B Tutorial on PTTI Frequency Standards

ATOMIC FREQUENCY STANDARDS AND CLOCKS

General

Atomic frequency standards are all based on particular frequency resonances in, ideally, isolated atoms of an isotope of an element such as cesium-133, which happens to be the basis for the present definition of frequency or time interval. The term atom is used here, but for present purposes, it can also mean ion or molecule. All atoms of the same element and isotope are absolutely identical in structure and therefore have identical resonances and resonance frequencies. This is a fundamental fact of nature and is described by quantum mechanics. It explains why atomic frequency standards can be so accurate.

Quantum mechanically, isolated atoms have a number of energy levels, or states, determined by their structure. Atoms in a given state can be induced to make transitions to another state through interaction with an electromagnetic field. The interaction is not strong unless the frequency v of the field is close to ΔE/h, where h is Planck’s constant and ΔE is the difference in energy of the two states. This is the basis for atomic resonances.

At room temperature isolated atoms are usually in the lowest state, called the ground state. The ground state may be split into a number of substates of small energy separation, called hyperfine splitting, by interactions of the atomic electrons with the nuclear magnetic moment. These substates, in turn, may be split by an applied magnetic field. These energy splittings are typically small compared with the thermal energy at room temperature, so the atoms are almost equally liable to be in any of these states. The frequencies associated with these hyperfine splittings are in the microwave or millimeter-wave range. Atoms with splitting in the ground state are typically used for microwave atomic frequency standards. They include hydrogen, the alkalis, and some singly ionized atoms such as mercury. Figure B.1 shows the energy levels in the ground state of atomic hydrogen as a function of external magnetic field.

If the atoms are excited to higher-energy electronic states, they relax with a characteristic relaxation time to some lower energy state by emitting electromagnetic radiation at frequency v = ΔE/h. The



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An Assessment of Precision Time and Time Interval Science and Technology B Tutorial on PTTI Frequency Standards ATOMIC FREQUENCY STANDARDS AND CLOCKS General Atomic frequency standards are all based on particular frequency resonances in, ideally, isolated atoms of an isotope of an element such as cesium-133, which happens to be the basis for the present definition of frequency or time interval. The term atom is used here, but for present purposes, it can also mean ion or molecule. All atoms of the same element and isotope are absolutely identical in structure and therefore have identical resonances and resonance frequencies. This is a fundamental fact of nature and is described by quantum mechanics. It explains why atomic frequency standards can be so accurate. Quantum mechanically, isolated atoms have a number of energy levels, or states, determined by their structure. Atoms in a given state can be induced to make transitions to another state through interaction with an electromagnetic field. The interaction is not strong unless the frequency v of the field is close to ΔE/h, where h is Planck’s constant and ΔE is the difference in energy of the two states. This is the basis for atomic resonances. At room temperature isolated atoms are usually in the lowest state, called the ground state. The ground state may be split into a number of substates of small energy separation, called hyperfine splitting, by interactions of the atomic electrons with the nuclear magnetic moment. These substates, in turn, may be split by an applied magnetic field. These energy splittings are typically small compared with the thermal energy at room temperature, so the atoms are almost equally liable to be in any of these states. The frequencies associated with these hyperfine splittings are in the microwave or millimeter-wave range. Atoms with splitting in the ground state are typically used for microwave atomic frequency standards. They include hydrogen, the alkalis, and some singly ionized atoms such as mercury. Figure B.1 shows the energy levels in the ground state of atomic hydrogen as a function of external magnetic field. If the atoms are excited to higher-energy electronic states, they relax with a characteristic relaxation time to some lower energy state by emitting electromagnetic radiation at frequency v = ΔE/h. The

