Effect that limits performance of advanced atomic clocks
The Dick effect (hereinafter; "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 formula_1, a measure of the expected statistical variation of fractional frequency as a function of averaging time formula_2. 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 formula_3 dependence for times longer than an "attack time" for the feedback loop. For increasing values of the measuring time formula_4, the stability approaches the limiting formula_5 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.
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.
Methodology.
Introduction.
Modern atomic frequency standards or clocks typically comprise a local oscillator (LO), an atomic system that is periodically interrogated by the LO, and a feedback loop to correct the frequency errors in the LO based on the results of that interrogation; thus locking the frequency of the LO to that of the atomic system. The effect describes a process that makes for imperfect locking, one that depends on details of the atomic interrogation protocol. Two steps are required in order to calculate this newly recognized impact of LO noise on the frequency stability of the "locked local oscillator" (LLO) that provides useful output for the frequency standard. These are: 
In contrast to other examples of photonic excitation of atoms or ions, this excitation process is a slow business, taking milliseconds or even seconds to accomplish on account of the extremely high Q factors involved. Instead of a photon striking a solid and ejecting an electron, here is a process where a coherent EM field consisting of many photons (slowly) drives each atom or ion in a cloud from their ground state into a mixed state, typically one with equal amplitudes in the ground and excited states.
Functional forms for formula_6 during the time that the atoms are being exposed to interrogating microwave or optical fields can be calculated using a "fictitious spin" model for the quantum mechanical state-transition process or by using an algebraic approach. These forms, in combination with constant values (typically zero or unity) during times when no signal is applied, allow the sensitivity function to be well defined over the entire interrogation cycle.
The discriminating power for both Ramsey Interrogation and Rabi Interrogation of atomic systems had been previously calculated, based on the quantum-mechanical response of an atom or ion to a slightly detuned drive signal. These previously calculated values are now seen to correspond to a time average of the sensitivity function formula_8, taken over the interrogation cycle.
Calculation of the sensitivity function.
"The concepts and results of calculations presented below can be found in the first papers describing the effect".
Each interrogation cycle in an atomic clock typically begins with preparation of the atoms or ions in their ground states. Let P be the probability that any one will be found in its excited state after an interrogation. The amplitude and time of the interrogating signal are typically adjusted so that tuning the LO exactly to the atomic frequency will give formula_16, that is, all of the atoms or ions being in their excited state. P is determined for each measurement by then exposing the system to a different signal that will generate fluorescence only for the (e.g.) excited state atoms or ions.
In order to obtain effective feedback using periodic measurements of P, the protocol must be arranged so that P has a sensitivity to frequency variations. The sensitivity to frequency variation formula_17 can then be defined as 
formula_18
where formula_19 is the interrogation time, so that the value of formula_17 characterizes the sensitivity of a measurement of P to a variation in frequency of the LO. Since P is maximized (at formula_16) when the LO is exactly tuned to the atomic transition frequency, the value of formula_17 would be zero for that case. Thus, for example, in