Indian Journal of Pure & Applied Physics Vol. 45, December 2007, pp. 945-949

Determination of Allan deviation of Cesium atomic for lower averaging

P Banerjee, Arundhati Chatterjee & Suman Time and Frequency Section, National Physical Laboratory, New Delhi 110 012 E-mail: [email protected] Received 15 January 2007; revised 11 September 2007; accepted 8 October 2007

Absolute Allan deviation of the Cesium clock for averaging time ( τ) of 5 days or more may be calculated from the corresponding data of circular T published by Bureau International des Poids et Mesures (BIPM). For lower values of τ, the Allan deviation may indirectly be found from the extrapolation of these values through τ-1/2 fit as recommended by CCTF- WGMRA guidelines. Absolute Allan deviation may also be directly found out by inter-comparison of minimum three assuming that the noise in all clocks is fully uncorrelated. This paper analyses the values of Allan deviation determined by the direct method keeping in mind the limitation of the measurement system. These values of Allan deviation tally well with those found from the data of circular T. Keywords : Cesium , Allan deviation, Lower averaging time

1 Introduction Allan deviation [ σ(τ)] of the frequency departure A stable signal generator in various fields such as averaged over time. Another method of characterizing physics, radio communication, radar, space vehicle, noise in the frequency fluctuations is by means of the tracking, navigation, timing applications etc is the key power spectral density [ Sf(f)]. After the signing of component. An oscillator has some nominal MRA, the determination of uncertainties of reference frequency at which it operates. The frequency frequency source to establish the calibration stability of an oscillator is a term used to characterize measurement capabilities (CMC) has gained how small the frequency fluctuations are of the momentum. The Allan deviation is normally used to oscillator signal. Frequency stability is the degree to quantify the uncertainties of time and frequency which an oscillating signal produces the same value reference source against which time and frequency of frequency for any interval, throughout a specified devices are calibrated. So it becomes necessary for period of time. It is of interest to note that the practical requirement of calibration and testing to frequency stability actually refers to the frequency determine the Allan or Allan deviation [ σ(τ)] instability. The frequency instability is the of the source averaged over the time ( τ) ranging from spontaneous and/or environmentally caused frequency few seconds to few tens of seconds. change within a given time interval. Characterization Spectral power densities are theoretical concept of phase and frequency instabilities of high-quality involving infinite duration process, infinite frequency time and frequency sources has been of great range and true average. In practice only finite duration importance and has drawn the attention of scientists process are available. The spectrum analyzers have 1- 4 since mid sixties of last century . Given a set of data non-zero bandwidth frequency window with lower of the fractional frequency or time fluctuations and upper frequency limit. A lower limit of 1 Hz and between a pair of oscillators, it is useful to upper limit of few kilohertz may be reached using characterize these fluctuations with reasonable and present-day technology of FFT analyzer. The spectral tractable models of performance. The fluctuations that power density may be used to find two sampled are random in nature have been considered. These can variance σ2(τ) through the standard formulae 5. But usually be best characterized statistically. If we the boundary condition of lower limit of bandwidth of survey the literature, two major methods of specifying FFT does not allow us to find σ2(τ) for τ larger than the frequency stability of an oscillator may be one or two seconds. It may, thus, be necessary to 2 identified. One is time domain method namely the attempt to find σ y(τ) for average time τ of few 946 INDIAN J PURE & APPL PHYS, VOL. 45, DECEMBER 2007

seconds to few tens of seconds directly [i.e. in time . The most commonly used measure is the domain]. two-sample (also known as Allan This paper attempts to address the problems of deviation) which is the square root of the two-sample determining the frequency stability in time domain for zero dead-time variance (also designated as Allan lower averaging time. These problems have more variance). The normal Allan, or 2- sample variance is relevance particularly to a precision oscillator like defined as: Rubidium clocks, Cesium clocks and Hydrogen masers which may normally be used to maintain the 1 M −1 2 yt yt 2 … (4) national time standard. σy() τ=∑ [( i + τ )()] − i 2(M − 1) t=1

