The Dick effect (hereinafter referred to as "the effect") is an important limitation to frequency stability for modern atomic clocks such as atomic fountains and optical lattice clocks. It is an aliasing effect: High frequency noise in a required local oscillator (LO) is aliased (heterodyned) to near zero frequency by a periodic interrogation process that locks the frequency of the LO to that of the atoms. The noise mimics and adds to the clock's inherent statistical instability, which is determined by the number of atoms or photons available. In so doing, the effect degrades the stability of the atomic clock and places new and stringent demands on LO performance.
For any given interrogation protocol, the effect can be calculated using a quantum-mechanical sensitivity function, together with the spectral properties of the LO noise. This calculational methodology, introduced by G. John Dick, is now widely used in the design of advanced microwave and optical frequency standards, as well as in the development of methodologies for atomic-wave interferometry, frequency standard comparison, and other areas of measurement science.
Background
General Frequency stability
The frequency stability of an atomic clock is usually characterized by the Allan deviation ,Allan, D. Statistics of Atomic Frequency Standards, pages 221–230. Proceedings of the IEEE, Vol. 54, No 2, February 1966. a measure of the expected statistical variation of fractional frequency as a function of averaging time . Generally, short-term fluctuations (frequency or phase noise) in the clock output require averaging for an extended period of time in order to achieve high performance. This stability is not the same as the accuracy of the clock, which estimates the expected difference of the average frequency from some absolute standard.
Excellent frequency stability is crucial to a clock's usability: Even though it might have excellent accuracy, a clock with poor frequency stability may require averaging for a week or more for a single high precision test or comparison. Such a clock would not be as useful as one with a higher stability; one that could accomplish the test in hours instead of days.
Stability and operation of atomic clocks
Instability in the output from an atomic clock due to imperfect feedback between atoms and LO was previously well understood. This instability is of a short-term nature and typically does not impact the utility of the clock. The effect, on the other hand gives rise to frequency noise which has the same character as (and is typically much larger than) that due to the fundamental photon– or atom–counting limitation for atomic clocks.
With the exception of hydrogen and ammonia (hydrogen maser, ammonia maser), the atoms or ions in atomic clocks do not provide a usable output signal. Instead, an electronic or optical local oscillator (LO) provides the required output. The LO typically provides excellent short-term stability; long-term stability being achieved by correcting its frequency variability by feedback from the atoms.
In advanced frequency standards the atomic interrogation process is usually sequential in nature: After state-preparation, the atoms' internal clocks are allowed to oscillate in the presence of a signal from the LO for a period of time. At the end of this period, the atoms are interrogated by an optical signal to determine whether (and how much) the state has changed. This information is used to correct the frequency of the LO. Repeated again and again, this enables continuous operation with stability much higher than that of the LO itself. In fact, such feedback was previously thought to allow the stability of the LO output to approach the statistical limit for the atoms for long measuring times.
The effect
The effect is an additional source of instability that disrupts this happy picture. It arises from an interaction between phase noise in the LO and periodic variations in feedback gain that result from the interrogation procedure. The temporal variations in feedback gain alias (or heterodyne) LO noise at frequencies associated with the interrogation period to near zero frequency, and this results in an instability (Allan deviation) that improves only slowly with increasing measuring time. The increased instability limits the utility of the atomic clock and results in stringent requirements on performance (and associated expense) for the required LO: Not only must it provide excellent stability (so that its output can be improved by feedback to the ultra-high stability of the atoms); it must now also have excellent (low) phase noise.
A simple, but incomplete, analysis of the effect may be found by observing that any variation in LO frequency or phase during a dead time required to prepare atoms for the next interrogation is completely undetected, and so will not be corrected. However, this approach does not take into account the quantum-mechanical response of the atoms while they are exposed to pulses of signal from the LO. This is an additional time-dependent response, calculated in analysis of the effect by means of a sensitivity function.
Quantitative
The graphs here show predictions of the effect for a trapped-ion frequency standard using a quartz LO. In addition to excellent stability, quartz oscillators have very well defined noise characteristics: Their frequency fluctuations are characterized as flicker frequency over a very wide range of frequency and time. Flicker frequency noise corresponds to a constant Allan deviation as shown for the quartz LO in the graphs here.
