Frequency metrology in quantum degenerate helium

We report the first direct observation and high precision measurement of the triplet to singlet transition between the metastable states of helium. We obtain a precision of 8×10<sup>−12</sup>, providing a stringent test of two-electron quantum electrodynamic theory and of nuclear few-body theory. We use either the <sup>4</sup>He or the <sup>3</sup>He isotope, and from the isotope shift derive a new value for the <sup>3</sup>He charge radius.


1
Introduction are based on the interaction between the electric field of a laser and the electricdipole moment it induces in an atom (an extensive review of optical dipole traps for neutral atoms is presented in Ref. [25]). Optical dipole traps can be tuned to confine atoms in either the electric fields maxima or minima, and can be shaped into lattice-type potentials originating from the interference between two or more laser beams. Optical lattices are used to study a wide variety of quantum many-body physics in periodic potentials [26]. Optical dipole traps, as an alternative to magnetic traps, provide the freedom of trapping additional spin states of the atoms. Contrary to magnetic traps, where only low-field seeking spin states are trapped, in optical dipole traps the magnetic quantum number is a free experimental parameter, allowing the stability of various spinmixtures to be investigated. The last two chapters of this thesis are related to experiments involving the stability of He * atoms in different spin states, and the lifetime of a BEC, in the case of metastable helium.
A valuable tool which becomes available with the freedom of an external magnetic field is the control over interactions between atoms via Feshbach resonances [27,28] (for an extensive review see Ref. [29]). Through the tuning of external magnetic fields, the strength of interactions between various atomic species can be controlled. Prominent examples for the applications of Feshbach resonances are the study of the crossover between a BEC of molecules and the Bardeen-Cooper-Schrieffer (BCS) state of Bose-Einstein condensed Cooper pairs [30,31], and the study of Efimov states and universal few-body physics [32,33].
Specific to the metastable noble gases are the large internal energies of the trapped atoms and the additional methods for detection this entails. Upon impact with a metal surface, an electron is expelled from the surface which can subsequently be detected. This provides a unique way of atom detection, with time-resolved and position-sensitive counting of atoms using a micro-channel plate detector. Notable experiments making use of this observation method are studies of pair correlations in atomic four wave mixing [34,35], and the direct observation of the Hanbury-Brown Twiss effect for both bosons [36] and fermions [37], as well as higher-order correlations [38].

Precision spectroscopy in helium
Trapping atoms at extremely low temperatures is advantageous for precision and quantum-optic measurements. Precision spectroscopy is an invaluable ingredient in the most accurate optical atomic clocks. With atoms (or ions) residing nearly motionless in a trap, extremely weak and narrow transitions can be interrogated in the same atoms or ions for prolonged periods of time, which is required to reach the exceedingly high accuracies. The most accurate clock to date is a quantum logic clock based on a single aluminum ion [39], reaching a relative precision of 8.6 × 10 −18 . Experiments involving optical atomic clocks based on neutral atoms, the magic-wavelength optical lattice clocks [40], are aiming for similar precision [41]. Frequency metrology in helium is conventionally performed in gas-cells or atom beams. A very accurate measurement of the 3 He hyperfine structure of the metastable 2 3 S 1 state (∆f = 6 739 701 177(16) Hz) was performed in the 1970s by Rosner and Pipkin [42], using the atomic-beam magnetic-resonance method. Current precise measurements of helium fine structure and hyperfine be considered as a ground state, excitations to the n 3 P manifolds test calculations of the fine-structure splitting. When high-precision spectroscopy is combined with high-precision theory, the 2 3 P splitting may be used to derive a competitive value for the fine-structure constant [45,56]. On the other hand, comparing isotope shift measurements with high-precision QED theory, one can determine the difference in the nuclear charge radii [57]. In the calculation of the 3 He and 4 He ionization energies, the mass-independent terms are common between the isotopes and therefore cancel, which reduces the uncertainty to the sub-kHz level in the isotope shift calculations. At this precision, the influence of the finite nuclear charge radius of the nuclei becomes apparent. By comparing with similarly accurate experimental determinations of the isotope shift, the difference between the squared nuclear charge radii of the isotopes can be deduced. As the 4 He nuclear charge radius is one of the most precisely known of all nuclei [58], this procedure can provide the 3 He nuclear charge radius with similar precision.
This thesis presents the first observation and measurement of the extremely weak 2 3 S 1 -2 1 S 0 magnetic-dipole transition, in both 3 He and 4 He (see Fig. 1.1(a)). This required the combination of ultracold atoms, pinpointed in an optical dipole trap to provide long interrogation times, and a stable absolute frequency reference in the form of a femtosecond frequency comb laser. The measured transition frequencies, at a fractional uncertainty of 10 −11 , provide a stringent test of two-electron QED theory. Relying on high-precision calculations of the isotope shift, a value for the nuclear charge radius of 3 He is determined.

