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We explore the properties of ultra-narrow spectral holes in ensembles of solid-state emitters in crystals over a range of sub-kelvin temperatures, with a focus on their potential application in frequency stabilization schemes as an alternative to ultrastable cavities. We investigate how the parameters used to burn the spectral hole impact its shape, and how these factors determine the minimum achievable linewidth. In addition to the stability of the hole's center frequency, the linewidth and contrast play a crucial role in frequency locking. At sub-kelvin temperatures, the temperaturedependent T^7 broadening from two-phonon Raman scattering is expected to be negligible, and the spectral hole's linewidth should therefore remain constant in this interval. We observe however a linear broadening with increasing temperature, highlighting the need for further investigation into the mechanisms governing the linewidth at ultra-low temperatures.

We investigate the frequency response of narrow spectral holes in a doped crystal structure as a function of temperature below 1 K. We identify a particular regime in which this response significantly deviates from the expected two-phonon Raman scattering theory. Namely, near 290 mK, we observed a behaviour exhibiting a temperature-dependent frequency shift of zero, to first-order. This is of particular interest for applications which require high frequency-stability, such as laser frequency stabilization, as by operating the scheme at this specific point would result in the spectral hole frequency being highly immune to temperature fluctuations, providing the potential for a laser fractional frequency instability as low as $\mathrm{\sim6\times10^{-22}}$ at 1 s., Comment: 6 pages, 3 figures

The possibility of generating an narrow spectral hole in a rare-earth doped crystal opens the gateway to a variety of applications, one of which is the realization of an ultrastable laser. As this is achieved by locking in a pre-stabilized laser to the narrow hole, a prerequisite is the elimination of frequency fluctuations of the spectral hole. One potential source of such fluctuations can arise from temperature instabilities. However, when the crystal is surrounded by a buffer gas subject to the same temperature as the crystal, the effect of temperature-induced pressure changes may be used to counterbalance the direct effect of temperature fluctuations. For a particular pressure, it is indeed possible to identify a temperature for which the spectral hole resonant frequency is independent of the first-order thermal fluctuations. Here, we measure frequency shifts as a function of temperature for different values of the pressure of the surrounding buffer gas, and identify the ``magic'' environment within which the spectral hole is largely insensitive to temperature., Comment: 6 pages, 5 figures

The agile generation and control of multiple optical frequency modes combined with the realtime processing of multi-mode data provides access to experimentation in domains such as optomechanical systems, optical information processing, and multi-mode spectroscopy. The latter, specifically spectroscopy of spectral-hole burning (SHB), has motivated our development of a multi-mode heterodyne laser interferometric scheme centered around a software-defined radio platform for signal generation and processing, with development in an entirely open-source environment. A challenge to SHB is the high level of shot noise due to the laser power constraint imposed by the spectroscopic sample. Here, we have demonstrated the production, detection, and separation of multiple optical frequency modes to the benefit of optical environment sensing for realtime phase noise subtraction as well as shot noise reduction through multi-mode averaging. This has allowed us to achieve improved noise performance in low-optical-power interferometry. Although our target application is laser stabilization via SHB in cryogenic temperature rare-earth doped crystals, these techniques may be employed in a variety of different contexts.

We report a series of measurements of the effect of an electric field on the frequency of the ultranarrow linewidth $^7F_0 \rightarrow$ $^5D_0$ optical transition of $\rm Eu^{3+}$ ions in an $\rm Y_2SiO_5$ matrix at cryogenic temperatures. We provide linear Stark coefficients along two dielectric axes and for the two different substitution sites of the $\rm Eu^{3+}$ ions, with an unprecedented accuracy, and an upper limit for the quadratic Stark shift. The measurements, which indicate that the electric field sensitivity is a factor of seven larger for site 1 relative to site 2 for a particular direction of the electric field are of direct interest both in the context of quantum information processing and laser frequency stabilization with rare-earth doped crystals, in which electric fields can be used to engineer experimental protocols by tuning transition frequencies., Comment: https://aip.scitation.org/doi/full/10.1063/5.0025356