2 Definition of Allan Deviation where y(t) is the ith of M fractional frequency values One must also realize that any frequency averaged over the measurement interval τ. measurement involves two oscillators. It is impossible In terms of phase data, the Allan variance may be to purely measure only one oscillator. In some calculated as: instances ,one oscillator may be enough better than the other that the fluctuations measured may be 1 considered essentially those of the latter. However, in σ2 ( τ ) = y 2(N − 2) τ 2 general, because frequency measurements are always N−2 … (5) 2 dual, it is useful to define the normalized frequency ×∑[(xti +− 2)2(τ xt i ++ τ ) x i ()] τ off set as: t=1

where x(t) is the i th of the N = M+1 phase values ν− ν y( t ) = 1 0 … (1) spaced by the measurement interval τ. ν0 The terms x(t+τ) and x(t) are proportional to instantaneous phase (time) difference obtained from y(t) is a dimensionless quantity and useful in the comparison between two clocks at date ( t+τ) and t describing oscillator and clock performance. The time and have the dimension of time. In any frequency and deviation, x(t), of an oscillator is related to y(t) by: phase measurement, one clock is compared with respect to a reference clock. So, the measurement is a t φ(t ) relative one, so also Allan deviation. But the focus of x() t= y () t dt = … (2) this paper relates to the absolute Allan deviation of an ∫ 2πν 0 0 oscillator.

Since it is impossible to measure instantaneous 3 Allan Deviation through Data of Circular T frequency, any frequency or fractional frequency International Bureau of Weights and Measures measurement always involves some sample time, (Bureau International des Poids et Mesures i.e. BIPM) some time window through which the oscillators are in Paris, has been coordinating the task of realizing observed; whether its a picosecond, a second, or a International Atomic Time (TAI) and Universal day, there is always some sample time. So when Coordinated Time (UTC). To achieve this, the remote determining a fractional frequency, y(t), the time clocks are compared through common-view GPS deviation is measured say starting at some time t and method. These data are sent to BIPM through again at a later time, t + τ . The difference in these two internet/E-mail at regular interval of time. BIPM uses time deviations divided by τ gives the average all these data to generate a smooth time scale through fractional frequency over that period: robust software, which is virtually the weighted average of clocks of all participating laboratories. xt(+τ ) − xt () y( t ) = … (3) This software generated time scale combining all the τ received data is named as Universal Coordinated Time 6 (UTC). The software also generates the status Tau, τ, may be called the averaging time. of the time scale of contributing laboratories with The random frequency stability of an oscillator, in respect to UTC through a circular T (published by time domain, may be estimated by several sample BIPM). BANERJEE et al. : ALLAN DEVIATION OF CESIUM ATOMIC CLOCK 947

National Physical Laboratory, New Delhi, India The above method has the following practical (NPLI) maintains the time scale of Indian Standard difficulties. Time (IST) with the help of a commercial Cesium 1. It is necessary that the clock should be linked to atomic clock (model HP5071A). The time scale BIPM through GPS network to make the availability maintained by NPLI is designated as UTC(NPLI). of xi ( t)’s values with respect to the virtually ideal The status of time scale UTC (NPLI) is, thus, also clock. available through circular T which records (UTC- 2. One should also have the feedback data from UTC(NPLI)) at the interval of 5 days. Time scale circular T for several months to get sufficient number generated by UTC is for long term maintenance of of data to find σy(τ ). time scale. Long term normally refers to the averaging time of more than one day. BIPM, thus, has decided 4 Direct Method to generate data in Circular T every 5 days. The data Using the data of circular T, the determination of circular T corresponding to UTC(NPLI) defines stability is quite time consuming. Further, it is [UTC(NPLI)-UTC] which is the local values of x(t). necessary to find the Allan deviation for lower UTC, being the average of more sixty atomic clocks, averaging time directly. One possible direct method is may be considered to be a virtual noiseless oscillator. to intercompare few clocks directly. In the direct Thus, it may be assumed that these x(t) values method, the phases of two Cesium clocks are correspond to the time differences (i.e. x(t)) with compared at a regular interval of time τ with the help respect to an ideal oscillator. So Allan deviation, of a time interval counter (TIC). The measured data determined out of these data, may be assumed to be may be assumed to correspond to xjk ( t)’s of one clock the absolute Allan deviation of the Cesium clock, j with respect to the other clock k. With the help of not a relative one . So, these values of x(t)’s are used intercompared data, one may, thus, find out σ (τ) (the σ τ jk in Eq. (2) to find 1( ) (implying absolute value of value of σ (τ) for the clock #j with respect to that of σ τ τ y y( ) for the Cesium clock #1) for averaging time of clock #k) using Eq. (5). These values of Allan 5 days or more. These have been plotted in Fig. 1. deviation are, again, relative ones, not absolute ones. The status UTC (NPLI) is updated every 5 days in But we may note that the noise in all the cesium Circular T. So Allan deviation for the averaging time clocks is absolutely un-correlated. So one may write: less than 5 days cannot be directly found out through 7 these data. But CCTF-WGMRA guideline 2 2 2 - -1/2 σjk (τ) = σj (τ) + σk (τ) j, k=1,2,3… … (6) recommends that τ fit may be applied to calculate the values of Allan deviation for lower averaging σ σ τ (e.g. j(τ) is absolute value of y(τ) for clock #j) time. The Allan deviation for lower values of may The Eq.(6) may be used to find out independent be obtained from the extraploted line as shown in (absolute) value of the Allan deviation (i.e. σ (τ)) for Fig. 1. k each of the clocks by intercomparing several clocks. The Eq.(6) dictates that this method requires minimum three clocks to find unique solutions for