Outline of this thesis
Introduction Finally, a new computer program has been written to control the experimental sequences.
In chapter 3 the frequency-metrology measurement is discussed to which this thesis lends its title. The experimental procedures are discussed, as well as the various systematic effects which perturb the measured transition frequency. The measurements are compared with state-of-the-art two-electron QED theory, and through the isotope shift (i.e., the difference between the 3 He and 4 He transition frequency) the difference of the squared nuclear charge radii is obtained.
Chapter 4 reports on a measurement which confirms long-standing calculations of the magnetic field dependence of the two-body loss rate in a Bose-Einstein condensate of 4 He in the 2 3 S 1 , m = +1, state. An improved three-body loss rate coefficient was measured for atoms in the m = −1 state.
Finally, in chapter 5 an experiment demonstrating the different one-body decay rates of the condensate-and thermal-fraction of a trapped atomic sample is discussed. This effect is attributed to the transfer of condensate atoms to the thermal cloud during its decay.

Experimental setup
The experimental setup, used for cooling and trapping of metastable helium atoms, forms the basis of the experiments described in this thesis. The apparatus was built by Rooijakkers in 1992Rooijakkers in -1996 [59] for the production of magneto-optically trapped metastable helium [60] and has since been under constant modification, evolving into a machine capable of reliably producing mixtures of a 4 He * Bose-Einstein condensate and a 3 He * degenerate Fermi gas. Extensive descriptions of earlier stages of the the experimental setup can be found in the theses of Herschbach [61] and Tol [62]. A more recent account is presented in the thesis of Tychkov [63], describing the introduction of a new magnetic trap into the apparatus which culminated in the production of a Bose-Einstein condensate in 2005 [15]. The work of Stas [64] and McNamara [65] paved the way for the introduction of ultracold Fermi gases of 3 He * into the system a year later [18]. A collaboration with the cold helium group of the Institute d'Optique in Orsay resulted in an experiment demonstrating the contrasting bunching and anti-bunching behavior for 4 He * and 3 He * , respectively, attributed to the different quantum statistics of these species (elaborated on in the thesis of Jeltes [66]).
For the most recent experiments, the laser systems have been upgraded to stabler and more powerful versions. The introduction of an optical dipole trap has opened up the possibility to perform measurements on various spin states. In contrast, in a magnetic trap only the states with a positive magnetic quantum number are trapped. The use of an optical trap makes the magnetic trapping potential redundant, thus allowing complete freedom in tuning of the magnetic field. The source chamber is filled with helium, of which a small fraction enters the collimation section through a skimmer. The remainder is recycled, so as to waste a minimal amount of 3 He. In the collimation section, helium atoms in the metastable 2 3 S 1 state are collimated into a beam and deflected around a knife-edge (not shown), which blocks the groundstate helium atoms. Through the Zeeman slower, the atoms enter the magneto-optical trap (MOT) in the trap chamber.

Atomic beamline
As the apparatus has been extensively documented, the following will be a brief overview with an emphasis on the differences between the current state of the apparatus and the incarnation of the setup as documented in the thesis of Tychkov [63].