Material strain has recently received growing attention as a complementary resource to control the energy levels of quantum emitters embedded inside a solid-state environment. Some rare-earth ion dopants provide an optical transition which simultaneously has a narrow linewidth and is highly sensitive to strain. In such systems, the technique of spectral hole burning, in which a transparent window is burnt within the large inhomogeneous profile, allows to benefit from the narrow features, which are also sensitive to strain, while working with large ensembles of ions. However, working with ensembles may give rise to inhomogeneous responses among different ions. We investigate experimentally how the shape of a narrow spectral hole is modified due to external mechanical strain, in particular, the hole broadening as a function of the geometry of the crystal sites and the crystalline axis along which the stress is applied. Studying these effects are essential in order to optimize the existing applications of rare-earth doped crystals in fields which already profit from the more well-established coherence properties of these dopants such as frequency metrology and quantum information processing, or even suggest novel applications of these materials, for example as robust devices for force-sensing or highly sensitive accelerometers.

Rare-earth ion doped crystals for hybrid quantum technologies is an area of growing interest in the solid-state physics community. We have earlier theoretically proposed a hybrid scheme of a mechanical resonator which is fabricated out of a rare-earth doped mono-crystalline structure. The rare-earth ion dopants have absorption energies which are sensitive to crystal strain, and it is thus possible to couple the ions to the bending motion of the crystal cantilever. Here, we present the design and fabrication method based on focused-ion-beam etching techniques which we have successfully employed in order to create such microscale resonators, as well as the design of the environment which will allow to study the quantum behavior of the resonators.

A challenge of modern physics is to investigate the quantum behavior of a bulk material object, for instance a mechanical oscillator. We have earlier demonstrated that by coupling a mechanical oscillator to the energy levels of embedded rare-earth ion dopants, it is possible to prepare such a resonator in a low phonon number state. Here, we describe how to extend this protocol in order to prepare momentum- and position squeezed states, and we analyze how the obtainable degree of squeezing depends on the initial conditions and on the coupling of the oscillator to its thermal environment.

We present an experimental technique for realizing a specific absorption spectral pattern in a rare-earth-doped crystal at cryogenic temperatures. This pattern is subsequently probed on two spectral channels simultaneously, thereby producing an error signal allowing frequency locking of a laser on the said spectral pattern. Appropriate combination of the two channels leads to a substantial reduction of the detection noise, paving the way to realizing an ultra-stable laser for which the detection noise can be made arbitrarily low when using multiple channels. We use such technique to realize a laser with a frequency instability of $1.7\times 10^{-15}$ at 1 second, not limited by the detection noise but by environmental perturbation of the crystal. This is comparable with the lowest instability demonstrated at 1 second to date for rare-earth doped crystal stabilized lasers.

Rare-earth doped crystals have numerous applications ranging from frequency metrology to quantum information processing. To fully benefit from their exceptional coherence properties, the effect of mechanical strain on the energy levels of the dopants - whether it is a resource or perturbation - needs to be considered. We demonstrate that by applying uniaxial stress to a rare-earth doped crystal containing a spectral hole, we can shift the hole by a controlled amount that is larger than the width of the hole. We deduce the sensitivity of $\rm Eu^{3+}$ ions in an $\rm Y_2SiO_5$ matrix as a function of crystal site and the crystalline axis along which the stress is applied.

We present a brief review of our progress towards measuring parity violation in heavy-metal chiral complexes using mid-infrared Ramsey interferometry. We discuss our progress addressing the main challenges, including the development of buffer-gas sources of slow, cold polyatomic molecules, and the frequency-stabilisation of quantum cascade lasers calibrated using primary frequency standards. We report investigations on achiral test species of which promising chiral derivatives have been synthesized.

