Fundamentals of Time and Frequency

2 Japan Standard Time and its Related Research 2-1 Fundamentals of Time and Frequency KAJITA Masatoshi, KOYAMA Yasuhiro, and HOSOKAWA Mizuhiko Time (frequency) is one of the fundamental physical quantities, that can be measured most accurately among them. In this paper, the principles for the measurement of the time & frequency and the estimations of its uncertainties are summarized. Keywords Time and frequency measurement, International system units, Frequency measurement, Allan variance 1 History and signifi cance of addition to these, the standard for acoustic fre- frequency quency has been provided since a tuning fork was invented in the 18th century. Following Frequency is how many times a periodic this, improvements of frequency precision phenomenon is repeated per unit time. It has made two major leaps. The fi rst was the devel- been known from antiquity that using an ap- opment of crystal oscillators originating with propriate natural phenomenon enables us to the discovery of the piezoelectric effect of maintain this quantity extremely accurately. quartz crystal by Paul-Jacques Curie and Pierre What human beings fi rst recognized as such Curie in 1880. The second was the develop- would be the periodicity and accuracy of revo- ment of atomic frequency standards starting lutions of celestial bodies such as the sun, the with the utilization of inversion transition of moon and stars. Utilizing this, the clock time ammonia molecules in 1949. The accuracy of was determined and the calendar was orga- atomic frequency standards has reached 16 dig- nized. However, these are extremely large- its until today since it was fi rst developed with scaled movements that are beyond human con- an exceptional accuracy of 10 digits about 60 trol. While one could enumerate pitch control years ago. Furthermore, it can be said that with stringed instruments and wind instruments atomic frequency standards in the optical re- from such ancient times as one of what humans gion, whose accuracy has been dramatically created by themselves and could control, it is improved from the outset of the 21st century, is considered that full-fl edged art started from the the third leap. For the history of frequency discovery of the isochronism of the pendulum standards, refer to [1] listed in the bibliography by Galileo from the 16th century that the fre- below. quency of a pendulum depends only on its In this manner, fairly high-accuracy fre- length regardless of bob weights. In the 17th quency signals are easily available today due to century, Huygens invented pendulum clocks highly accurate oscillators. Now, how can we and furthermore mechanical clocks utilizing measure and evaluate its precision? Phases and the accurate frequency of machinery vibration amplitudes are the basic elements of frequency caused by a hairspring and a balance wheel. In and noise is further added to real signals. As KAJITA Masatoshi et al. 3 will be discussed below, since amplitude fl uc- Soon after the advent of atomic clocks, the tuations not usually problematic with good fre- accuracy of clocks was improved by about 3 quency signals, it is necessary to measure and digits compared with the crystal clocks avail- evaluate accuracy, stability, noise etc., for the able up until that time. Atomic clocks take the phase as a function of time with appropriate frequency of electromagnetic waves absorbed measures. This paper will give an explanation and emitted by atoms (neutral, ions) as a stan- of the measuring methods for frequency sig- dard. Originally called “atomic frequency stan- nals obtained from high-accuracy frequency dards,” atomic clocks do not function as clocks oscillators etc. and evaluating methods for until they are used together with crystal clocks these measures. 2 provides an overview the de- etc. and feedback errors. The differences be- gree of the accuracy of frequency oscillators tween atomic clocks and conventional clocks and defi nes quantity as the basis for frequency are detailed below: evaluation. 3 discusses the principle and basis (i) Electrons ranging around a nucleus can for frequency measurement and 4 explains its only acquire energy intermittently. The energy evaluating methods. 5 offers commentary on conditions are unique since they can only be classifi cations of noise and spectral expansion determined by Coulomb force between elec- and 6 presents a conclusion. trons and a nucleus. An atom can absorb and emit electromagnetic waves with frequency (v) 2 Accuracy of time/frequency corresponding to difference in each level ener- standards gy (Ei) (ν = Ei − Ek / h. Here, h stands for Planck constant). 