Quantum Frequency Standards

devices in which the quantum transitions of particles (such as atoms, molecules, and ions) from one energy state to another are used for precise measurement of the frequency of oscillations or to generate oscillations with extremely stable frequency. Quantum frequency standards make possible measurement of the frequency of oscillations and, consequently, of their period (duration) with much higher accuracy than is possible with other frequency standards. This led to their introduction in metrology.

Quantum frequency standards are the basis for national frequency and time standards and secondary frequency standards, which are close to the national standard with respect to their degree of accuracy and metrological capabilities but must be calibrated with respect to it. Quantum frequency standards are used as laboratory frequency standards that have a broad range of output frequencies and are equipped with a device that compares the measured frequency with the standard frequency, as well as reference frequencies, which make possible observation of a selected spectral line without introducing any significant distortion, and comparison (with high accuracy) of a measured frequency with a frequency defined by a spectral line. The quality of quantum frequency standards is characterized by their stability—their ability to hold a selected frequency for a long period.

Quantum laws impose extremely rigid limitations on the state of atoms. Under the action of an external electromagnetic field of a certain frequency, atoms may either be excited, in which case they jump from a state of lower energy ε₁ to a state of higher energy ε₂ absorbing in the process a portion (quantum) of the energy of the electromagnetic field equal to

hv = ε₂ – ε₂

or they may move to a state of lower energy, radiating electromagnetic waves of the same frequency.

Quantum frequency standards are commonly divided into two classes. In active standards the quantum transitions of atoms and molecules lead directly to the radiation of electromagnetic waves whose frequency serves as a standard or reference frequency. Such instruments are also called quantum generators. In passive standards the measured frequency of the oscillations of an external generator is compared with the frequency of the oscillations that correspond to a certain quantum transition of the selected atoms—that is, with the frequency of the spectral line. Passive quantum frequency standards using beams of cesium atoms (cesium frequency standards) were the first to achieve technical perfection and to become generally available. In 1967 by international agreement the duration of the second was defined as 9, 192, 631, 770.0 periods of oscillations corresponding to a certain energy transition of atoms of the only stable isotope of cesium, ¹³³Cs. The zero after the decimal point means that this number is not subject to further change. A contour of the spectral line of ¹³³Cs that corresponds to the transition between the two selected energy levels ε₁ and ε₂ may be observed in the cesium frequency standard. The frequency that corresponds to the peak of this line is fixed, and frequencies being measured are compared with it by means of special devices.

An atom-beam tube in which a high vaccuum is maintained is the main part of a quantum frequency standard using a beam of cesium atoms. At one end of the tube is the source of the beam of atoms, a cavity in which a certain quantity of liquid cesium is placed (Figure 1). The cavity is connected with the rest of the tube by a narrow channel or a set of parallel channels. The source is maintained at a temperature of about 100°C, when the cesium is in the liquid state (the melting point of cesium is 29.5°C) but its vapor pressure is still low, and the cesium atoms emitted from the source rather seldom pass through the channels without colliding with each other. As a result of this a slightly divergent beam of cesium atoms is formed in the tube.

Figure 1. Diagram of an atom-beam tube: (1) source of cesium beam, (2) and (4) deflecting magnets that generate the nonuniform magnetic fields H₁, and H₂, (3) cavity resonator (in the permanent and uniform magnetic field H) in which electromagnetic waves are excited, (5) incandescent tungsten filament, (6) cesium-ion collector, (7) measuring device, (8) domain of the permanent uniform magnetic field H (bounded by broken line)

At the other end of the tube is an extremely sensitive receiver (detector) of cesium atoms, which is capable of recording negligible changes in the intensity of the atom beam. The detector consists of an incandescent tungsten filament (5) and a collector (6), between which is placed a voltage source (the positive pole is attached to the wire, the negative pole to the collector). As soon as a cesium atom touches the incandescent filament, it gives off its outer electron (the ionization energy of cesium is equal to 3.27 electron volts [eV], and the electron work function for tungsten is 4.5 eV). The cesium ion is attracted to the collector. If a sufficiently large number of cesium atoms strike the incandescent tungsten, an electric current is generated in the circuit between the collector and the tungsten filament. The intensity of the cesium beam striking the detector may be estimated by measuring this current.

In passing from the source to the detector, the beam of cesium atoms passes between the pole shoes of two strong magnets. The nonuniform magnetic field H₁ of the first magnet splits the beam of cesium atoms into several beams in which atoms of various energies (at various energy levels) travel. The second magnet (the field H₂ ) focuses on the detector only atoms that belong to one pair of energy levels ε₁ and ε₂, deflecting all others.