There is an increasing demand for precise molecular spectroscopy, in particular in the mid-infrared fingerprint window that hosts a considerable number of vibrational signatures, whether it be for modeling our atmosphere, interpreting astrophysical spectra or testing fundamental physics. We present a high-resolution mid-infrared spectrometer traceable to primary frequency standards. It combines a widely tunable ultra-narrow Quantum Cascade Laser (QCL), an optical frequency comb and a compact multipass cell. The QCL frequency is stabilized onto a comb controlled with a remote near-infrared ultra-stable laser, transferred through a fiber link. The resulting QCL frequency stability is below 10-15 from 0.1 to 10s and its frequency uncertainty of 4x10-14 is given by the remote frequency standards. Continuous tuning over ~400 MHz is reported. We use the apparatus to perform saturated absorption spectroscopy of methanol in the low-pressure multipass cell and demonstrate a statistical uncertainty at the kHz level on transition center frequencies, confirming its potential for driving the next generation technology required for precise spectroscopic measurements.

We report on the first comparison of distant caesium fountain primary frequency standards (PFSs) via an optical fiber link. The 1415 km long optical link connects two PFSs at LNE-SYRTE (Laboratoire National de m\'{e}trologie et d'Essais - SYst\`{e}me de R\'{e}f\'{e}rences Temps-Espace) in Paris (France) with two at PTB (Physikalisch-Technische Bundesanstalt) in Braunschweig (Germany). For a long time, these PFSs have been major contributors to accuracy of the International Atomic Time (TAI), with stated accuracies of around $3\times 10^{-16}$. They have also been the references for a number of absolute measurements of clock transition frequencies in various optical frequency standards in view of a future redefinition of the second. The phase coherent optical frequency transfer via a stabilized telecom fiber link enables far better resolution than any other means of frequency transfer based on satellite links. The agreement for each pair of distant fountains compared is well within the combined uncertainty of a few 10$^{-16}$ for all the comparisons, which fully supports the stated PFSs' uncertainties. The comparison also includes a rubidium fountain frequency standard participating in the steering of TAI and enables a new absolute determination of the $^{87}$Rb ground state hyperfine transition frequency with an uncertainty of $3.1\times 10^{-16}$. This paper is dedicated to the memory of Andr\'{e} Clairon, who passed away on the 24$^{th}$ of December 2015, for his pioneering and long-lasting efforts in atomic fountains. He also pioneered optical links from as early as 1997.

In this paper we report the evaluation of an optical lattice clock based on neutral mercury down to a relative uncertainty of $1.7\times 10^{-16}$. Comparing this characterized frequency standard to a Cs atomic fountain we determine the absolute frequency of the $^1S_0 \rightarrow \phantom{}^3P_0$ transition of $^{199}$Hg as $\nu_{\mathrm{Hg}} = 1 128\,575\,290\,808\,154.62\,$Hz $\pm\,0.19\,$Hz (statistical) $\pm\,0.38\,$Hz (systematic), limited solely by the realization of the SI second. Furthermore, by comparing the mercury optical lattice clock to a Rb atomic fountain, we determine for the first time to our knowledge the ratio between the $^{199}$Hg clock transition and the $^{87}$Rb ground state hyperfine transition. Finally we present a direct optical to optical measurement of the $^{199}$Hg/$^{87}$Sr frequency ratio. The obtained value of $\nu_{\mathrm{Hg}}/\nu_{\mathrm{Sr}}=2.629\,314\,209\,898\,909\,15$ with a fractional uncertainty of $1.8\times10^{-16}$ is in excellent agreement with the same measurement obtained by Yamanaka et al. (arXiv:1503.07941). This makes this frequency ratio one of the few physical quantities agreed upon by different laboratories to this level of uncertainty. Frequency ratio measurements of the kind of those reported in this paper have a strong impact for frequency metrology but also for fundamental physics as they can be used to monitor putative variations of fundamental constants., Comment: 23 pages, 12 figures, updated version including journal reference