Vacuum system
Experiments on ultracold, trapped, gases are necessarily performed under ultra-high vacuum (UHV) conditions to provide an environment where collisions with background gas (leading to trap loss) are infrequent, i.e., collision rates much lower than one per second. Therefore, the apparatus is built up in several sections of differential pumping (see Fig. 2.1), separating the highvacuum source chamber (5 × 10 −5 mbar) from the ultra-high vacuum trap chamber (at approximately 1 × 10 −10 mbar) where the experiments are conducted. Along the beam line are a total of six turbomolecular pumps: at the source, the collimation section, the start and the end of the Zeeman slower, and two are connected to the trap chamber (for detailed information see [18]).
Separating the source chamber and the collimation section is a 1 mm diameter skimmer, which allows the pressure in the latter to be 10 −7 mbar. The small fraction of helium atoms, entering through the skimmer, which is in the metastable state, is collimated into a beam and deflected around a knifeedge. The knife-edge blocks the direct line of sight between the nozzle and the entry to the Zeeman slower, such that the beam of ground-state helium atoms is blocked from entering the UHV trap chamber. A 10 cm long, 3 mm diameter, tube had previously been present at the entry of the Zeeman slower -in line with the atomic trajectory -with the purpose of differential pumping, resulting in a lifetime of a trapped metastable helium sample of 180 s. The tube had, however, slightly misaligned over time, effectively blocking a part of the atomic trajectory and thus reducing the flux of metastable helium. For this reason the tube has been removed and replaced by a flange with a 3 mm diameter hole at its center, resulting in a recovered flux yet at the expense of a reduced efficiency of the differential pumping stage. Because of this, the lifetime of a trapped metastable helium sample is reduced to approximately 100 s, which is still sufficiently long for any of the current, and planned, experiments.

Source
Through electron bombardment in a DC-discharge, a small fraction of helium atoms is excited to the metastable 2 3 S 1 state. Helium is inserted into the source chamber through a quartz tube, which houses a tantalum needle and a boron nitride nozzle (0.25 mm inner diameter). The nozzle is located between the needle and the skimmer (separating the source chamber from the collimation section), across which the discharge is maintained. Cooling the nozzle with liquid nitrogen, the most probable initial velocity is reduced by a factor of two to approximately 1.1 km/s for 4 He (1.3 km/s in the case of 3 He).
For experiments involving only 4 He, a constant flow of helium gas is provided directly from a pressurized tank, the excess of which is pumped out of the source chamber by the turbomolecular pump, backed by a diaphragm pump. When working with 3 He (or a mixture of 3 He and 4 He) however, a recycling apparatus is used to filter and recycle the helium gas, as 3 He is quite expensive. Instead of a diaphragm pump for backing, a helium tight scroll pump is employed. The exhaust of which is connected to the recycling apparatus. Inside the recycling apparatus are two LN 2 -cooled molecular sieves, made of small pored material (sodium zeolite, pore sizes of severalÅ) to capture anything except helium. An assembly of valves allows the recycling system to be filled with an arbitrary mixture of 3 He and 4 He, and to be isolated from the rest of the experiment when its use is not required.
The function of the collimation section of the apparatus is twofold: to collimate the divergent atomic beam coming through the source skimmer, and to deflect the collimated beam around a knife-edge, which blocks the ground state helium. A curved-wavefront technique [67] is employed to collimate and deflect the metastable 2 3 S 1 atoms in the beam, the ground-state atoms are not affected in this procedure. In short, laser beams resonant to the (1083 nm) 2 3 S 1 → 2 3 P 2 transition are elliptically stretched (providing a long interaction region) and enter the collimation chamber both vertically and horizontally, and are retro-reflected back out again. Being slightly convergent, with a radius of curvature of 12 m, the light at the interaction region (starting about 3 cm from the skimmer, into the collimation chamber) has a curved wavefront, forming trajectories which are followed by the metastable 2 3 S 1 atoms. The atomic beam is steered through the Zeeman slower and further into the magnetooptical trap (MOT). The radius of convergence and the alignment of the beams are empirically optimized to obtain the largest loading rate of the MOT.