σj(τ)’s. Thus, for three clocks, using all combinations of Eq. (6), it is quite possible to find out Allan deviation of each clock independently by using the following relations:

1 στ2()= (( στστστ 2 () + 2 () − 2 ()) … (7) 12 12 13 23

1 στ2()= (( στστστ 2 () + 2 () − 2 ()) … (8) 22 23 21 13

21 2 2 2 στ3()= (( στστστ 32 () + 31 () − 12 ()) … (9) Fig. 1 Allan deviation of Cesium clock 1 at NPLI 2 948 INDIAN J PURE & APPL PHYS, VOL. 45, DECEMBER 2007

NPLI has utilized three cesium clocks for this 5 Limitation of Measurement System purpose. A high resolution time interval counter (TIC) The universal counter (model HP53132A) has been to compare the phases of two clocks has also been used in this measurement as TIC and this TIC has used as shown in Fig. 2. The selector switch helps in RMS resolution of 300 ps 8. So the uncertainty (1 σ) of selecting one chosen pair of Cesium clocks for frequency measurement through this counter is measurements. Linking of TIC to the computer through RS232C port, a serial port, limits the speed to 300× 10 −12 one measurement every four seconds. So, minimum ∆y = … (10) 3τ value of τ (i.e. averaging time) has been four seconds. Each of three sets of measurement has been This fact and a close look at Eqs (1 −4), lead us to carried out for 6 hr continuously at the sampling rate conclude that the limiting values of Allan deviation of four seconds. Each set of these measured values caused by the limitation of measurement ( Appendix (i.e. xjk ( t)) are used in Eqs (5) to find σ12 (τ), σ13 (τ ) σ A) is: and 23 (τ). Using these values of σjk (τ)’s in Eqs (7 −9), σ s (i.e. for each of three Cesium atomic clocks) k’ 3× 10 −10 have been found out. For example, with help of ∆σ =3 ×∆=y … (11) 1 τ Eq.(7) σ1 (τ) has been find out as shown in Table 1 and Fig. 2. As UTC(NPLI) is maintained through Cesium Clock #1, so the emphasis has been given for Eq. (11) implies that the system cannot measure the stability whose value is actually lower than the σ1(τ). ∆σ corresponding 1. In other words, if any value of stability determined by the direct method becomes lower than the uncertainty limit for that averaging time, the stability value cannot be accepted. Thus, the line indicating the limit of the measurement system based on Eq. (11) has been shown in Fig. 1. So this line implies that accepted values of Allan deviation determined through the direct method should lie above this limiting line.