Leveraging the unrivaled performance of optical clocks in applications in fundamental physics beyond the standard model, in geo-sciences, and in astronomy requires comparing the frequency of distant optical clocks truthfully. Meeting this requirement, we report on the first comparison and agreement of fully independent optical clocks separated by 700 km being only limited by the uncertainties of the clocks themselves. This is achieved by a phase-coherent optical frequency transfer via a 1415 km long telecom fiber link that enables substantially better precision than classical means of frequency transfer. The fractional precision in comparing the optical clocks of three parts in $10^{17}$ was reached after only 1000 s averaging time, which is already 10 times better and more than four orders of magnitude faster than with any other existing frequency transfer method. The capability of performing high resolution international clock comparisons paves the way for a redefinition of the unit of time and an all-optical dissemination of the SI-second., Comment: 14 pages, 3 figures, 1 table

In this article, we report on the work done with the LNE-SYRTE atomic clock ensemble during the last 10 years. We cover progress made in atomic fountains and in their application to timekeeping. We also cover the development of optical lattice clocks based on strontium and on mercury. We report on tests of fundamental physical laws made with these highly accurate atomic clocks. We also report on work relevant to a future possible redefinition of the SI second.

Progress in realizing the SI second had multiple technological impacts and enabled to further constraint theoretical models in fundamental physics. Caesium microwave fountains, realizing best the second according to its current definition with a relative uncertainty of 2-4x10^(-16), have already been superseded by atomic clocks referenced to an optical transition, both more stable and more accurate. Are we ready for a new definition of the second? Here we present an important step in this direction: our system of five clocks connects with an unprecedented consistency the optical and the microwave worlds. For the first time, two state-of-the-art strontium optical lattice clocks are proven to agree within their accuracy budget, with a total uncertainty of 1.6x10^(-16). Their comparison with three independent caesium fountains shows a degree of reproducibility henceforth solely limited at the level of 3.1x10^(-16) by the best realizations of the microwave-defined second., Comment: 9 pages, 4 figures, 2 tables

With Hg atoms confined in an optical lattice trap in the Lamb-Dicke regime, we obtain a spectral line at 265.6 nm in which the full-width at half-maximum is <15Hz. Here we lock an ultrastable laser to this ultranarrow clock transition and achieve a fractional frequency stability of 5.4x10^(-15)/sqrt(tau) for tau<=400s. The highly stable laser light used for the atom probing is derived from a 1062.6 nm fiber laser locked to an ultrastable optical cavity that exhibits a mean drift rate of -6.0x10^(-17) s^(-1) (or -16.9 mHz.s^(-1) at 282 THz) over a five month period. A comparison between two such lasers locked to independent optical cavities shows a flicker noise limited fractional frequency instability of 4x10^(-16) per cavity.

We present an assessment of the (6s^{2})1S0 -> (6s7s)3P0 clock transition frequency in 199Hg with an uncertainty reduction of nearly three orders of magnitude and demonstrate an atomic quality factor, Q, of ~10^(14). The 199Hg atoms are confined in a vertical lattice trap with light at the newly determined magic wavelength of 362.5697 +/-0.0011 nm and at a lattice depth of 20Er. The atoms are loaded from a single stage magneto-optical trap with cooling light at 253.7 nm. The high Q factor is obtained with an 80 ms Rabi pulse at 265.6 nm. The frequency of the clock transition is found to be 1 128 575 290 808 162.0 +/-6.4 (sys.) +/-0.3 (stat.) Hz (fractional uncertainty = 5.7x10^(-15)). Neither an atom number nor second order Zeeman dependence have yet to be detected. Only three laser wavelengths are used for the cooling, lattice trapping, probing and detection., Comment: 5 pages, 6 figures

