Atomic Clocks A literature study Artwork credit: Brad Baxley and Ye Labs, JILA Cathelijne Glaser, BSc [email protected] supervised by Dr. Jeroen Koelemeij summer 2015 Abstract In atomic clocks, the frequency of the oscillator is determined by the energy difference between two quantum mechanical states in atoms, ions or molecules. This transition energy depends only on fundamental constants and thus provides a stable frequency reference. Disturbances of the system that may shift the frequency are discussed, as well as broadening of the peak due to dephasing and the uncertainty principle. This literature study gives an overview of the different types of both microwave and optical atomic clocks, as well as the different atom species that are suitable as clock atoms. A qualitative way to determine clock performance is given in terms of the fractional frequency uncertainty, and the Allan variance to determine the stability and characterise the noise of the clock. Various methods and procedures to decrease the uncertainties are discussed. An outlook on possible future developments is presented, including clocks based on nuclear transitions and techniques to increase the accuracy and stability beyond the curent state of the art. Front page: artist's view of an optical lattice clock. [1] 1 Table of contents Abstract 1 1 Introduction 3 2 Atomic transitions 4 2.1 Frequency shifts 4 2.2 Line broadening 5 3 The basic clock system 9 3.1 Active clocks 9 3.2 Passive clocks 10 3.3 Measuring and feedback 10 4 Characterizing the performance of clocks 12 4.1 Accuracy and reproducibility 12 4.2 Stability 12 4.3 Comparison and synchronisation 16 5 Active clocks 19 6 Passive clocks 25 6.1 Microwave regime 25 6.2 Optical regime 32 7 Future prospects 36 7.1 (Fundamental) limits 36 7.2 Suggested further research 38 8 Conclusion 39 References 40 2 1 Introduction Humans have, as far as known, kept track of time, or cyclic events, since the beginning of development. To measure time, a reference oscillator is needed. It is impossible to measure time itself. It is only possible to measure a frequency or duration. In time measurement, it is assumed that two identical phenomena acquire the same time to be produced, the so- called reproducibility postulate. [2] The rhythm of day and night and the four seasons is always present. But descriptions of more subtle recurring events, like solar eclipses and other astronomical phenomena, have been found dated at least 4000 years ago. The sundial was the earliest form of a clock that divides time into smaller segments than a day. The most sophisticated versions had an error of 24 seconds for each 0.1 degree of angle measurement. Other historical clocks were based on water or mechanical devices, such as the pendulum clock or the spring-balance- wheel clock. The quartz clock is based on the piezoelectric effect, in which vibrations are excited by applying an alternating voltage to a crystal. This was the first clock based on material properties instead of astronomical observations or mechanical movement. However, it still suffers from ageing and is very sensitive to environmental conditions. [3] In atomic clocks, the electromagnetic transition between two quantum mechanical states of an atom, ion or molecule determines the frequency of the oscillator. Frequency dividers provide pulses at a desired rate, for example with a frequency of 1 Hz. The development of these types of clock boosted the abilities of time measurement. They are so sensitive that relativistic effects can be measured. Precise time measurements are for example applied in metrology, fundamental constant research, the foundations of quantum mechanics, gravity, and geodetics. But it also finds applications in navigation and communication networks. It makes a precise measurement of position or length possible, especially since the meter has been defined in terms of the speed of light. [2] This literature study dives into the principle of atomic clocks, and what makes them intrinsically suitable to measure time. After defining a quantitative manner to measure the performance of a clock, several factors that determine the performance of an atomic clock are described. Although it is impossible to mention every atomic clock ever built, an overview of the different types is given, in both the microwave domain and the optical domain. These are mutually compared and their performance, advantages and disadvantages are discussed, as well as the different improvements that came with each new design. The currently best optical lattice clock is described, evoking a look into the future. What is there to be expected of future atomic clocks, and is there a limit to the potential performance of clocks? 