1: Principle of an atomic clock based on a “Rabi”, or one-cavity scheme. Deflection angles are greatly exaggerated. Actual clocks operate usually with two separated cavities (“Ramsey scheme”). 

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... that always c ≥ 0 implying repulsive interactions and therefore excluding the possibility of pairing investigated in a 1D δ -interacting Fermi gas in an external magnetic field [GBLB07]. Actually, the magnetization can be changed to an arbitrary value using the microwave technique and remains constant in a given experiment. Indeed, experiments often employ the lowest Zeeman sublevel of the atom, say | F = 9 / 2 , m F = − 9 / 2 for 40 K, in combination with the next lowest state ( m F = − 7 / 2); for which spin-changing collisions are energetically disfavored. When tuning the p-wave interactions, the atoms are prepared in the | F = 9 / 2 , m F = − 7 / 2 state from which depolarization into m F = − 9 / 2 , − 5 / 2 also is suppressed by the second order Zeeman effect [GTSE05, Köh07]. The even-wave interactions are constrained by the transverse frequency of the waveguide and linear density and therefore γ e is considered as a parameter. The linear densities accessible in current experiments vary within the range n = 0 . 2 2 μ m , the transverse frequency ω ⊥ ∼ 100 kHz and the background 3D even- wave scattering length of 40 K is a bg = 104 a 0 . Using Olshanii’s relation a e 1 D = − 2 a a 2 ⊥ bg (1 − 1 . 4603 a a bg ⊥ ) where a ⊥ = 2 /mω ⊥ , it follows that the typical values of γ e ≈ 0 . 1 − 10. In Fig. 6.3 and 6.4 g 2 , which determines the photoassociation rate, and g 3 , which determines the rate of three-body recombination, are shown as a function of the magnetization σ = S/N and γ o for different values of γ e . It is clear that states of lower magnetization are more stable and for larger values of γ e , that is, stronger s-wave interactions, the region of stability increases. Using the mapping between a 1D spinor Fermi gas and the Lieb-Liniger-Heisenberg model, the two and three-body correlation functions g 2 and g 3 have been calculated. It is found that g 3 is small enough in a wide area of the γ o , γ e -plane, encompassing both ferromagnetic and antiferromagnetic phases, to ensure stability of this system against three-body recombination over experimental lifetimes. g 2 is small in the large region where g 3 is small, suggesting that photoassociation experiments would be difficult. The limiting case of the fermionic Tonks-Girardeau gas, falls however within the region of instability. Finally, by means of a variational ansatz the results are extended for an arbitrary spin polarization. We propose a solvable generalization of the Tonks-Girardeau model that describes a coherent 1D gas of cold two-level Bosons which interact with two external fields in a Ramsey interferometer. They also interact among themselves by infinitely strong contact potentials, with interchange of momentum and internal state. We study the corresponding Ramsey fringes and the quantum projection noise which, essentially unaffected by the interactions, remains that for ideal Bosons. The dual system of this gas, an ideal gas of two-level Fermions coupled by the interaction with the separated fields, produces the same fringes and noise fluctuations. The cases of time-separated and spatially-separated fields are studied. For spatially separated fields the fringes may be broadened slightly by increasing the number of particles, but only for large particle numbers far from present experiments with Tonks-Girardeau gases. The uncertainty in the determination of the atomic transition frequency diminishes, essentially with the inverse root of the particle number. The difficulties to implement the model experimentally and possible shortcomings of strongly interacting 1D gases for frequency standards and atomic clocks are discussed. A clock is a device that, in a sense, “produces” time by counting stable oscillations, for example of a pendulum. Atomic clocks, in particular, count the oscillations of the field resonant with an atomic transition, most frequently a hyperfine transition of caesium-133 (which defines officially the second as the time required for 9192631770 periods of the resonant microwave field). An external quartz oscillator is locked by a servo loop to a resonance excitation curve between two hyperfine states | g and | e so that its frequency is always adjusted to the maximum of the curve, and thus to the natural frequency of the selected transition. In more detail, see Fig. 7.1, caesium atoms are heated in an oven to produce a beam. A first magnet filters out the excited atoms so that only ground state atoms enter into the microwave cavity in which the field wavevector is