In this subsection, after a general introduction to the metrics of evaluating a clock, we will briefly discuss some common noise sources of our clock. A detailed analysis of the TACC-2 stability is given in Sec. 3.1.
1.1.3.1 Figures of merit
In general, locking the LO to the atomic transition realises a clock signal
νclk(t) = νat0(1 + + y(t)) (1.16) where νat0 denotes the unperturbed atomic frequency. We will use frequencies in Hz for convenience. We identify two terms that are commonly used for evaluating clocks.
Accuracy denotes a systematic shift from the unperturbed atomic frequency. It de-pends on the particular realisation and its uncertainty quantifies the accuracy of the clock.
It is less of a concern for secondary standards for applications, as long as the systematic inaccuracies can be calibrated with the primary standards. However, the uncertainty of the systematic error cannot be distinguished from the random error of a measurement, therefore also contributes to the clock instability. The clock accuracy is not studied in this thesis.
Stability y(t) is the fractional frequency fluctuation and quantifies the stability of the clock. More specifically, one also distinguishes short-term (seconds) and long-term (hours, days) stability, depending on the application. The noise mechanisms and limits are also very different. In this thesis, we focus on the applications of quantum technologies, which target on improving the short-term stability.
Allan variance The Allan variance is the standard way to characterise the clock stability.
It resolves the problem that the standard variance is not well defined at low frequencies if the clock signal has flicker noise or drift. Specifically, the Allan variance (AVAR) is defined as [74]:
σy2(τ) = 1 2(M − 1)
M −1
X
i=1
(¯yi+1−¯yi)2 (1.17)
It is the expectation value of the two-sample variance, where ¯yi(τ)’s are contiguous samples
which reveals the noise at different time scales.
If we consider the noise spectrum (a function of the Fourier frequency), the Allan variance can be understood as a bandpass filter near the frequency 1/(2τ). The scaling of σy2(τ) as a function of τ then reflects different noise sources in the power spectrum of the clock signal [75]. For example, white frequency noise, which usually dominates in timescales between seconds to hours, scales as τ−1. The flicker frequency noise appears flat in σy2 versus τ, and random walk frequency noise diverges as ∼ τ.
We will be later using the Allan deviation which is the square root of the Allan variance.
We assume that the clock has white frequency noise in the timescale of interest such that
σy(τ) = σy,shot
s Tc
τ (1.19)
where σy,shot is the Allan deviation of one clock cycle, with cycle time Tc.
In each clock cycle, the atomic frequency is only probed by the LO during the Ramsey time. The frequency difference is affected by the sensitivity function g(t) of the Ramsey sequence [76]: includes shift of the ensemble atomic frequency that is subject to fluctuations, contributing to clock instability. Overall, the uncertainty of a clock measurement (one cycle) is given by the uncertainty of ∆ν, σ∆ν, and the uncertainty in the detection (of P↑), σP↑: where Qc is the clock quality factor (Eq. 1.9). We will briefly discuss some major noise sources in the following.
1.1.3.2 Noise in P↑ measurement
Uncertainties in determining P↑ originate from different mechanisms:
1In practice, for limited clock measurement samples, one uses the overlapping Allan variance or Total variance [74].
Quantum projection noise As we have already discussed in the introduction, projec-tive measurements of uncorrelated atoms lead to the binomial distribution of the outcome.
Operating at mid-fringe with P↑ = 12, the variance of N↑ from measuring N atoms reads:
Var(N↑) = P↑(1 − P↑)N = N/4. Hence the noise in P↑ due to QPN reads σP↑ = 1/(2√ N).
One can only fight with it by increasing the atom number in an unfavourable scaling N−1/2. Moreover, density-related frequency shift and technical difficulties also limit this approach, as we will see in particular for TACC (Sec. 3.1).