We report on the characterization of a commercial-core fiber-based frequency comb equipped with an intracavity electro-optic modulator (EOM). We investigate the relationship between the noise of the pump diode and the laser relative intensity noise (RIN) and demonstrate the use of a low noise current supply to substantially reduce the laser RIN. By measuring several transfer functions, we evaluate the potentiality of the EOM for comb repetition rate stabilization. We also evaluate the coupling to other relevant parameters of the comb. From these measureemnts, we infer the capabilities of the femtosecond laser comb to generate very low phase noise microwave signals when phase locked to a high spectral purity ultra-stable laser., Comment: Submitted to IEEE UFFC Special issue

In this letter we report on an all optical-fibre approach to the synthesis of ultra-low noise microwave signals by photodetection of femtosecond laser pulses. We use a cascade of Mach-Zehnder fibre interferometers to realize stable and efficient repetition rate multiplication. This technique increases the signal level of the photodetected microwave signal by close to 18 dB, that in turn, allows us to demonstrate a phase noise level of 118 dBc/Hz at 1 Hz and -160 dBc/Hz at 10 MHz from 12 GHz signal. The noise floor of the fiber multiplier and photodetection system alone is around -164 dBc/Hz at the same offset frequency, which is very close to the fundamental shot noise floor, Comment: 3 pages, 4 figures, in press Optics Letters

When a photo-diode is illuminated by a pulse train from a femtosecond laser, it generates microwaves components at the harmonics of the repetition rate within its bandwidth. The phase of these components (relative to the optical pulse train) is known to be dependent on the optical energy per pulse. We present an experimental study of this dependence in InGaAs pin photo-diodes illuminated with ultra-short pulses generated by an Erbium-doped fiber based femtosecond laser. The energy to phase dependence is measured over a large range of impinging pulse energies near and above saturation for two typical detectors, commonly used in optical frequency metrology with femtosecond laser based optical frequency combs. When scanning the optical pulse energy, the coefficient which relates phase variations to energy variations is found to alternate between positive and negative values, with many (for high harmonics of the repetition rate) vanishing points. By operating the system near one of these vanishing points, the typical amplitude noise level of commercial-core fiber-based femtosecond lasers is sufficiently low to generate state-of-the-art ultra-low phase noise microwave signals, virtually immune to amplitude to phase conversion related noise., Comment: 7 pages, 6 figures, submitted to Applied Physics B

We report what we believe to be the lowest phase noise optical-to-microwave frequency division using fiber-based femtosecond optical frequency combs: a residual phase noise of -120dBc/Hz at 1 Hz offset from a 11.55GHz carrier frequency. We furthermore report a detailed investigation into the fundamental noise sources which afflicts the division process itself. Two frequency combs with quasi-identical configurations are referenced to a common ultrastable cavity laser source. To identify each of the limiting effects we implement an ultra-low noise carrier-suppression measurement system, which circumvents the detection and amplification noise of more conventional techniques. This technique now allows the suppression of these unwanted sources of noise to very low levels. In the Fourier frequency range of ~200 Hz to 100 kHz, a fed-forward technique based on a voltage-controlled phase shifter delivers a further noise reduction of 10 dB. For lower Fourier frequencies, optical power stabilization is implemented in order to reduce the relative intensity noise which causes unwanted phase noise through power to phase conversion in the detector. We implement and compare two possible control schemes based on an acousto-optical modulator and comb pump current. We also present wideband measurements on the relative intensity noise of the fiber comb., Comment: 16 pages, 10 figures accepted in

We have developed an ultra-stable source in the deep ultraviolet, suitable to fulfill the interrogation requirements of a future fully-operational lattice clock based on neutral mercury. At the core of the system is a Fabry-P\'erot cavity which is highly impervious to temperature and vibrational perturbations. The mirror substrate is made of fused silica in order to exploit the comparatively low thermal noise limits associated with this material. By stabilizing the frequency of a 1062.6 nm Yb-doped fiber laser to the cavity, and including an additional link to LNE-SYRTE's fountain primary frequency standards via an optical frequency comb, we produce a signal which is both stable at the 1E-15 level in fractional terms and referenced to primary frequency standards. The signal is subsequently amplified and frequency-doubled twice to produce several milliwatts of interrogation signal at 265.6 nm in the deep ultraviolet., Comment: 7 pages, 6 figures