6 Conclusions Values of σ (τ) of Cesium atomic clock calculated 1 by the direct method for τ higher than 100’s of Fig. 2 Experimental set up for determination of Allan deviation seconds have been found to lie above the limit line by direct method (Fig. 2) and thus, they can be accepted to be reliable ones (i.e. for τ-values of 256 s, 512 s, 1024 s and 2048 s ). But for averaging time lower than 100 s, Table 1Calculation of absolute value of Allan deviation of Cs #1 values of σ 1(τ)’s have been found to lie almost on the limit line and thus, these values cannot be reliable. τ σ12 σ 13 σ 32 σ 1 The reliably calculated values of Allan deviations (i.e. for τ higher than 100 s) may be used for extrapolation 4 1.05E-10 4.13E-11 4.34E-11 7.36E-11 through τ−1/2 law to find Allan deviations for lower 8 5.29E-11 2.29E-11 2.38E-11 3.71E-11 τ -1/2 16 2.85E-11 1.30E-11 1.37E-11 1.99E-11 values of τ. It is of interest to note that line fit by 32 1.33E-11 7.66E-12 8.14E-12 9.20E-12 circular T method almost coincides with that by direct 64 7.95E-12 5.02E-12 5.22E-12 5.53E-12 method confirming the reliability of the measurement 128 4.50E-12 3.37E-12 3.70E-12 2.99E-12 by direct method. For example, for τ of 100s, Allan -12 -12 256 2.62E-12 2.43E-12 2.27E-12 1.95E-12 deviation is 2.58 ×10 and 2.04 ×10 by direct 512 1.38E-12 1.85E-12 1.75E-12 1.06E-12 method and circular T method ,respectively. 1024 9.58E-13 1.12E-12 8.56E-13 8.48E-13 So, the direct method may be used to find the 2048 6.35E-13 7.04E-13 5.59E-13 5.41E-13 absolute Allan deviation for lower values of τ. This BANERJEE et al. : ALLAN DEVIATION OF CESIUM ATOMIC CLOCK 949

does not require the linking of clocks to BIPM and 2 Barnes J A, Chi A R, Cutler L S, Healey D J, Leeson D B, thus, one need not wait for a long time to get sufficient McGunigal T E, Mullen J A, Smith W L, Sydnor R, Vessot R F & Winkler G M R, IEEE Trans Instr Meas , IM-20 (1971) data from BIPM. But the limitation of the measurement 105. system has to be kept in mind. For example, to get 3 Cutler L S & Searle C L, IEEE , 54 (1983) 136. Allan deviation for averaging time lower than 100 s 4 Rutman J, Proc. IEEE, 66 9 (1978) 48. directly (i.e. without extrapolation) particularly for 5 Kartaschoff P, Frequency and Time , Academic Press, 1978. precise clocks like atomic clocks, one needs to have a 6 Annual report of the BIPM Time Section , 18 (2005) 89. TIC with an rms resolution better than 300 ps. 7 Uncertainty extrapolation for T&F CMC entries , CCTF WGMRA Guideline 3 (Rev. 20021210). References 8 Operating Guide, Universal Counter HP 53131A/53132A, 1 Allan D W, Proc. IEEE , 54 (1966) 221. 1999.

Appendix A

Uncertainty of Allan deviation determined through direct ∆σ ∆ σ ∆ σ ∆σ 12≈ 13 ≈ 23 ≈ y … (3A) method σ σ σ σ Using Eq. (1) for Allan deviation and noting that y(t+τ) and y(t) 12 13 23 y are correlated, one may write : Thus, with the help of Eqs(3A) and (1A), (2A) reduces to : 2   2 ∆σ ∆y  2 y  = … (1A)     2   ∆σ1 ∆ y σ y  y    =3  … (4A) σ  y  1 ∆ represents the uncertainty of the corresponding parameter and ∆y is, thus, the uncertainty of the frequency measurement through Eq. (4A) simplifies to: TIC. σ σ It may be noted that all ’s in Eq. (4) are partially uncorrelated. ∆σ =3 1 ∆ y … (5A) So it follows: 1 y

2 2 2 2  ∆σ   ∆ σ   ∆ σ  ∆ σ  σ1 21 = 2  12  + 2 13  + 2 23   … (2A) Assuming, ≈1 , one may write: σ σ σ σ  y 1   12   13  23  

For the practical purpose, one may assume: ∆σ1 =3 ∆ y … (6A)