detuning, and ground-state decoherence on linewidth, which directly impact the performance of the Raman clock. The basic three-level model was also used to model pulsed CPT excitation and measure light shift in Ramsey interference which imposes a fundamental limit on the long-term frequency stability of the Raman clock. The theoretical calculations illustrate reduction (or suppression) of light shift in Ramsey interference as an important advantage over CPT for Raman clock development. To make the model more accurate than an ideal three-level system, I developed a comprehensive atomic model using density-matrix equations including all sixteen Zeeman sublevels in the D1 manifold of 87Rb atoms in a vapor medium. The multi-level atomic model has been used for investigating characteristics of CPT and Ramsey interference under different optical excitation schemes pertaining to the polarization states of the frequency-modulated CPT beam in a Raman clock. It is also used to study the effects of axial and traverse magnetic fields on the contrast of CPT and Ramsey interference. More importantly, the multi-level atomic model is also used to accurately calculate light shift in Ramsey interference in the D1 manifold of 87Rb atoms by taking into account all possible off-resonant excitations and the ground-state decoherence among the Zeeman sublevels. Light shift suppression in Ramsey interference with pulse saturation is also found to be evident in this comprehensive model.
In the experimental component of the research, I designed a prototype of the Raman clock using a small (2 cm in length), buffer-gas filled, and isotopically pure 87Rb cell. A fiber-coupled waveguide electro-optic modulator was used to generate the frequency-modulated CPT beam for the experiments. The experimental setup was operated either by continuous excitation or pulsed excitation for experimentally characterizing CPT and Ramsey interference under different experimental conditions and for testing different optical excitation schemes which were investigated theoretically. Several iterations of the clock physics package were developed in order to attain better frequency stability performance in the Raman clock. The experimental work also provided a basis to develop a new repeated-query technique for producing an ultra-narrow linewidth central fringe with a high S/N ratio, and suppressing the side fringes in Ramsey interference.
The above described research was carried out keeping in mind compact, high-performance clock development, which relies on technologies that can be miniaturized. Vapor cell based atomic clocks are ideal candidates for compact clock technology. The CPT phenomenon, observed by Raman excitation in a vapor medium, is a promising candidate for compact, high-performance Raman clock development. However, atom-field interaction involved in a vapor medium is often more complex than other media such as cold atom or atomic beam. It is difficult to model this interaction in order to predict its influence on CPT characteristics and, hence, the performance of the Raman clock. This dissertation addresses one such problem by developing a comprehensive atomic model to investigate light shift and modification of light shift in the Raman clock, particularly with pulsed excitation. It demonstrates a clear possibility of reducing (or suppressing) the light shift associated with Ramsey interference in a vapor medium for achieving higher frequency stability in the Raman clock. Additionally, theoretical comparisons of various optical excitation techniques have been calculated to demonstrate the relative strengths and weaknesses of different schemes for Raman clock development.
Chapter 1 outlines a brief history of atomic clock technologies related to this current work. Principles of common types of atomic beam and vapor cell clocks are described, along with their applications. Determining precision of the clock requires measurement of the output frequency over time, so the clock stability is also discussed, addressing sources of noise and errors in the measured frequency. The requirements and challenges for developing portable, miniature clocks are discussed.
Chapter 2 describes an idealized three-level Λ-system, which is used to model CPT and Ramsey interference. A fundamental basis for theoretical modeling is developed in this chapter by formulating density-matrix equations. The simulations are performed with numerical solvers in both steady-state and time-dependent regimes for studying CPT phenomenon in continuous excitation and Ramsey interference in pulsed excitation. The general behaviors concerning the lineshapes and resonance properties of CPT and Ramsey interference were established. Characteristics of CPT have also been studied in density-matrix elements relating to the coherence between the ground-states and excited state. Ramsey interference is modeled to investigate how the phenomenon responds to different pulse parameters. The three-level model serves as a simple basis for investigating CPT and Ramsey interference for these Raman clock studies.