We use two fiber-based femtosecond frequency combs and a low-noise carrier suppression phase detection system to characterize the optical to microwave synchronization achievable with such frequency divider systems. By applying specific noise reduction strategies, a residual phase noise as low as -120 dBc/Hz at 1 Hz offset frequency from a 11.55 GHz carrier is measured. The fractional frequency instability from a single optical-to-frequency divider is 1.1E-16 at 1 s averaging down to below 2E-19 after only 1000 s. The corresponding rms time deviation is lower than 100 attoseconds up to 1000 s averaging duration., Comment: 4 pages, 3 figures

In this letter, we report on all-optical fiber approach to the generation of ultra-low noise microwave signals. We make use of two erbium fiber mode-locked lasers phase locked to a common ultra-stable laser source to generate an 11.55 GHz signal with an unprecedented relative phase noise of -111 dBc/Hz at 1 Hz from the carrier.The residual frequency instability of the microwave signals derived from the two optical frequency combs is below 2.3 10^(-16) at 1s and about 4 10^(-19) at 6.5 10^(4)s (in 5 Hz bandwidth, three days continuous operation)., Comment: 12 pages, 3 figures

We present two ultra-stable lasers based on two vibration insensitive cavity designs, one with vertical optical axis geometry, the other horizontal. Ultra-stable cavities are constructed with fused silica mirror substrates, shown to decrease the thermal noise limit, in order to improve the frequency stability over previous designs. Vibration sensitivity components measured are equal to or better than 1.5e-11 per m.s^-2 for each spatial direction, which shows significant improvement over previous studies. We have tested the very low dependence on the position of the cavity support points, in order to establish that our designs eliminate the need for fine tuning to achieve extremely low vibration sensitivity. Relative frequency measurements show that at least one of the stabilized lasers has a stability better than 5.6e-16 at 1 second, which is the best result obtained for this length of cavity., Comment: 8 pages 12 figures

We demonstrate the use of a fiber-based femtosecond laser locked onto an ultra-stable optical cavity to generate a low-noise microwave reference signal. Comparison with both a liquid Helium cryogenic sapphire oscillator (CSO) and a Ti:Sapphire-based optical frequency comb system exhibit a stability about $3\times10^{-15}$ between 1 s and 10 s. The microwave signal from the fiber system is used to perform Ramsey spectroscopy in a state-of-the-art Cesium fountain clock. The resulting clock system is compared to the CSO and exhibits a stability of $3.5\times10^{-14}\tau^{-1/2}$. Our continuously operated fiber-based system therefore demonstrates its potential to replace the CSO for atomic clocks with high stability in both the optical and microwave domain, most particularly for operational primary frequency standards., Comment: 3 pages, 3 figures

We present a theoretical model for the propagation of non self-interacting atom laser beams. We start from a general propagation integral equation, and we use the same approximations as in photon optics to derive tools to calculate the atom laser beam propagation. We discuss the approximations that allow to reduce the general equation whether to a Fresnel-Kirchhoff integral calculated by using the stationary phase method, or to the eikonal. Within the paraxial approximation, we also introduce the ABCD matrices formalism and the beam quality factor. As an example, we apply these tools to analyse the recent experiment by Riou et al. [Phys. Rev. Lett. 96, 070404 (2006)].

Optical atomic clocks promise timekeeping at the highest precision and accuracy, owing to their high operating frequencies. Rigorous evaluations of these clocks require direct comparisons between them. We have realized a high-performance remote comparison of optical clocks over km-scale urban distances, a key step for development, dissemination, and application of these optical standards. Through this remote comparison and a proper design of lattice-confined neutral atoms for clock operation, we evaluate the uncertainty of a strontium (Sr) optical lattice clock at the 1x10-16 fractional level, surpassing the best current evaluations of cesium (Cs) primary standards. We also report on the observation of density-dependent effects in the spin-polarized fermionic sample and discuss the current limiting effect of blackbody radiation-induced frequency shifts., Comment: To be published in Science, 2008