2.1 Accuracy of frequency standards (ii) An atomic clock, unlike conventional (Why are atomic clocks accurate?) clocks, takes a single atom (in conditions of Along with technological developments for gas) as a standard and accordingly its condition oscillators discussed in 1, the accuracy of time is unique (interactions between neighboring at- displayed on a clock has been improved. The oms and molecules etc., are not unique in a frequency of the pendulum of a pendulum solid state). clock developed in the 17th century can be ex- In short, atomic clocks are the adaptation of pressed by: microcosmic phenomena to macro world. As a result, even commercially available atomic （1） clocks ensure an accuracy of more than 12 digits. This means that it will take several tens of Here, l stands for the length of the pendulum thousands of years to generate an error of one and g for gravitational acceleration. Still, this second. The advent of atomic clocks has en- period cannot be always stable as (a) the length abled us to experimentally demonstrate the of the pendulum varies at a rate of about 10−6 relativistic effect that time passes more slowly with a change in temperature by 1K, and (b) by force of movements and gravity. gravitational acceleration varies at a rate of However, also atomic clocks have limited about 10−7 with a locational change on the accuracy. What was shown in (i) above is not earth, especially with a change in altitude by completely correct since there is a width of Δν 1m. It holds true also with crystal clocks, whose = 1/2πτ in accordance with a limited time of τ oscillating frequencies depend on temperature because of uncertainty principle for energy and etc., that clocks gain or lose as temperature time (a fundamental principle of quantum me- changes. However, crystal clocks are more sta- chanics). τ is sometimes limited by the interac- ble than pendulum clocks as the rate of vibration time between atoms and electromagnetic tion frequency to frequency width (called Q waves and sometimes determined by the life- value) for the former is higher than that for the span of quantum state with a fi xed phase (lim- latter by 5–6 digits. ited by spontaneous emission, and collision 4 Journal of the National Institute of Information and Communications Technology Vol.57 Nos.3/4 2010 etc.). Also, electron orbits can be distorted by um and reaches 18 digits only from cumulative external electric fi elds and magnetic fi elds to uncertainty factors [2]. cause transition frequency shifts (one caused by electric fi elds is called Stark shift and one 2.2 Defi nition of frequency caused by magnetic fi elds Zeeman shift). What As the accuracy of clocks improves, it be- was shown in (ii) is also not completely accu- comes necessary to provide a defi nition for de- rate; even gaseous atoms collide against one termining certain length of time. Time is one of another or the wall of a container. Consequent- base quantities as well as length, mass, electric ly, the transition frequency shift at that moment current, thermodynamic temperature, amount appears as collision shift. There are also fre- of substance and luminous intensity that are de- quency shifts caused by relativistic effects from fi ned in the International System of Units (SI), the movements and gravity of atoms that freely and its basic unit is the second (s) [3]. For ex- fl y around (second-order Doppler shift and pressing time units longer than a second though, gravitational shift). the kilosecond (ks: 103 sec.) etc., could be used Once laser cooling could decrease the tem- but conventional units such as minute (min: perature of atoms as far as several μK, they be- 60sec.), hour (hour: 3600sec.) etc. are used in came sluggish and could be made to stand still most cases. Time units shorter than a second, at one place, which enabled us to take longer millisecond (ms: 10−3 sec.) and microsecond measuring time τ. In addition, with further im- ( μs: 10−6sec.), etc., are used as defi ned in SI. provements of the accuracy of clocks, the sec- Originally, one second was defi ned as ond-order Doppler shift decreased below 18 “1/86400 of the average rotation period of the digits, cesium fountain atomic clocks have earth”. The short-term variation at that time been able to measure with an accuracy of 16 was approximately 10−8. digits. Consequently, following the new defi ni- Previous clocks took microwave frequency tion: “one second shall be 1/331556925.9747 1‒50GHz as a standard that required a mea- of the revolution period of the earth” by the In- surement period of more than 10 days in order ternational Committee of Weights and Mea- to gain an accuracy of 16 digits. With the fre- sures in 1956, the Measurement Act of Japan quency shift and uncertainty on the same level, was also revised [4]. The accuracy of observa- taking high frequencies as a standard is more tion at this time was approximately 10−9. advantageous for shortening measuring time, The world’s fi rst atomic clock was devel- and many atomic quantum transitions in the oped with reference to the inversion transition optical region have narrow line width and are frequency of the ammonia molecule in 1949.

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