perpendicular to the atomic motion, and the frequency, locked to an external quartz oscillator, is very close to the transition frequency. This excites some atoms which are selected with a second magnet and later detected. The excitation probability compared with previous runs at slightly different frequencies tells us how far or close is the field frequency to the transition frequency, and this information is used to modify the excitation frequency so that it stays as close as possible to the maximum of the excitation curve, see Fig. 7.2. The clock includes the appropriate counting electronics which we shall not discuss here. A very sharp peak is clearly desirable to minimize the fluctuation of the external oscillation frequency around the natural one. This contributes to the stability of the clock and explains why a beam configuration is chosen: a perpendicular excitation with respect to the atomic motion avoids the Doppler effect and its associated line broadening. In actual clocks things are slightly more complicated because instead of a single cavity (Rabi scheme) there are actually two (Ramsey scheme). The reason for the two cavities is that they produce a quantum interference between two possible paths corresponding to excitation in the first cavity or excitation in the second, and this interference may be used to sharpen the resonance and to make it less dependent of inhomogeneities of the fields. A basic feature of the observed interference fringes in a standard Ramsey experiment is that their width is determined by the inverse of the time taken by the atoms to cross the intermediate drift region. For precision measurement purposes, as in atomic clocks, this motivates the use of very slow (ultracold) atoms, and therefore the development of laser cooling techniques has changed the entire prospects of frequency standards [Phi98]. Experimentally, atomic velocities of the order of 1 cm/s and smaller can be achieved, and space-based clocks are in development to elimi- nate gravitational effects in the motion of such slow particles [LLS + 98, SDA + 01]. Laser cooled atoms are also interesting in metrology and interferometry because of the possibility to achieve narrow velocity distributions and avoid averaging effects. In addition, fundamentally new effects may arise by using coherent few-body or many-body states as input in the form of condensates or otherwise: for example, there exist proposals to beat the limitations imposed by quantum projection noise using entanglement [WBI + 92, HMP + 97]. In spite of the above, the motto “the slower the better” in the context of atomic clocks has actually a limited domain beyond which quantum motional phenomena may affect strongly and eventually deform totally the usual Ramsey pattern. If the slow atom moves initially along the x -axis and the fields are oriented perpen- dicularly along the y -axis, there are two origins of modification of the standard Ramsey result [Ram50,Ram56]. First, the absorption of a photon leads to a transverse momentum transfer on the atom, such that the excited state separates in space from the ground state. This is negligible for microwaves but not for optically induced (one or two-photon) transitions. The effect can be understood classically by means of energy conservation and momentum conservation in y -direction. It has been studied in detail by Bordé and coworkers [Bor01, WBC + 02, Bor02] and multi-beam setups have been implemented to correct for this separation in order to observe quantum interference [BSA + 84, KC91, KC92, Mor92]. Second, the field acts as a barrier for the longitudinal motion of the atom, and quantum reflection and tunneling may occur. Thus, momentum in x -direction is not con- served as a consequence of the x -dependence of the fields. For microwave fields and the corresponding Rabi frequencies these quantized motion effects are tiny for present atomic velocities but may become important for deeply ultracold particles. Moreover, in view of a proclaimed near-future accuracy of frequency standards of 10 − 18 [SGN + ], even those tiny effects have to be studied beyond the limits of the standard theoretical description of the Ramsey pattern. Recently, an exact quantum result has been given for the Ramsey fringes of guided atoms as a function of the detuning including quantum tunneling and reflection by means of two-channel recurrence relations [SM07]. Apart from quantum motion effects affecting ensembles of independent particles, other effects are due to the importance of quantum statistics and interactions. The use of a Bose-Einstein condensate for an atomic clock immediately comes to mind, but the improvements associated with low velocities and narrow velocity distribution may be compensated by negative ...