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An Assessment of Precision Time and Time Interval Science and Technology FIGURE B.1 Ground-state energy levels of atomic hydrogen as a function of magnetic H′. The transition used for the frequency standard is (0,0) – (1,0), which depends on magnetic field only in second order. relaxation time from an excited state depends on the atomic structure and the energy of the state and varies from nanoseconds for moderate state energies and highly allowed transitions to seconds for what are called forbidden transitions. These transitions are in the optical, infrared, and ultraviolet ranges. Those with long relaxation times are useful for isolated atom optical frequency standards. The frequency linewidth Δv associated with a transition is of the order 1/(πT), where T is the relaxation time. The relaxation time for transitions between ground-state levels is typically years for isolated atoms, so the relaxation can be neglected for isolated atom microwave standards. However, in gas cell microwave standards or other standards in which atomic collisions occur, collisions shorten the relaxation time and determine the linewidth. In isolated atom microwave standards, the linewidth is roughly proportional to the reciprocal of the interaction time of the electromagnetic field with the atoms. This can be understood from Fourier transform relationships. The quality factor of a resonance is Ql = v/Δvl, where Δvl is the linewidth determined by collisions, interaction time, or the natural linewidth, whichever gives the largest linewidth. The achievable accuracy (small numbers are desired here) of an isolated atom frequency standard (essentially no collisions) is proportional to 1/Ql, so a high-frequency v and a small-linewidth Δvl are desired. This is what makes the optical standards attractive. Their accuracy can be orders of magnitude better than microwave standards. With regard to the interaction with the electromagnetic field, atomic transitions to a higher energy state absorb a quantum of energy hv, called a photon, from the field. Transitions to a lower energy state add a photon to the field. This is simply conservation of energy. Typically, there is a small, homogeneous, static magnetic field in the interaction region to provide what is called a quantization axis and also to separate magnetic-field-dependent states from those used for the frequency standard. The frequency standard states usually have only a small, second-order dependence on magnetic field. A technique is provided to detect either the transition probability of the atom or the change in the electromagnetic field and convert it to an electrical signal. This allows the atomic resonance to be used

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An Assessment of Precision Time and Time Interval Science and Technology in a frequency standard. To get a large signal-to-noise ratio, necessary for good short-term stability, as described below, many atoms must be interrogated per unit time, particularly for microwave standards. Since in equilibrium atoms are distributed essentially uniformly in the ground state, on interaction with the electromagnetic field roughly as many upward transitions as downward transitions would occur, so very little signal can be observed. A technique for state-selection must be used to prepare the atoms to be mainly in one state before (or during) the interaction. This produces the necessary large signal. The functions of state selection, detection, atom containment, and means for interaction with the electromagnetic field form the atomic resonator and state detector. There must also be magnetic shielding and means for supplying the required small magnetic field, typically between 1 and 50 milligauss. A typical electrical signal from the resonance as a function of frequency is shown in the lower part of Figure B.2. It is used as described below, in “Passive Standards.” If a number of atoms are state-selected or optically pumped to be in the upper level in the ground state or an excited state, and if they are in a suitable, low-loss cavity resonator, an oscillation at the transition frequency can build up. Here, the atoms supply energy to the cavity field by making transitions to the lower level. The atoms in turn are stimulated to make the transition by the cavity field. The oscillation builds up to where the power (energy per unit time) supplied by the atoms is equal to the power dissipated in the cavity and any load connected to the cavity. This is the principle of maser and laser devices. The maser is used as described below, in “Active Standards.” Atomic frequency standards are so accurate that relativistic effects must be taken into account. For example, a frequency standard 1 km above the surface of the oceans is 1.1 × 10−13 higher in frequency than one at sea level due to the change in gravitational potential. Velocity effects are also important. The velocity of the atoms in a cesium beam standard at room temperature causes it to be about 1 × 10−13 lower in frequency than a standard built with stationary atoms. Both gravitational and velocity effects must be taken into account for the standards in GPS satellites. Passive Standards The atomic resonator described above is used to make a microwave frequency standard, as shown in FigureB.2. The goal is to produce a useful output signal the frequency of which is tied to the atomic resonance at vo. Since the resonance is, to a good approximation, an even function of frequency about vo, it is impossible to stabilize directly to the peak value of the resonance with any accuracy. The derivative of the signal with respect to frequency would work since it vanishes at vo. The technique most often used to approximate this is to frequency modulate the microwave excitation signal, often with a square wave to the inflection points of the resonance. The electrical output signal will then contain a signal at the frequency of the modulation oscillator roughly proportional to the error in the average frequency of the microwave excitation and whose phase depends on the direction of the error. This signal is detected in a synchronous detector and gives a DC and low-frequency (baseband) error signal that is amplified and filtered and applied to the electronic frequency control (EFC) input of a voltage-controlled oscillator (VCO), also known as the local oscillator (LO), whose frequency output is converted to the microwave excitation frequency by the frequency synthesizer, which effectively multiplies the LO frequency by a rational number. This is a frequency servoloop that forces the average microwave excitation frequency to be at the center of the atomic resonance. When the system is locked, the frequency output 1 is the atomic resonance frequency divided by the synthesizer fraction, which may be chosen along with the LO frequency to produce a useful frequency such as 10 MHz. The fractional frequency resolution of the

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An Assessment of Precision Time and Time Interval Science and Technology FIGURE B.2 Block diagram of a passive microwave atomic frequency standard.