Detection noise The error in counting atoms in the two states also leads to an uncertainty in P↑. It is usually of technical origin and is subject to the experimental methods. It may also boil down to some form of shot noise: for example, the photon shot noise of the imaging beam or the fluorescence. In TACC, the primary detection is absorption imaging. The detection noise is nearly limited by the photon shot noise.
Other technical noise In a Ramsey sequence, we assume two π/2 pulses. However, noise in the pulse area of the π/2 pulses due to e.g. power fluctuations of the LO field directly leads to error in the final P↑.2 Apart from technical improvements, slow variations can be rejected by e.g. probing alternately on both sides of the Ramsey fringe and only extracting the frequency from the differential signal.
1.1.3.3 Local oscillator noise
Most naively, as a phase modulo π is measured in a Ramsey scheme, the phase is subject to an ambiguity when its deviation might exceed π. Moreover, a large frequency deviation reduces the sensitivity of the Ramsey spectroscopy (cf. Eq. 1.9, with P↑ different from 1/2).
In other words, the clock has a fairly narrow bandwidth in correcting the LO frequency.
More profoundly, as we can see from Eq.1.20, how precisely the LO frequency is measured is also determined by the sensitivity function. The clock is blind to the LO noise outside the Ramsey time. With the TR only a small fraction of the cycle time Tc, the clock resembles a discrete data acquisition that periodically samples the LO frequency and its fluctuations, suffering from aliasing such that high frequency LO noise (multiples of the sampling frequency 1/Tc) can further degrade the clock stability.
This is known as the Dick effect, which has been one of the most important noise sources for optical clocks today. The Cs fountain clock at SYRTE is supported by a cryogenic sapphire oscillator, which is not accessible for broader metrological applications. In fact, the Dick effect can be the most prominent limit for compact devices where only a quartz crystal is affordable.
This also motivates new techniques such as non-destructive measurements to use the same atoms in multiple clock measurements [56]. In the case where the excess LO noise limits the Ramsey time, multiple short Ramsey sequences sharing a single phase of atom preparation effectively improve the clock duty cycle (TR/Tc), alleviating the Dick effect.
In TACC, the Dick effect is a major contribution to clock instability. Although recycling atoms with non-destructive measurements has not been studied in this thesis, it is one of the two major objectives of TACC-2 and experiments will be carried out in the near future.
2Although in principle errors in the preparation also influence the collisional shift during the Ramsey sequence, see e.g. [77].
1.1.3.4 Collisional shift
As introduced in Eq.1.13, the collisional shift is one of the most important systematic effects in atomic fountains and has a major contribution to the uncertainty [78,79]. The situation is much severer for trapped-atom clocks in which the density can be 4 orders of magnitude higher.
Correction to the systematic error can be applied shot-to-shot based on the measurement of the total atom number which determines the atomic density. However, this correction is compromised by the detection noise in the atom number. Furthermore, in the presence of atom loss, only the final atom number is known. The statistical nature of atom loss imposes an uncertainty in the average atom number during the interrogation time, hence in the collisional shift. In fact, as we will see, for TACC-2 the uncertainty of this correction can be an important source of clock instability, if a large part of the atoms is lost.
1.1.3.5 Other instabilities in TACC
There are other technical fluctuations which deteriorate the clock stability in TACC. Here I only take an overview and a detailed analysis will be given in Sec. 3.1.
Magnetic field fluctuations As I mentioned, the pseudo-magic trap for TACC requires a bias field slightly lower than the magic field. In this case, however, the Zeeman shift is more sensitive to magnetic field variations. Further is the bias field away from the magic field, the bigger the contribution from the magnetic field fluctuation. As we will see, this is one of the major noise sources in TACC.
Atom temperature and its fluctuations Both the Zeeman shift and the density shift depend on the atom temperature, but an optimum field exists at which the total shift is first-order insensitive to temperature fluctuations. However, this optimum field differs from the magic field hence the insensitivity to the temperature fluctuation and that to the magnetic field fluctuation can not reconcile. The overall optimum bias field inevitably suffers from noise both in magnetic field and in atom temperature.