The "expected" curve on the plot shows how stability of the LO is improved by feedback from the atoms. As measuring time is increased (for times longer than an attack time) the stability steadily increases, approaching the inherent stability of the atoms for times longer than about 10,000 seconds. The "actual" curve shows how the stability is impacted by the effect. Instead of approaching the inherent stability of the atoms, the stability of the LO output now approaches a line with a much higher value. The slope of this line is identical to that of the atomic limitation (minus one half on a log-log plot) with a value that is comparable to that of the LO, measured at the cycle time, as indicated by the small blue (downwards) arrow. The value (the length of the blue arrow) depends on the details of the atomic interrogation protocol, and can be calculated using the sensitivity function methodology.
The second graph here indicates how various performance aspects of the LO impact achievable stability for the atomic clock. The dependence labeled "Previously Analyzed LO Impact" shows the stability improving on that of the LO with an approximately dependence for times longer than an "attack time" for the feedback loop. For increasing values of the measuring time , the stability approaches the limiting dependence due to statistical variation in the numbers of atoms and photons available for each measurement.
The effect, on the other hand, causes the available stability of the frequency standard to show a counter-intuitive dependence on high-frequency LO phase noise. Here stability of the LO at times less than the Cycle Time is shown to influence stability of the atomic standard over its entire range of operation. Furthermore, it often prevents the clock from ever approaching the stability inherent in the atomic system.
History
Within a few years of the publication of two papers laying out an analysis of LO aliasing, the methodology was experimentally verified, generally adopted by the Time and Frequency community, and applied to the design of many advanced frequency standards. It was also clarified by Lemonde et al. (1998) with a derivation of the sensitivity function that used a more conventional quantum-mechanical approach, and was generalized by Santarelli et al. (1996) so as to apply to interrogation protocols without even time symmetry.
Where performance limits for atomic clocks were previously characterized by accuracy and by the photon– or atom–counting limitation to stability, the effect was now a third part of the picture. This early stage culminated in 1998 in the publication of four papers in a Special Issue on the Dick effect for the journal IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control.
Impact
Perhaps the most significant consequence of the Dick analysis is due to its presentation of a mathematical framework that enabled researchers to accurately calculate the effect based on the methodology and technology used for many very different atomic clocks. Since the effect is generally the most significant limitation to stability for advanced frequency standards, a great deal of work since that time has focused on amelioration strategies. Additionally, the effect methodology and the sensitivity function have enabled significant progress in a number of technical areas.
The value for the limiting instability due to the effect is determined by the interrogation protocol combined with phase noise properties of the LO. Consequences of the theory have been worked out for several different kinds of atomic clocks.
Laboratories working on microwave Atomic fountain clocks have turned to cryogenic or optical LO techniques to replace the quartz ultrastable oscillator (USO) previously used as a reference for microwave atomic frequency standards. While the instability of the quartz USO could be reduced by feedback to effectively realize the inherent atomic stability in a clock, its phase noise, transformed by the effect, was now the primary source of the clock's instability, as shown by the graph in the previous section. Cryogenic and optical techniques can provide both the stability and phase noise required to realize the inherent stability of the atomic standard. These atomic clocks typically operate by tossing a ball of laser-cooled atoms upward through a microwave cavity that acts to start the clock in each individual atom. As the atoms return downward, they again traverse this same cavity where they receive a second microwave pulse that stops their clocks. The ball then falls through an optical interrogation apparatus below the cavity that "reads out" the phase difference between the microwaves (the LO) and the atoms that developed during their flying time. This is repeated again and again; a sequential process that gives rise to the effect. On a smaller scale, the stability of Rb vapor clocks using a pulsed optically pumped (POP) technique has improved to such an extent that the effect due to a quartz USO LO has become the limiting performance factor. Developments in a combined USO--DRO (dielectric resonator oscillator) LO technology now enable improved performance.
Optical clocks have achieved the highest stability of any clocks, and are on track to replace Cesium fountain clocks as the definition of the second. However, as (Katori, 2011) states: "In optical lattice clocks, however, owing to the significantly low QPN (quantum projection noise), the Dick effect becomes the major obstacle in achieving higher stability". Analysis of the effect and its consequence as applied to optical standards has been treated in a major review (Ludlow, et al., 2015) that lamented "the pernicious influence of the Dick effect", and in several other papers.