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An Assessment of Precision Time and Time Interval Science and Technology synthesizer can easily be 1 × 10−15 or better. The synthesizer may also produce another useful output, frequency output 2. This type of servoloop acts as a low-pass filter to the atomic resonator frequency noise and a high-pass filter to the LO frequency noise. The loop time constant is chosen to optimize overall frequency stability performance. Noisier local oscillators require a longer loop time constant, which is often undesirable. The frequency stability, characterized by the Allan deviation and determined by the noise in the atomic resonator, is approximately 1/(S/N Qlτ1/2), where S/N is the effective electrical signal-to-noise ratio in a 1 Hz bandwidth centered at the modulation frequency at the output of the resonator, Ql is the quality factor mentioned above, and τ is the averaging time of the measurement. Because this is low-pass filtered by the loop, it applies for times that are long compared with the loop time constant until other effects such as flicker noise or aging take over. It is clear that a large S/N and Ql are desirable. In a good design, the noise of the atomic resonator will be dominant. The frequency stability at times short compared with the loop time constant, as mentioned above, is determined by the LO. There is also an effect due to frequency noise at the second harmonic of the modulation frequency that effectively adds to the atomic resonator noise. Low-noise atomic resonators thus require good LOs. In modern standards, much of the electronics is digital, and microprocessors are used for much of the control and filtering functions, providing greatly improved performance. Some of the environmental effects will be discussed below. Figure B.2 can also apply to optical frequency standards. The LO in that case is a narrow linewidth, very stable laser, which is a formidable engineering task. Optical pumping with lasers does most of the state preparation and transition detection. Accurate transfer of optical frequencies to microwave frequencies is very important and is discussed later. Active Standards Figure B.3 shows a typical active microwave atomic standard such as a hydrogen maser. The power output of the atomic oscillator is typically quite small, on the order of picowatts, and its frequency is very susceptible to changes in output loading. For these reasons, the signal is amplified and heterodyned down to some convenient frequency by mixing with an output from the synthesizer. The signal is compared in phase with another output from the synthesizer. The phase error is amplified and filtered and applied to the EFC of a LO, which in turn supplies the reference frequency for the synthesizer. This is a phase-lock loop that forces the LO to run at a frequency that is a rational fraction, determined by the synthesizer, of the atomic oscillator. The loop time constant is chosen to provide overall best noise performance. At the present time, there are no really accurate active optical frequency standards (lasers). The reasons for this are the high optical gain and low optical cavity loss required to get oscillation. The high optical gain requires many atoms with low Ql. This, coupled with the low cavity loss, makes the laser oscillate close to the cavity resonance frequency rather than the atomic resonance frequency. Clocks A clock consists of two parts: first, a good oscillator such as the atomic standards described here and second, a means for counting the cycles of the oscillator or, equivalently, monitoring its elapsed phase and converting the phase to convenient units of time. In a mechanical clock, the oscillator is the

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An Assessment of Precision Time and Time Interval Science and Technology FIGURE B.3 Block diagram of an active microwave atomic standard, such as a hydrogen maser.

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An Assessment of Precision Time and Time Interval Science and Technology pendulum or balance wheel and the elapsed-phase device is the combination of gear train and clock hands. The elapsed-phase device in atomic clocks is the frequency synthesizer and an electronic counter. The clock’s time must be set initially to agree with some chosen time scale; this is synchronization. The clock’s frequency must also be set so that its rate matches the chosen time scale; this is syntonization. An error in synchronization gives a time offset to the clock. An error in syntonization gives a magnitude of time error in the clock that increases linearly with passing time (assuming the clock rates are constant). As mentioned above, in the most accurate clocks relativistic effects are very important and must be taken into account. MICROWAVE ATOMIC FREQUENCY STANDARDS Laboratory Cesium Beam These standards use a beam of cesium-133 that is state separated and detected either by magnetic deflection or optical pumping. Optically pumped standards give the best performance. The state-separated atoms are interrogated by an applied microwave magnetic field in a cavity, configured to subject them to two microwave regions leading to a double microwave pulse in time known as Ramsey interrogation (named after Norman Ramsey, who won a Nobel Prize for his work in atomic frequency standards). Such a cavity is called a Ramsey cavity. If the microwave frequency is at the atomic hyperfine resonance (~9192.631 MHz) and the field is at the correct amplitude, the probability of transitions in the ground state is maximized. The atoms in the beam are then subjected to detection of the probability of transition leading to an output electrical current proportional to the transition probability. This is used in a servo system like that shown in Figure B.2 to keep the frequency of the microwave excitation at the center of the probability versus frequency dependence. Useful output frequencies are obtained by frequency synthesis. The frequency width of the atomic resonance is proportional to the inverse of the time between the microwave pulses and is on the order of 70 Hz in typical laboratory systems, and the accuracy capability is about ±1 × 10−14. The relativistic shift is a few parts in 1013 for the thermal beam in the apparatus, to which must be added the shift due to elevation above sea level. Cesium Fountain In this recently developed standard, cesium atoms are cooled and trapped at microkelvin temperatures by laser cooling and then state selected by optical pumping. Balls of these trapped atoms are launched upward sequentially and pass through a microwave cavity, where they interact with the applied microwave magnetic field. They continue upward for perhaps a meter and then fall back under gravity through the microwave cavity a second time, giving the double microwave pulse for Ramsey interrogation. The probability of state transition is then determined optically and appears as an electrical signal, which is used as described above to control the frequency of the microwave excitation. The frequency width of the atomic resonance is on the order of 1 Hz in typical fountains because of the long time between the microwave pulses, leading to an accuracy of about ±1 part in 1015, much better than the beam standards. The shift due to relativistic effects in the apparatus is about 4 × 10−17 for a 1-meter toss height. This is much smaller than the shift in the cesium beam. The shift due to elevation above sea level must be added to this. The main limitation of accuracy is the frequency shift caused by cesium-cesium collisions