We apply direct frequency-comb spectroscopy, in combination with precision cw spectroscopy, to measure the ${\rm 4s4p} ^3P_1 \to {\rm 4s5s} ^3S_1$ transition frequency in cold calcium atoms. A 657 nm ultrastable cw laser was used to excite atoms on the narrow ($\gamma \sim 400$ Hz) ${\rm 4s^2} ^1S_0 \to {\rm 4s4p} ^3P_1$ clock transition, and the direct output of the frequency comb was used to excite those atoms from the ${\rm 4s4p} ^3P_1$ state to the ${\rm 4s5s} ^3S_1$ state. The resonance of this second stage was detected by observing a decrease in population of the ground state as a result of atoms being optically pumped to the metastable ${\rm 4s4p} ^3P_{0,2}$ states. The ${\rm 4s4p} ^3P_1 \to {\rm 4s5s} ^3S_1$ transition frequency is measured to be $\nu = 489 544 285 713(56)$ kHz; which is an improvement by almost four orders of magnitude over the previously measured value. In addition, we demonstrate spectroscopy on magnetically trapped atoms in the ${\rm 4s4p} ^3P_2$ state., Comment: 4 pages 5 figures

We have performed sub-Doppler spectroscopy on the narrow intercombination line of cold calcium atoms using the amplified output of a femtosecond laser frequency comb. Injection locking of a 657-nm diode laser with a femtosecond comb allows for two regimes of amplification, one in which many lines of the comb are amplified, and one where a single line is predominantly amplified. The output of the laser in both regimes was used to perform kilohertz-level spectroscopy. This experiment demonstrates the potential for high-resolution absolute-frequency spectroscopy over the entire spectrum of the frequency comb output using a single high-finesse optical reference cavity., Comment: 4 pages, 4 Figures

We study the propagation of a non-interacting atom laser distorted by the strong lensing effect of the Bose-Einstein Condensate (BEC) from which it is outcoupled. We observe a transverse structure containing caustics that vary with the density within the residing BEC. Using WKB approximation, Fresnel-Kirchhoff integral formalism and ABCD matrices, we are able to describe analytically the atom laser propagation. This allows us to characterize the quality of the non-ideal atom laser beam by a generalized M$^2$ factor defined in analogy to photon lasers. Finally we measure this quality factor for different lensing effects.

We measure the angular divergence of a quasi-continuous, rf-outcoupled, free-falling atom laser as a function of the outcoupling frequency. The data is compared to a Gaussian-beam model of laser propagation that generalizes the standard formalism of photonic lasers. Our treatment includes diffraction, magnetic lensing, and interaction between the atom laser and the condensate. We find that the dominant source of divergence is the condensate-laser interaction., Comment: 5 pages, 3 figures. V3: as will appear in Phys. Rev. Lett.; RevTeX 4.0

We demonstrate a way to circumvent the interruption of evaporative cooling observed at high bias field for $^{87}$Rb atoms trapped in the (F=2, m=+2) ground state. Our scheme uses a 3-frequencies-RF-knife achieved by mixing two RF frequencies. This compensates part of the non linearity of the Zeeman effect, allowing us to achieve BEC where standard 1-frequency-RF-knife evaporation method did not work. We are able to get efficient evaporative cooling, provided that the residual detuning between the transition and the RF frequencies in our scheme is smaller than the power broadening of the RF transitions at the end of the evaporation ramp., Comment: 12 pages, 2 figures

We present the design of our iron-core electromagnet for BEC, and how to solve the specific experimental problems raised by this technique. After presenting the experimental set-up, we address the interruption of runaway evaporative cooling when the Zeeman effect is not linear. We present the ways to circumvent this problem, use of multiple RF frequencies, sympathetic cooling and show some applications of these high magnetic fields (cavity coupling, high confinement)., Comment: 22 pages, 15 figure