Development of a Prototype Atomic ClockBased on Coherent Population TrappingA thesis submitted in partial fulfillment of the requirementfor the degree of Bachelor of SciencePhysics from the College of William and Mary in Virginia,byNathan T. BelcherAccepted for: BS in PhysicsAdvisor: I. NovikovaW. J. KosslerWilliamsburg, VirginiaMay 2008AbstractThe goal of this project is to construct a prototype atomic clock usingcoherent population trapping (CPT) in a rubidium vapor cell. We have createda laser system to lock the laser’s frequency to an atomic resonance in rubidiumand utilized radiofrequency (rf) modulation to create a sideband and carriercomb. With the system assembled, we have observed C PT and locked the CPTto our rf source to create a prototype atomic clock. In the future, we willcontinue to study the CPT res onances to make the clock better.1 IntroductionIn the last decade, advances have been made in creating miniature (sub-cubic cen-timeter) atomic clocks, based on laser probing of an atomic vapor. The laser employedin these clocks is a vertical-cavity surface-emitting laser (VCSEL), which has usefulcharacteristics for this application: low power consumption, ease of current modula-tion with radiofrequency (rf) signals, and use in minature atomic clocks.The atomic clocks contain three parts: a hyperfine transition (in the microwaveregion) betwe en two long-lived spin states of an alkali metal (that has a hydrogen-likespectrum), a frequency modulator with counter, and a laser. The element of use inthis atomic clock is rubidium, which is an alkali metal. The frequency modulator pro-vides phase modulation, creating two electro-magnetic fields at different frequenciesout of one physical laser. A counter matches the frequency driven by the modulatorto give a reference for time, and the laser drives the entire process. To keep thelaser at a set frequency corresponding to the atomic transition, another part calleda dichroic atomic vapor laser locking (DAVLL) system is needed. The DAVLL usesan error signal determined by the difference of signal in two photodetectors to keepthe laser’s frequency on a rubidium resonance. In addition, electronics of the fre-quency modulator keep the mo dulation frequency at the “clock” frequency, allowingan atomic clock to be created for an extended period of time.The overall goal of this project is to create a prototype atomic clock, which wehave done successfully. This paper describes the following: theories underlying theatomic clock in section 2; the experimental setup in section 3; creating and measuringcoherent population trapping (CPT) in section 4; creating and measuring the clockin section 5; further CPT studies in section 6.2 Theory2.1 Coherent Populati on TrappingCoherent population trapping (CPT) drives the clock, so it is important to understandthe underlying theory. We will begin with the theory of a two-level system, and thenmove to a three-level system. The treatment of this theory will be semiclassical, withthe atoms treated as quantum objects in a classical electromagnetic field. For the1two-level system, most of the underlying mathematics will be omitted, and only theequations needed to describe the three-level system will be introduced.Figure 1: A two-level system. Level g1is a state in 5S1/2, and level e is the excitedstate 794.7 nm away from the 5S1/2state. The level g2is included to c ompare withthe Λ system.We first be gin with figure 1, where the electromagnetic field interacts with a two-level atom. In this case, the Schr¨odinger equation to find the wavefunction of theunpaired electron in a rubidium atom in the electromagnetic field is:i!˙ψ(t) = (ˆH0+ˆHint)ψ (1)The first termˆH0is the atomic part of the Hamiltonian, and represents the electronicenergy levels in the field of the nuclei. The second term describes the interaction ofan atom with the electromagnetic field, and can be defined as:ˆHint= −eˆ"E ·ˆ"r = −eˆEˆx (2)The wavefunction ψ is defined for the two-level interaction as:|ψ(t)" = α(t) |e" + β(t) |g" , (3)and it is convenient to write the Hamiltonian operators in the bases of the states|e" and |g". Following the definition for the matrix elements for the electron dipolemoments we have:ˆH0= !ωe|e" #e| + !ωg|g" #g| (4)ˆHint= ℘eg(E |e" #g| + E∗|g" #e|) (5)where#g| − e · x|e" = #e| − e · x|g" = ℘eg(6)#g| − e · x|g" = #e| − e · x|e" = 0 (7)2The energies of the ground and e xcited states are !ωgand !ωe, respectively, and itis convenient to set the energy of the ground state as a zero level (ωg= 0). We thendefine ωeg= ωe− ωgas the frequency of atomic transition.We next write down the expression for the electric field of a monochromatic elec-tromagnetic wave and make some simplifications, arriving at equations for α and β.By solving these equations for a constant amplitude of the electromagnetic field andresonant interaction, the equation for α(t) becomes that of a harmonic oscillator¨˜α = −℘egE2!2˜α, (8)where E is the amplitude of a monochromatic electromagnetic wave. The atomicpopulation |α|2oscillates between the ground and excited states with a frequency:Ω =℘egE!(9)This effect is known as Rabi oscillations, and frequency Ω is called the Rabi fre-quency. To find¨˜α, we use a technique known as the rotating wave approximation.Two approximations are made: one to neglect fast oscillating terms due to a weakelectromagnetic field; another to assume that the electromagnetic field’s frequency isnear the resonant frequency.While the calculations up to this point are correct theorectically, they are some-what incomplete without taking into account any decoherence mechanisms. The mostfundamental decoherence mechanism for a two-level system is spontaneous emission,caused by the interaction of atoms with vacuum fluctuations. Spontaneous emissionlimits the time an atom can spend in the excited state. By adding a relaxation termΓ to the excited state amplitude, we can represent the loss of atoms in the excitedstate. This term is added in the equations˙˜α = −(Γ2+ i∆)˜α +iΩ2β (10)˙β =Γ2+iΩ∗2α (11)where ∆ = ωeg− ω is the difference between the frequency of the atomic transitionand the laser frequency. This term is also often referred to as the detuning of thelaser from the atomic transition.For simplicity, we assume that the total population of both levels is always 1. Weuse the normalization condition of α∗α + β∗β