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An Assessment of Precision Time and Time Interval Science and Technology and uncertainty in the local gravitational potential. The collisional shift is much smaller with rubidium, and work on a rubidium fountain standard is ongoing. Cesium fountain standards have been and are being built in a number of laboratories around the world, including the USNO. Trapped Mercury Ion In the most advanced microwave trapped-mercury-199 ion frequency standard on which work is being done, ions are confined in a linear radio-frequency quadrupole trap partitioned electrically along its length so that ions can be shuttled from one end to the other. Ions are moved to one end of the extended trap for optical state selection and detection and then moved to other end for the atomic resonance, where interaction with the approximately 40.5-GHz electromagnetic field takes place. Helium at a pressure of 10−5 torr is used to cool the ions so that a large number can be trapped. In addition, the interaction time with the microwave field can be long, leading to a narrow linewidth. Optical pumping with a mercury-202 lamp is used for state selection and detection. The spectrum of the mercury-202 ion is such that the mercury-199 ions are pumped into the lower hyperfine state. Detection is accomplished by monitoring the fluorescence with photomultiplier tubes after interaction with the microwaves. Maximum fluorescence occurs when the microwave frequency matches the ionic resonance if the microwave amplitude and interaction time are optimized. The electronics for the standard are similar to that shown in Figure B.2. With the large number of trapped ions and the long interaction time, very good short-term stability should be achieved, about 4 × 10−14τ−1/2, which requires an extremely good local oscillator. The accuracy of this standard should also be good. There are relativistic effects due to the velocities of the ions in the radio-frequency trap. There is also a small frequency shift due to collisions with the helium. This shift can be estimated by varying the helium pressure, observing the frequency changes, and extrapolating to zero pressure. Hydrogen Maser Molecular hydrogen is dissociated into atoms and state separated in a magnetic field so that the higher energy state atoms are directed into a suitably coated bulb inside a high quality-factor (Qc) cavity resonant at the hyperfine frequency (~1420.4 MHz). If the number of atoms in the bulb and in the cavity Qc are both high enough, oscillation can take place in the cavity by stimulated emission of the atoms at the hyperfine frequency. This oscillation is coupled out and used to phase-lock a voltage-controlled oscillator/frequency synthesizer combination to provide the maser frequency output. There are frequency-pulling effects associated with (1) mistuning of the cavity with respect to the atomic hyperfine frequency and (2) collisions with the coated walls, known as wall shift. Some of the masers have a technique for automatically tuning the cavity to remove the cavity pulling effect. The short-term frequency stability of the hydrogen maser is excellent, but its accuracy is inferior to that of the cesium devices owing to the uncertainty associated with the wall shift. There is also a small frequency aging effect caused by changes of the wall shift with time. The relativistic shift due to the velocity of the hydrogen atoms, called second-order Doppler shift, is about −1.4 × 10−13 per kelvin, so temperature must be very carefully controlled to achieve good stability. Work is proceeding on a cryogenic hydrogen maser operating at 0.5 K. In this maser, the wall coating is liquid helium in the superfluid state. The frequency stability may be as good as 1 × 10−18. There is also a passive hydrogen maser in which the hyperfine transition is interrogated by a signal injected into the cavity. Early work was done at NIST, and there was an effort, later abandoned, to