Laser cooling and trapping
From the collimation section the atoms travel through a 3 mm hole and into two meters of Zeeman slower before entering the MOT. The most probable velocity of over 1 km/s of the atoms entering the slower is to be reduced to below the maximal capture velocity of the MOT, approximately 60 m/s. The atoms are decelerated by the radiation pressure force induced by a circularlypolarized laser beam, detuned by -250 MHz from the 2 3 S 1 → 2 3 P 2 cycling transition, counter-propagating the mean atomic trajectory. With each absorption/spontaneous-emission cycle, the momentum of an atom is on average reduced by the recoil of the absorbed photon. The atoms remain resonant with the laser light by compensating the detuning, ∆, and the Doppler shift, kv(z), by a position-dependent magnetic field, B(z), which induces a differential Zeeman shift between the 2 3 S 1 (m = +1) and 2 3 P 2 (m = +2) states of µ B B(z)/ . The resonance condition, µ B B(z)/ = kv(z) − ∆, where k is the wave vector of the laser and µ B is the Bohr magneton, is fulfilled as atoms move into and out of resonance as they are decelerated through the Zeeman slower. The maximum magnetic field is kept below 563 G (corresponding to a resonant velocity of 1.13 km/s) because of a level crossing in the 2 3 P manifold, which could lead to a depopulation of the cycling transition. The maximal capture velocity of the MOT (60 m/s) corresponds to a field of -138 G. In practice, the fields are empirically optimized to obtain the maximum loading rate of the MOT. A much more complete description of the dynamics of the deceleration process in the Zeeman slower can be found in the theses of Stas [64] and Tol [62].
From the Zeeman slower the atoms travel into the UHV trap chamber, where they are captured in the MOT. Magneto-optical trapping is a well established technique [4], which allows atoms to accumulate and cool before being loaded into a purely magnetic trap, necessary for further evaporative cooling. In short, three pairs of large sized (3 cm diameter) counter-propagating laser beams, perpendicular to each other, intersect at the trap center. The beams are red detuned from the atomic transition such that a velocity-dependent force is exerted to the atoms. A quadrupole magnetic field is applied, using a set of coils in anti-Helmholtz configuration, to provide a spatial dependence to the force. The two beams of one counter-propagating pair carry opposite circular polarization, such that they address different magnetic substates. The combination of red-detuning and state-dependent interaction results in the radiation pressure force being exerted towards the trap center. At the center of the trap, where the magnetic field is small, only the velocity-dependent force is of relevance. With velocities averaging out to about 3 m/s, the temperature of this optical molasses is approximately 1 mK. Special to the helium MOT is the relatively large detuning of -35 MHz (-22Γ) which is used to create a large trapping volume, avoiding high densities which lead to Penning ionization (He * + He * → He + He + + e − ) and associative ionization (He * + He * → He + 2 + e − ), and at the same time results in a low absorption probability at the trap center leading to a smaller loss rate due to light-assisted ionizing collisions.