In the gap between the magnets the atoms pass through the cavity resonator (3)—a cavity with conducting walls—in which electromagnetic oscillations of a certain frequency are excited by means of a stable quartz generator. If a cesium atom with energy ε₁ moves to the energy state ε₂ under the influence of these oscillations, the field of the second magnet rejects it from the detector, since for an atom that has moved to the state ε₂ the field of the second magnet no longer will be a focusing field and the atom will bypass the detector. Thus, the current through the detector will prove to be less by a quantity proportional to the number of atoms that make energy transitions under the influence of the electromagnetic resonator. The transitions of cesium atoms from the state ε₂ to the state ε₁ will be fixed in the same manner.

The number of atoms making an induced transition per unit time under the influence of an electromagnetic field is at a maximum if the frequency of the electromagnetic field acting on the atom coincides precisely with the resonance frequency v₀ == (ε₂ − ε₁)/h. As the noncoincidence or detuning of these frequencies increases, the number of such atoms decreases. Therefore, by smoothly changing the field frequency near v₀ and plotting the frequency v along the horizontal axis and the change in the detector current along the vertical, we obtain the contour of the spectral line that corresponds to the ε₁ → ε₂ transition and, conversely, ε₂ →ε₁ (Figure 2, a).

The frequency v₀, which corresponds to the peak of the spectral line, is also a reference point on the frequency scale, and the oscillation period corresponding to it is accepted as 1/9, 192, -631.0 sec.

The accuracy of determination of the frequency corresponding to the peak of the spectral line is usually several percent or, at best, fractions of a percent of the width of the line. The narrower the spectral line, the higher the accuracy; thus explaining the desirability of eliminating or at least weakening all factors that lead to broadening of the spectral lines used.

In cesium standards the broadening of the spectral line (Figure 2, a) is determined by the time of interaction between the atoms and the electromagnetic field of the resonator: the shorter this time, the broader the line. The time of interaction coincides with the duration of the atom’s passage through the resonator. It is proportional to the length of the resonator and inversely proportional to the speed of the atoms. However, the resonator cannot be made very long, because the dispersion of the atom beam would increase. A substantial reduction in the speed of the atoms by lowering the temperature is also impossible, since the intensity of the beam decreases in the process. Increasing the dimensions of the resonator is made more difficult since the resonator must lie within a magnetic field H of extremely uniform magnitude and direction. This condition is necessary because the energy transitions used in cesium atoms are due to the change in the orientation of the magnetic moment of the nucleus of the cesium atom with respect to the magnetic moment of its electron shell. Transitions of this type cannot be observed outside a magnetic field, and the frequency corresponding to such transitions depends, although only slightly, on the intensity of the field. It is difficult to create such a field in a large volume.

A narrow spectral line is produced by using a horseshoe-shaped resonator (Figure 3), in which the beam passes through the aperture near its ends and only there interacts with the high-frequency electromagnetic field. Therefore, uniformity and stability of the magnetic field H are required only in these two

Figure 2. Shape of spectral line in cesium frequency standards: (a) with an ordinary resonator, (b) with a horseshoe-shaped resonator; v) resonance frequency, Δv) width of spectral line

small regions. Before the second entry into the resonator the atoms “retain” the result of their first interaction with the field. In the case of a horseshoe-shaped resonator the spectral line takes on a more intricate shape (Figure 2, b) that reflects both the passage time through the electromagnetic field within the resonator (a broad pedestal) and the total passage time between the two ends of the resonator (a narrow central peak). It is the narrow central peak that serves to fix the frequency.

Figure 3. Diagram of an atom-beam tube with horseshoe-shaped resonator (symbols are the same as in Figure 1)

In quantum frequency standards with a beam of cesium atoms the error in the value of the frequency v₀ occurs only in the 13th digit for one-of-a-kind devices (frequency standards) and in the 12th digit for series-manufactured high-precision devices (secondary standards).

In addition to an atom-beam tube and a quartz-crystal oscillator, quantum frequency standards using a beam of cesium atoms also incorporate special radio circuits that make possible highly accurate comparison of the measured frequency of external generators with the frequency defined by the quantum frequency standards. In addition, the cesium standard is usually supplemented with devices that generate a set of “integral” standard frequencies whose stability is equal to the stability of the standard. These systems sometimes also generate precise time signals. In such cases the quantum frequency standard is converted into a quantum clock.

One-of-a-kind laboratory quantum frequency standards that operate with beams of cesium atoms and are part of the national frequency and time standards make possible reproduction of the duration of a second—and consequently of the entire complex of measurements of frequency and time—with a relative error of less than 10⁻ⁿ. In practice this relative error does not exceed 10⁻¹², but by international agreement extended observations are necessary to fix this value. A significant advantage of quantum frequency standards using beams of cesium atoms is that industrial designs make possible reproduction of the nominal value of a frequency (or time) with an error of 10⁻¹¹, that is, they are not inferior to the standard with respect to accuracy. Even small instruments of this type, which are suitable for use under ordinary laboratory conditions and in mobile facilities, operate with an error not greater than 10⁻¹⁰, and some models have an error of 10^ℒ11.