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An Assessment of Precision Time and Time Interval Science and Technology commercialize the device in the United States. Such masers are available commercially now from Russia. The main advantage of the passive maser is its much smaller frequency pulling due to cavity mistuning. However, the short-term stability is much poorer than that of the active maser. Rubidium Gas Cell Rubidium-87 atoms, along with a buffer gas in a glass cell surrounded by a microwave cavity, are optically pumped by light from a rubidium lamp/filter combination to perform state selection between the hyperfine levels. The transmitted pumping light is monitored by a photodetector to provide the electrical output signal. In the absence of microwave excitation at the hyperfine resonance frequency (~6834 MHz), equilibrium between the optical pumping and hyperfine relaxation gives a particular transmitted light intensity. If microwave excitation at the resonance frequency is present, additional hyperfine transitions are induced, which allows more optical pumping to take place. Then, more pumping light is absorbed, and the transmitted light and output signal decrease. The frequency width of this resonance is typically a few hundred hertz. The buffer gas confines the rubidium atoms, largely preventing collisions with the cell walls and also reducing the Doppler width. However, collisions of rubidium atoms with buffer gas atoms cause a frequency shift that depends on pressure, temperature, and gas mixture. In addition, there is a frequency shift due to the simultaneous irradiation of the rubidium atoms by the pumping light and the microwave excitation. This so-called light shift depends on both light intensity and spectral distribution. Thus the rubidium gas cell standard has much poorer accuracy than the cesium beam standard and has frequency aging due to drifts in the light intensity, diffusion of helium into the cell, and other changes in the buffer gas that are due to diffusion from the cell walls. However, if the rubidium gas cell is properly designed, its short-term stability can be very good An Allan deviation of 1 × 10−13τ−1/2 has been demonstrated at NIST, which believes that an order of magnitude improvement is possible. This would be a very good local oscillator for laboratory microwave frequency standards. Rubidium in a gas cell can also oscillate as a maser if the cavity Q is high enough and the spectrum of the optical pumping source is appropriate. Work was done on this in the late 1960s. Work is ongoing on a double-bulb version in which the optical pumping is done in one of the bulbs outside the microwave cavity and the maser action takes place in the other bulb, which is inside the microwave cavity and connected to the pumping bulb. This arrangement virtually eliminates the light shift. The power output of the rubidium maser is considerably larger than that of the hydrogen maser. It might serve as an LO for a rubidium fountain standard. SPACEBORNE Cesium Beam The spaceborne cesium beam standards used in, for example, GPS satellites are very similar to the magnetically state-selected laboratory devices described above but have much shorter beam tubes, leading to a much larger linewidth and a lower signal-to-noise ratio, with consequent lower accuracy and poorer short-term stability. Great attention is paid to ruggedness, reliability, small size, and low power. There is at least one U.S. effort to develop a small, optically pumped beam tube, which should give improved short-term stability. The Navy is supporting this work.

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An Assessment of Precision Time and Time Interval Science and Technology Rubidium Gas Cell The spaceborne rubidium gas cell standard is similar to the laboratory rubidium standard described above, except that small size, ruggedness, long life, and moderately good short-term stability are emphasized. In the block IIR GPS satellites, these standards are providing better short-term stability than the cesium beam standards. In MILSTAR and the Advanced Extremely High Frequency (AEHF) military satellite communications systems, the emphasis is again on low power, small size, and environmental issues. Clocks on the ISS The primary atomic reference clock in space (PARCS) standard is an ongoing joint effort of NIST, JPL, University of Colorado, and the Smithsonian Astrophysical Observatory, with funding from NASA. It uses laser cooling and trapping to generate and launch balls of cold, state-selected cesium atoms in a way similar to that described above for the cesium fountain. However, in the microgravity environment of space, the balls are in free, unaccelerated motion and are directed through a Ramsey cavity, like that in the beam apparatus described above, to achieve Ramsey interrogation. The atoms have very low velocity, so that the time between their encounters with microwave pulses is about 10 seconds, leading to a very small linewidth, on the order of 0.1 Hz. The expected accuracy is on the order of ±1 × 10−16. This standard will require an excellent LO. Present plans are to use the Smithsonian spaceborne maser discussed below. PARCS will be used, not only as an excellent standard in space but also for a number of scientific experiments. The rubidium atomic clock experiment (RACE), a joint project between Pennsylvania State University and JPL, is a spaceborne rubidium cold-atom clock expected to have an accuracy of ±1 part in 1017. It is to fly on the ISS after PARCS, circa 2007. It will be used to improve the classic clock tests of general relativity. The atomic clock ensemble in space (ACES) is a European Space Agency mission on the ISS. It consists of a laser-cooled cesium clock that, like PARCS, takes advantage of the microgravity environment to achieve low linewidth and high accuracy. It is scheduled to fly at about the same time as PARCS (2005 or 2006). Hydrogen Maser There was an effort at Hughes to develop a small, spaceborne hydrogen maser a number of years ago, but the effort was abandoned. The Smithsonian Astrophysical Observatory also had a program that is now again active. This maser weighs about 100 kg and has demonstrated stability of 5 × 10−15 at 100 seconds averaging time. It is planned to be used as the LO for the PARCS mentioned above. Commercial Cesium Beam There are five commercial cesium beam standards available at the present time, three of them made by U.S. companies. They are all similar to the magnetically state-selected laboratory standard described above but have shorter beam tubes and consequent larger linewidth. They vary in their accuracy, short-term stability, environmental performance, and price. The best have an accuracy of plus or minus a few