Evaporative cooling
From the MOT, the helium atoms are transferred into a purely magnetic trap where the temperature can be further decreased, and the phase-space density increased. In the process of switching off the MOT and switching on the magnetic trap, a spin polarizing laser beam is applied. Since only atoms with a positive magnetic moment are trapped in a magnetic field minimum, the equally distributed spin population of the 2 3 S 1 manifold in the MOT, i.e., m J = {−1, 0, +1} in the case of 4 He * , is optically pumped to the m J = +1 state (m F = +3/2 for 3 He * ). Besides being trapped in a local field minimum, these states are also much more stable against Penning ionization [21] (the stability of different spin populations is discussed in chapter 4), a necessity for inner coil pair (pinch coils) provides a confining field in the axial direction. The planar cloverleaf coils are responsible for radial confinement. The large coil pair (compensation coils) is in Helmholtz configuration, providing an offset to the magnetic field magnitude at the trap center and allowing the trap to be compressed. Figure  from Ref. [63].
the production of quantum degenerate samples of metastable helium. The spin population is optically pumped by applying a slightly detuned σ + -polarized pulse of 1.2 ms duration at a certain, empirically chosen, time in between switching the two traps when the local magnetic field has the right magnitude to fulfill the resonance condition. The atoms are then loaded into the magnetic trap, which is generated by a Ioffe-Pritchard type cloverleaf configuration of current carrying solenoids, suspended in tightly closed plastic containers for water cooling (implemented by Tychkov, for details about the design the reader is referred to his thesis [63]). Fig. 2.2 depicts the coil configuration. The inner pair of coils (pinch coils) generates a saddle-point potential, providing confinement in the axial direction. Each pinch coil is surrounded by a set of four planar cloverleaf coils which are responsible for a quadrupole magnetic field, providing radial confinement. An additional field is generated by a Helmholtz coil pair (compensation coils), which compensates the field in the center of the trap due to the pinch coils, but has little effect to the curvature in the axial direction, thus providing a tunable offset at the center of the trap close to zero magnetic field (although not exactly zero so as to avoid losses due to Majorana spin flips [61,68]). At the same time, the compensation coils allow for a compression of the trap in the radial direction.
After the transfer, the temperature of the cloud is approximately 1.2 mK. Subsequently, by means of one-dimensional Doppler cooling the temperature is reduced to 0.12 mK, which is three times the Doppler limit ( Γ 2kB = 39 µK). A weak, σ + -polarized beam is retroreflected along the axial direction of the trap, red-detuned from the 2 3 S 1 → 2 3 P 2 transition. Due to the red detuning, atoms counter-propagating the beam are more likely to absorb photons than those co-propagating, thus reducing the atoms average momentum in the axial direction. Spontaneous emission of the photons in random directions and reabsorption within the cloud additionally reduces the momentum along the radial directions. During this process, which requires two seconds, nearly all atoms remain trapped while the phase-space is increased by roughly three orders of magnitude.
During the last cooling stage quantum degeneracy is realized by application of radio frequency (RF) induced evaporative cooling. Before starting the evaporative cooling stage the trap is adiabatically compressed to increase the density, and hence elastic collision rate, of the gas. To adiabatically compress the trap, the current through the compensation coils is slowly 1 increased, thereby lowering the magnetic field at the trap center (to approximately 10 G), whilst the radial trap frequency is increased from 0.07 kHz to 0.4 kHz (for 4 He * , in the case of 3 He * the trap frequencies are higher by a factor 4/3 due to the mass difference). In short, during RF-induced evaporative cooling the most energetic atoms are removed from the trap, while the remaining atoms rethermalize at a lower average temperature. Although atoms are lost from the trap, the density of the gas is increased due to the decreasing temperature, resulting in an increasing phase-space density as well as a higher elastic collision rate. The RF field induces spin-flip transitions, thereby coupling trapped and non-trapped states of 4 He * (m J = +1 → m J = 0, −1). By slowly decreasing the radio frequency, the magnetic field at which the transitions are resonantly driven (i.e., when 2µ B B(r) = ω RF ) decreases as well. Thus, when starting at a relatively high frequency of the RF field, only the most energetic atoms are able to enter the region of the trap where the resonance condition is fulfilled, and are expelled from the trap. Gradually lowering the radio frequency then cools the sample down until quantum degeneracy is reached.
Evaporatively cooling pure gases of 3 He * using this method is not an option, as s-wave collisions are forbidden for identical fermions. The higher order partial wave expansions only contribute at higher temperatures: the centrifugal barrier for p-wave collisions is 10 mK [20]. A 3 He * -4 He * mixture, however, does experience s-wave collisions, allowing 3 He * to be sympathetically cooled by remaining in thermal equilibrium with evaporatively cooled 4 He * .

Optical dipole trap
After the evaporative cooling process the atoms are loaded into a far offresonant optical dipole trap. This kind of trap has several advantages over magnetic traps, most notably, it allows trapping of atoms in the lowest internal energy state of the 2 3 S 1 manifold, thereby reducing two-body losses. Furthermore, it allows for tuning of magnetic bias fields without affecting the trapping, which is of importance when studying magnetic field dependent interaction properties such as Feshbach resonances. Another advantage is the ability to rapidly switch off the trap and have the atoms ballistically expand, unaffected by decaying magnetic field gradients.