3 2 Atomic transitions According to quantum mechanics, the energy of an atom, molecule, or ion consists of discrete values. Transitions between these levels occurs by emitting or absorbing electromagnetic radiation of a specific frequency. This resonance frequency ν0 depends only on fundamental physical constants and is the same for all atoms of a particular element, which makes it a reproducible standard. Two other aspects make atomic transitions particularly suitable for timekeeping. The properties of the atoms do not, as far as known, change over time. Additionally, atoms do not wear out, as mechanical clocks all do. [2][4] 2.1 Frequency shifts The resonance frequency is an intrinsic property of the unperturbed atom, but several factors may cause the actual frequency to be shifted. It is important to accurately characterize the shifts that occur in the clock, to be able to state the correct output frequency. Frequency shifts can either be diminished by reducing the environmental perturbations acting on the clock system, or corrected for when accurate data about the shifts are available. Inhomogeneities and fluctuations in the perturbing fields should also be taken into account. Additionally, a shift in some level may influence the energy of a nearby level. [5] Depending on the respective design, each type of atomic clock will encounter these and other, minor disturbances, in varying degrees. This is described in more detail in the corresponding paragraphs in chapter 5 and 6. External magnetic fields Magnetic moments are associated with the various angular momenta of atoms: orbital, spin and nuclear angular momenta. These magnetic moments interact with an external magnetic field. The energy of atomic levels with different magnetic moments depend on their orientation with respect to the field. The external magnetic field therefore lifts the degeneracy of these levels. This is called the Zeeman effect. The energy shifts can be calculated using perturbation theory. The nature of the shifts depends on the strength of the external magnetic field relative to the internal magnetic field of the atom, generated by the moving charges. The latter causes spin-orbit coupling, which can also be considered a perturbation The shift in frequency for a certain transition due to an external magnetic field can be written as a Taylor expansion. For small magnetic fields, only the first two terms need to be considered. [5] Only constant magnetic fields contribute to the Zeeman effect as long as the energy shift is linear, because the time average of an oscillating (AC) field is zero. However, in the intermediate field regime, the quadratic Zeeman effect cause AC fields to play a role as well. [5][6] 4 External electric fields The electrical analogue to the Zeeman effect is the Stark effect. For a static electric field Ɛ in the direction of the z-axis, the perturbation operator for N electrons is given by: N = =− H ' e ∑ Ɛ zi Ɛ Dz (2.1) i=1 with e the electron charge, zi the spatial coordinate of electron i, and Dz the z-component of the electric dipole moment of the atom. Since this is an odd operator regarding parity, the first order perturbation energy is zero for wavefunctions of definite parity. Otherwise stated, atoms do not have a permanent dipole moment in non-degenerate states. The linear Stark effect exists only for hydrogen-like orbitals with n>1. The degeneracy is only partly removed. For weak fields, the Stark splittings are negligible compared to the fine structure effects. Intermediate field effects can be calculated using full perturbation theory. [7] The quadratic Stark shift is generally very small. According to perturbation calculations, the shifting of the levels depends on the neighbouring levels of opposite parity and the corresponding parity. [7] The quadratic Stark shift depends on the polarizibility of the atom, which is in turn dependent on the electron configuration. Note that both DC and AC electric fields may give rise to a quadratic Stark shift. [8] A gradient in the present electric field interacts with the electric quadrupole moment of the atom. The resulting energy shift is usually very small and even zero for many energy levels. [8] Gravitational red shift According to relativity, when two clocks experience different gravitational potentials, their clock rates differ. The frequency shift depends on the mutual height difference and the local value of the gravitational acceleration. [5] 2.2 Line broadening Atoms will absorb or emit not only electromagnetic radiation with exactly the resonance frequency, but over a small frequency range surrounding ν0. This range is called the resonance width or linewidth Δν. The ratio of resonance frequency to linewidth is called the quality factor Q: ν 0 Q= Δν (2.2) All other parameters equal, the stability of the atomic oscillator is proportional to Q. An important route to increased stability is thus to narrow the linewidth. [4] A perfect oscillator displays a pure sine wave. In the frequency domain, this is represented 5 by a Dirac delta function. Line broadening can originate from either homogeneous or inhomogeneous processes.