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An Assessment of Precision Time and Time Interval Science and Technology parts in 1013 and Allan deviation of about 8 × 10−12τ−1/2. One application is as clocks in international laboratories contributing to the international atomic time scale (for example, the USNO uses about 40 units in its time-scale ensemble). Another is the clock ensemble at the GPS ground control station that uploads time to the GPS satellites. They are also used aboard Navy ships, and there are a number of communications applications. As mentioned above, there is at least one U.S. effort under way to develop a small, optically pumped beam tube, which should give improved short-term stability. The Navy is supporting this work. Rubidium Gas Cell This is by far the best-selling commercial atomic standard, with sales in excess of 40,000 units per year by the two main U.S. suppliers. The units are all very similar to the rubidium units described above. Emphasis is on low cost and moderately small size (~100 cm3). Their main use is in cellular base stations and other telecommunications applications. Hydrogen Maser There is at present only one U.S. commercial supplier of active hydrogen masers. Both passive and active masers are also available from Russia. They are being used in increasing numbers as members of time-scale ensembles because they have excellent short-term stability. OPTICAL FREQUENCY STANDARDS Mercury Ion An optical frequency standard is being developed at NIST in which a single mercury-199 ion is contained in a quadrupole trap. This ion has a narrow (~1-Hz) optical transition at about 1.06 × 1015 Hz that is interrogated by a highly stabilized laser at 564 nm that is frequency doubled to reach the optical transition. Work done so far indicates that accuracy considerably better than ±1 × 10−16 is achievable. Calcium NIST is working on an optical frequency standard using trapped calcium atoms. The clock transition frequency is about 4.54 × 1014 Hz with a narrow natural linewidth of about 400 Hz. About 106 atoms are trapped in a magneto-optic trap (MOT). The trap is turned off, and then two separate pulsed lasers are used to interrogate the atoms and detect the transition probability. The trap is then turned on again and the process repeated. Because the linewidth is much larger than that of the mercury ion standard, the achievable accuracy is much poorer. However, the large number of atoms being interrogated gives very good short-term stability. Other A number of other ions and atoms are also under investigation as candidates for optical standards. Among these are single trapped ions of indium and ytterbium. Work is also being done on standards using trapped neutral atoms of strontium. Neutral atoms, like the calcium atoms mentioned above, have the advantage that many can be held in the trap, giving good signal-to-noise ratio and, consequently,

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An Assessment of Precision Time and Time Interval Science and Technology good short-term stability. However, the transitions currently being investigated in neutral atoms have much larger linewidths than those of the ions, so the achievable accuracy is much poorer. COHERENT POPULATION TRAPPING Although discussed in the early 1970s, this approach has only recently received much attention, mainly because lasers suitable for pumping are now available. Several groups are now working in this field. The excitation of the atoms in a cell is all optical, using two coherent optical signals to pump on two ground-state levels of the atoms. If the optical signals are at the frequencies to pump to a single excited state and the difference between their frequencies is at the resonance frequency between the ground states, the atoms are trapped in a coherent superposition of the ground states. Under these conditions, the excited state is not populated, and the fluorescence disappears. No microwave cavity is needed in the case of a passive device, so it is much easier to miniaturize. Monitoring the fluorescence or the transmission through the cell does the resonance detection for this type of passive microwave standard. If a cavity tuned to the resonance frequency between the ground states surrounds the cell, microwave power radiated by the atoms can be detected. The frequency of this radiation is the difference frequency between the optical signals. QUARTZ OSCILLATORS The simplified circuit diagram in Figure B.4 shows the basic elements of a modified Pierce crystal oscillator. A quartz crystal acts as a stable mechanical resonator, which, by its piezoelectric behavior and high Q, determines the frequency generated in the oscillator circuit. The crystal resonator (also called the “crystal unit”) is also the primary determinant of frequency stability in the oscillator. In the manufacture of quartz resonators, wafers are cut from a quartz crystal along precisely controlled directions with respect to the crystallographic axes. The properties of the device depend strongly on the angles of cut. After shaping to required dimensions, metal electrodes are applied to the quartz wafer, which is mounted in a holder structure. The assembly is hermetically sealed, usually into a metal or glass enclosure. To cover the wide range of frequencies, different cuts vibrating in a variety of modes can be used. When cut along certain directions, resonators whose temperature coefficient is zero may be produced (the resonator frequencies still vary slightly with temperature; the resonators possess zero temperature coefficient at two specific temperatures only). For PTTI applications, the AT-cut and SC-cut are used, both of which vibrate in the “thickness shear” mode. For the applications that demand the highest precision, the SC-cut has important advantages over the AT-cut; however, the SC-cut is more difficult to produce, so it is generally more expensive than the AT-cut. The three categories of crystal oscillators, based on the method of dealing with the crystal unit’s frequency versus temperature (f versus T) characteristic, are the uncompensated crystal oscillator, XO, the temperature-compensated crystal oscillator, TCXO, and the oven-controlled crystal oscillator, OCXO. A simple XO does not contain means for reducing the crystal’s f versus T variation. A typical XO’s f versus T stability may be ±25 parts per million (ppm) over a temperature range of −55°C to +85°C. In a TCXO, temperature-dependent reactance variations produce frequency changes that are equal and opposite to the frequency changes resulting from temperature changes—that is, the reactance variations compensate for the crystal’s f versus T variations. For example, the output signal from a