Basic principles of dipole traps
When placed in an oscillating electric field, an atom obtains an induced electric dipole moment which oscillates at the driving frequency of the field. The ratio between the amplitude of the induced electric dipole moment and the magnitude of the electric field is given by the complex polarizability α. Classically, following Ref. [25], the polarizability can be obtained from the Lorentz model, i.e., by considering an electron to be bound elastically to the core with eigenfrequency ω 0 . The system is damped through dipole radiation of the oscillating electron with a on-resonance damping rate Γ. It is given by: where ω is the driving frequency of the electric field. The potential energy associated with the induced dipole moment is given by the real, dispersive, part of the complex polarizability, which describes the magnitude of the in-phase oscillation of the dipole moment: with I(r) the intensity of the radiation field. Associated with the dipole potential is a conservative force, proportional to the intensity gradient of the light field, which allows for atoms to be trapped. The magnitude of the outof-phase oscillation is related to the imaginary part of the polarizability, and is associated with the rate with which photons are absorbed and spontaneously re-emitted: Considering the atom to be a two-level quantum system, the damping rate Γ corresponds to the radiative linewidth of the transition, and the dipole potential is essentially the AC Stark shift of the initial state. In the far-off resonant regime, i.e., when the detuning, ∆ ≡ ω − ω 0 , is much larger than the radiative linewidth, |∆| Γ, the evaluation of these relations results in the following expressions for the dipole potential and the scattering rate: In the case of far-red detuned laser light, irradiating on a two-level atom, the dipole potential has a minimum value when the intensity is at a maximum, constituting a trapping potential for the atom. When, on the other hand, the driving frequency is much larger than the resonance frequency (ω ω 0 , i.e., blue detuned), the real part of the polarizability is negative: the induced dipole moment is lagging behind the oscillating electric field, resulting in a dipole-potential minimum when the light intensity is at a minimum as well.
For a helium atom in the 2 3 S 1 state, irradiated by laser light at 1557 nm, the above approximation of a two-level atom works fairly well: it is far reddetuned from the nearest resonant transitions to the 2 3 P manifold at 1083 nm (by ∆/2π ≈ 84 THz), but even more so from the transitions to n 3 P with n ≥ 3, such that these transitions contribute much less to the polarizability. Also, the fine structure splitting of the 2 3 P 0,1,2 levels (on the order of 10 GHz) is not resolved as it is much smaller than the detuning, such that in this case, the helium atom in the 2 3 S 1 state can be considered as a two-level system. This results in a complex polarizability of the two-level system of: In a more rigorous approach, the AC Stark shift can be calculated by time-dependent second-order perturbation theory, see Sobel'man [69,70]. The dipole shift can be expressed in terms of dynamic polarizabilities, which take into account all the states coupled by the electric dipole operator, expressed in terms of the spontaneous decay rates A ki : , (2.8) where q = −1, 0, 1 represent the polarization components of the electric field and the summation over k runs over all excited states coupled to the 2 3 S 1 state by the electric dipole operator. Calculating the dynamic polarizability of the 2 3 S 1 , m = +1, state, irradiated with light at 1557.3 nm and taking linear polarization (q = 0), gives: where all electric dipole transitions up to n = 10 were included from the NIST Atomic Spectra Database. The real part of α 2 , calculated for the two-level system, deviates from α d by 1.3%, indicating the validity of the two-level model for this particular situation.

One-beam trap
Optical dipole traps allow for a wide range of trapping geometries. A straightforward configuration is a single focused, red detuned, Gaussian beam. The dipole potential resulting from such a beam, propagating along the y-axis, is given by: where w(y) = w 0 1 + (y/y R ) 2 , y R = πw 2 0 /λ is the Rayleigh range and w 0 is the beam waist. The depth of the dipole trap, U 0 , at the focus of the beam, is given by Eq. 2.2, with I(0, 0) = 2P πw 2 0 , where P is the power of the beam. The center of the trap is approximately harmonic. A power series expansion up to second order, around (r, y) = (0, 0) gives a potential of the form U (r, y) = −U 0 + 1 2 m(ω 2 r r 2 + ω 2 y y 2 ), with trapping frequencies: This shows that the axial confinement is weaker than the radial confinement by a factor √ 2πw 0 /λ, which, in case of a beam waist of w 0 = 85 µm and a wavelength of λ = 1.557 µm (our experimental parameters), gives rise to a very elongated trapping potential: ω r ≈ 250 × ω y .  an absorption image of atoms which are just loaded from the magnetic trap into a one-beam dipole trap, still in the process of moving outwards along the axial direction due to the much lower confinement in that direction, and will eventually collide with the re-entrant window of the vacuum chamber. The image on the right is a contour plot of the trapping potential, where the vertical axis was stretched by two orders of magnitude.