Chapter 3 expands on the three-level model by developing a comprehensive sixteen-level atomic model by incorporating all Zeeman sublevels in the D1 manifold of 87Rb atoms. Raman excitation in this multi-level model enables modeling of different polarization states for the CPT beam which in effect forms multiple three-level Λ-transitions depending on the polarization and the state selection. The model was also extended to include the effect of transverse magnetic field when calculating the resonances. Optical excitation schemes simulating different possibilities for Raman excitation were considered for the frequency-modulated beam in a Raman clock, and the drawbacks and benefits of each are discussed.
Chapter 4 specifically addresses light shift, a fundamental source of error in atomic clocks, which limits the long-term frequency stability. The slope of light shift as a function of laser intensity or laser detuning affects the long-term stability of the clock. Using the three-level model, described in Chapter 2, I calculated the light shift in Ramsey interference by modeling the interaction of the three-level atoms with bichromatic laser pulse sequence. The result obtained from this calculation revealed that light shift properties of Ramsey interference are different from CPT because of pulsed excitation. It showed an important property of light shift in Ramsey interference, that it can be reduced by strong interaction with the first pulse. Such an aspect of Ramsey interference constitutes a practically competitive scheme for compact, high-performance Raman clock development. Light shifts associated with coherent density-matrix terms are studied as they are relevant to the detection of Ramsey interference in transmission (or absorption) through the medium. For the single-velocity case, the numerically computed results from the three-level model were compared with the analytical results obtained using the adiabatic approximation. Later, light shift is more accurately calculated in the D1 manifold of 87Rb atoms by using the multi-level atomic model described in Chapter 3. Although, the light shift was found to be modified due to the presence of additional off-resonant excitations and decoherence introduced in the ground-state magnetic sublevels, the general characteristics of light suppression with pulse saturation remained valid. Effect of velocity averaging on Ramsey interference was also investigated, and shown to create a systematic frequency error which is not real in the atomic system under study.
Chapter 5 describes the design, development, and construction of the experimental atomic clock prototype. The iterations and versions of the experiment are described, leading to the most recent version. Experimental studies have been performed using a laboratory scale Raman clock employing a 2-cm long, isotopically pure rubidium cell, loaded with a buffer gas. Components involved in the experiment are chosen as to be suitable for compact clock development. A fiber-coupled waveguide EOM driven by an RF oscillator is used to produce a frequency-modulated laser beam for the experiments. The phase-locked-loops (PLL) are used as the RF frequency synthesizer for driving the EOM. Several laser lock techniques were explored for the tunable diode laser for finding one which is suitable for compact Raman clock development. Additionally, towards the end of this chapter, a brief discussion on vertical cavity surface emitting lasers (VCSELs) is presented, as they are a primary candidate for compact clock development. VCSELs can be directly modulated and thus do not require an external beam generation device such as an EOM. Methodology for generating CPT resonance and Ramsey interference fringes using the experimental setup are described. Chapter 5 reports on the experimental results obtained for CPT and Ramsey interference. These include results using various optical excitation schemes based on polarization states of the contributing CPT beams, light shift measurements, and contrast and linewidth measurements for CPT and Ramsey interference.
Chapter 6 reports on the clock frequency stability measurements using the phenomena of CPT and Ramsey interference. Conditions and parameters for achieving best clock stability are determined. Different clock schemes are compared to demonstrate short- and long-term stability improvements with CPT and Ramsey interference. A promising new interrogation method, known as repeated query technique, for acquiring Ramsey fringes is also discussed in this chapter. This technique enhances the contrast and the S/N ratio of the central fringe in Ramsey interference and significantly suppresses the side fringes by interference of multiple query signals. The frequency stability (or Allan deviation) of the clock is measured by employing the repeated query technique.
Chapter 7 concludes the dissertation with a summary of the results and methods, and addresses future avenues for improvement in the research. The future work described in this chapter pushes the experimental techniques and components closer to compact clock technologies. The future theoretical work incorporates more of the physical effects which occur in the real atomic system, leading to more accurate simulations, which will provide deeper insight into the requirements and limitations of vapor cell based compact clocks.
Appendices are provided to supplement the chapters. They present computer codes for simulation, and derivations showing magnetic field effects and equivalence of polarization schemes for the multi-level atomic model. The appendices also include supplements to the experiment section such as Helmholtz coil design simulation (COMSOL) and data collection methods (LabVIEW).