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An Assessment of Precision Time and Time Interval Science and Technology FIGURE B.4 The basic elements of a modified Pierce crystal oscillator.

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An Assessment of Precision Time and Time Interval Science and Technology temperature sensor (a thermistor) can be used to generate a correction voltage that is applied to a voltage-variable reactance (a varactor) in the crystal network. TCXOs generally use AT-cut crystal resonators. Analogue TCXOs can provide about a 20-fold improvement over the crystal’s f versus T variation. A good TCXO may have an f versus T stability of ±1 ppm over a temperature range of −55°C to +85°C. Hysteresis, i.e., the nonrepeatability of the f versus T characteristic, is the main TCXO stability limitation. In an OCXO, the crystal unit and other temperature-sensitive components of the oscillator circuit are maintained at an approximately constant temperature in an oven. The crystal is manufactured to have an f versus T characteristic that has zero slope at or near the desired oven temperature. To permit the maintenance of a stable oven temperature throughout the OCXO’s temperature range (without an internal cooling means), the oven temperature is selected to be about 15°C above the maximum operating temperature of the OCXO. OCXOs can provide more than a 1,000-fold improvement over the crystal’s f versus T variation. The best OCXOs may have an f versus T stability of ±1 × 10−10 over a temperature range of −55°C to +85°C. OCXOs require more power and are larger than and more expensive than TCXOs. A special case of a compensated oscillator is the microcomputer-compensated crystal oscillator, MCXO (in advanced development). The MCXO minimizes the two main factors that limit the stabilities achievable with TCXOs: thermometry and the stability of the crystal unit. Instead of a thermometer that is external to the crystal unit, such as a thermistor, the MCXO uses a more accurate self-temperature-sensing method. Two modes of an SC-cut crystal are excited simultaneously in a dual-mode oscillator. The two modes are combined such that the resulting beat frequency is a monotonic (and nearly linear) function of temperature. The crystal thereby senses its own temperature. To reduce the f versus T variations, the MCXO uses digital compensation techniques: pulse deletion in one implementation and direct digital synthesis of a compensating frequency in another. A good MCXO may have an f versus T stability of ±2 × 10−8 over a temperature range of −55°C to +85°C. As in TCXOs, hysteresis is the main MCXO stability limitation; however, the crystal resonators used in the MCXO exhibit significantly smaller hysteresis than the resonators used in TCXOs. LOCAL OSCILLATORS Quartz, Rubidium, and Hydrogen Relatively simple quartz oscillators as described above can serve as LOs for many of the lowerperforming atomic standards. The best commercial cesium beam standards require a good OCXO. Since the servoloop controls the low-frequency behavior of the LO, slow changes in its frequency caused by temperature variations or aging are removed from the output of the standard. This is particularly true if the loop filter contains at least two digital integrators. However, the LO determines the high frequency or very short-term performance of the standard. The oscillator may be chosen to meet this requirement, or another low-noise oscillator may be phase-locked to the standard’s output or to the output after frequency multiplication. Quartz oscillators are too noisy as LOs to realize the full capability of the high-performance laboratory microwave atomic standards. This is discussed further below. In some cases, a high-performance laboratory rubidium standard optimized for stability can serve the purpose. Hydrogen masers are also used as LOs. As an example, it is planned for the spaceborne PARCS cesium standard to use the spaceborne hydrogen maser mentioned above.