The hydrogen quantum generator (Figure 4) is the most important active quantum frequency standard. In a hydrogen generator a beam of hydrogen atoms is emitted from the source (1), where the hydrogen molecules are split into atoms at low pressure and under the influence of an electric discharge. The dimensions of the channels through which the atoms emerge from the source into the vacuum chamber are smaller than the distance traveled by the hydrogen atoms between collisions. Under this

Figure 4. Design of hydrogen quantum generator: (1) atom-beam source, (2) sorting system (multipole magnet), (3) resonator, (4) accumulator bulb

condition the hydrogen atoms are emitted from the source in the form of a narrow beam, which passes between the pole shoes of the multipole magnet (2). The action of the field generated by the magnet is such that it focuses atoms in the excited state near the axis of the beam and scatters atoms in the ground (unexcited) state.

The excited atoms pass through a small aperture into the quartz bulb (4) inside the cavity resonator (3), which is tuned to a frequency corresponding to the transition of hydrogen atoms from the excited state to the ground state. Under the influence of the electromagnetic field, the hydrogen atoms radiate as they pass into the ground state. The photons radiated by the hydrogen atoms during a comparatively long period of time, which is determined by the quality factor of the resonator, remain within it, giving rise again to stimulated emission of the identical photons by the hydrogen atoms that enter later. Thus, the resonator creates the feedback necessary for self-excitation of the generator. However, the intensity of the beam of hydrogen atoms that can be attained is still insufficient to provide self-excitation of such a generator if an ordinary cavity resonator is used. Therefore, a quartz bulb (4) whose walls are coated on the inside with a thin layer of polyfluoroethylene resin (Teflon) is placed inside the resonator. The excited hydrogen atoms may strike the Teflon film more than 10, 000 times without losing their excess energy. As a result, a large number of excited hydrogen atoms is collected in the bulb and the average time spent by each of them in the resonator increases to approximately 1 sec, which is sufficient for realization of the conditions of self-excitation and for the hydrogen quantum generator to begin to operate, radiating electromagnetic waves with an exceedingly stable frequency.

The bulb, which is made smaller than the generated wavelength, plays another extremely important role. The chaotic movement of the hydrogen atoms within the bulb should lead to broadening of the spectral line as a result of the Doppler effect. However, if the motion of the atoms is confined by a volume whose dimensions are less than the wavelength, the spectral line assumes the form of a narrow peak that rises above a broad, low pedestal. As a result, the width of the spectral line is only 1 hertz (Hz) in a hydrogen generator emitting radiation of wavelength » = 21 cm.

The extreme narrowness of the spectral line ensures the low error of the frequency of a hydrogen generator, which lies within the 13th digit. The error is due to the interaction of the hydrogen atoms with the Teflon coating of the bulb. The value of this frequency, measured by means of a quantum frequency standard using a beam of cesium atoms (see above), is equal to 1, 420, -405, 751.7860 ± 0.0046 Hz. The power of a hydrogen generator is extremely low (~ 10⁻¹² watts). Therefore, quantum frequency standards based on a hydrogen generator incorporate, in addition to circuits that compare and form the standard frequency grid, an extremely sensitive pickup.

Both types of quantum frequency standards operate in the superhigh-frequency (SHF) band. There are a number of other atoms and molecules whose spectral lines make it possible to devise active and passive quantum frequency standards in the radio band. However, they have not yet found practical application. Only rubidium quantum frequency standards, based on the method of optical pumping, are used extensively as a secondary frequency standard in laboratory practice and in navigation systems and quantum clocks.

Lasers, in which special measures are taken to stabilize the frequency of radiation, are optical quantum frequency standards. In the optical band the Doppler broadening of the spectral lines is very great, and, because of the short length of light waves, it is impossible to suppress it in the same way as in a hydrogen generator. Nor has there been any success as yet in developing an efficient laser using beams of atoms or molecules. Since several relatively narrow resonance lines of an optical resonator fall within the Doppler width of a spectral line, the generating frequency of the overwhelming majority of lasers is determined not so much by the frequency of the spectral line used as by the dimensions of the optical resonator, which define its resonance frequencies. But these frequencies do not remain constant; they change under the influence of changes in temperature and pressure, vibrations, aging, and so on. The lowest relative error of frequency in an optical quantum frequency standard (~ 10⁻¹³) has been achieved by using a helium-neon laser that generates at 3.39 microns (μ).

Optical quantum frequency standards are not yet connected (in a metrological sense) with radio-band quantum frequency standards, and consequently they are not connected with the unit of frequency (the hertz) or the unit of time (the second). Direct measurement of a frequency (comparison with the standard) is possible only in the longwave region of the infrared band (3.39 μ and longer).