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An Assessment of Precision Time and Time Interval Science and Technology Microwave A number of very high-performance microwave oscillators have been built using sapphire cavities. Some of these operate in what are called whispering gallery modes, in which the electromagnetic field is confined in the sapphire. These can have very high Q at cryogenic temperatures and, if some noise reduction techniques are used, can make an excellent, but expensive and complex, LO for a high-performance microwave standard. Oscillators of this type were first developed at Stanford University and at JPL. This technology is currently being pursued at the University of Western Australia and at JPL, and it has been used to demonstrate the highest performance of advanced atomic clocks, such as the mercury ion standard and the cesium fountain. A room-temperature version of the whispering-gallery-mode sapphire oscillator has been commercialized in Australia, which produces the highest spectral purity at 10 GHz by extensive use of the carrier-suppression technique. Work is under way on an optoelectronic oscillator using a fiber-optic delay line. The optical signal input to the delay line is modulated by a microwave signal that is an amplified, filtered signal obtained from a high-speed optical detector at the output of the delay line. With sufficient gain, the system oscillates at a frequency such that the roundtrip phase is an integer multiple of 2π. The equivalent Q of this system is πft, where f is the operating frequency and t is the delay of the line. For a 10-GHz oscillator with a 4-km delay line, the Q is about 6 × 105. This type of oscillator may hold promise as an LO for high-performance microwave standards. Laser The only LO suitable for probing very narrow optical transitions is a highly stabilized laser. Probably the best have been built at NIST for the mercury-199 ion optical transition. They consist of a dye laser stabilized by a rugged, large, very-high-Q optical cavity at 564 nm. Two of them have been built so that the linewidth can be measured by heterodyning them against each other. The measured optical linewidth is on the order of 0.1 Hz. Extreme care in terms of vibration and acoustic isolation is necessary to achieve this performance. Local Oscillator Effects Besides the noise effects of the LO mentioned above, there are other effects to consider. One of these is frequency noise density of the modulation frequency in passive standards at even harmonics, particularly the second harmonic. Depending on the type of resonator, the modulation waveform, and the synchronous detector, various amounts of the noise are downconverted to baseband and added to the noise of the atomic resonator. This is one of the main limitations of LO for high-performance atomic resonators. Another effect involves amplitude modulation of the atomic resonator output that is coherent with frequency modulation of the LO. This can occur, for example, in atomic beam standards as a result of changes in the gravitational acceleration direction caused by motion of a vehicle. Both the beam tube output and the crystal oscillator LO frequency can depend on the orientation in the gravitational field. Frequency components in this modulation higher than the loop bandwidth can cause a frequency error. The French navy first noticed this effect. It can be reduced by making the loop time constant short and having the mechanical layout such that the most sensitive axes of the atomic resonator and the LO are orthogonal.

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An Assessment of Precision Time and Time Interval Science and Technology CONNECTION BETWEEN MICROWAVE AND OPTICAL REGIONS Until recently, the absolute connection between microwave and optical frequencies, a range of four to five decades, required chains of phase-locked oscillators, many of them complex lasers. The equipment often filled rooms and would not function for very long periods. Very skilled people were needed to build and run them. Also, only a very small range of optical frequencies could be covered without having to redesign the whole chain. The technique of using regularly pulsed lasers to generate a comb of optical frequencies has been known for a long time. It follows from Fourier analysis that a uniform series of pulses in time spaced by τ leads to a uniform series of spectral lines in the frequency domain spaced by frequency 1/τ. However, unless there is something that constrains the optical phase to be the same from pulse to pulse, the origin of the spectral lines is offset from zero by a frequency Δɸ/2πτ , where Δɸ is the change in phase from pulse to pulse in radians. Therefore, all the comb frequencies are offset by the same amount, which is in general unknown. What is new is that a technique has been invented that allows the measurement and control of this offset frequency so that the absolute frequencies of the comb can be determined in terms of the pulse frequency, 1/τ, and the comb offset frequency. If these are measured or determined by, for example, a microwave frequency standard, the optical frequencies are known with the same accuracy (already tested to about ±1 × 10−17). By locking one of the comb lines to an optical frequency standard, a microwave frequency is generated that has the same accuracy as the optical standard. The technique for measuring and controlling the offset frequency involves broadening the comb spectrum with a nonlinear optical fiber so that it extends over a two-to-one frequency range, an octave. A group of lines from the low-frequency end of the comb is frequency doubled and frequency subtracted from a group of lines at twice the frequency of the first group. The result can be shown to contain a signal at the comb-offset frequency, thus allowing its frequency to be measured or controlled. The system can easily handle any optical frequency in the range of the comb. It is a simple and accurate way of connecting the microwave and optical frequency ranges. A number of laboratories in the United States, including NIST, are already building such combs, which are critical to realizing the performance capabilities of the new optical frequency standards as well as to measuring their frequencies.