Crossed-beam trap
A crossed-beam configuration is employed to increase the strength of the axial confinement. This is done by intersecting two single-beam dipole traps (ideally of equal intensity) at their foci, under a considerable relative angle, θ. In the region where the two beams intersect, the AC Stark shift doubles and a dimple in the dipole potential arises ( Fig. 2.4). In the, approximately harmonic, center part of the trap, the trapping frequency in the direction orthogonal to the plane of the two beams is: 13) similar to that of the one-beam trap above (although here the intensity, and hence U 0 , is twice as large). In the plane of the two beams the potential has two symmetry axes (assuming the beams are of equal shape and intensity), where the trap frequencies are approximately 2 : In practice the beams pass through the (uncoated) re-entrance windows of the vacuum chamber, with each pass decreasing in power by approximately 8%. The result is that the incident trap beam has 30% more power than the returning trap beam, which consequently leads to the effective dipole trap, i.e., the dimple, being only as deep as the dipole potential caused by the returning beam. Atoms for which the energy exceeds the trap depth will therefore leave the trap along the more powerful incident beam.

Experimental implementation
The crossed-beam optical dipole trap is generated by a commercial fiber laser (NP Photonics Scorpio), producing up to 2 W of power at a wavelength of 1557 nm. The crossed-beam trap consists of one beam, which is directed through the vacuum chamber twice and intersects with itself at the center of the magnetic trap, as depicted in Fig. 2.5. The incident beam and returning beam carry orthogonal linear polarizations to avoid standing waves due to interference. The waist of both of the beams is approximately 85 µm, and their mutual angle is 19 degrees, limited by the dimensions of the re-entrant windows. The power of the returning beam is measured on a thermopile power meter, and ranges between 50 mW and 700 mW depending on the experiment.
For a laser power of 100 mW (of the returning beam as measured on the power meter), the trap depth calculates to be approximately U 0 /k B = 3 µK, and the off-resonant scattering rate Γ sc = 1/300 s −1 . The corresponding trap frequencies, according to Eqs. 2.13, 2.14 and 2.15, are: (ω z , ω x , ω y ) = 2π × (304, 300, 50) Hz. To measure the trap frequencies, a dipole oscillation of the trapped atoms is excited by applying a weak magnetic-field gradient for a short time. The atoms are then held for a variable amount of time, after which the trap is turned off and the atoms ballistically expand, with a center-of-mass velocity resulting from the dipole oscillation. The oscillation frequency is recovered by imaging the atoms several ms after release from the trap, and plotting the center-of-mass position as a function of holding time (see Fig. 2.6). The trapping frequencies recovered using this approach are (ω x , ω y ) ≈ 2π × (365, 55) Hz. The discrepancy with the calculated values is likely due to uncertainties in estimating the light intensity at the trap center.

Optical setup 2.3.1 Laser systems
The laser system of the setup has been subject to large revision since the system as described in the theses of McNamara [65], Jeltes [66], Tychkov [63], Tol [62] and Herschbach [61]. An LNA laser 3 and two distributed-Braggreflector (DBR) diode lasers were used to generate the 1083-nm light required for collimation and deflection, laser cooling, spin polarization, Doppler cooling,  and absorption imaging of 4 He * . A commercial 2-Watt fiber laser 4 (IPG YLD-1BC-1083), a diode laser (Toptica DL100) and a DBR diode laser were used to generate light for the same purposes for 3 He * . With the introduction of commercial fiber lasers and amplifiers, which provide more power at a relatively narrow linewidth, the collection of lasers has been reduced to two seed/amplifier fiber-based systems operating at 1083 nm, considerably improving the stability of the setup. For 4 He, a Koheras Adjustik commercial fiber laser is used to seed a Keopsys amplifier, for 3 He, the aforementioned Toptica DL100 diode laser is used as a seed for a Nufern amplifier. The output power of these amplifiers (> 1 W) is more than sufficient to generate the required beams. An additional laser system, invaluable to the experiments described in this thesis, is the NP Photonics (Scorpio) 1557-nm fiber laser, which is used to generate an optical dipole trap and provides the right wavelength to excite the 2 3 S 1 → 2 1 S 0 transition.
The Toptica diode laser has been extensively described in the thesis of McNamara [65]. The Koheras is a single frequency distributed feedback (DFB) fiber laser system. It is based on an Ytterbium-doped single-mode optical fiber, in which population inversion is established by a semiconductor pump diode. A Bragg grating, imprinted in the active fiber, ensures the narrow emission linewidth. The laser frequency can be tuned around the central frequency, determined by the Bragg grating, via temperature adjustment of the active fiber. This results in a 6 GHz/K tuning ratio, for a maximum range of 180 GHz. For fast modulation a piezoelectric transducer (PZT, or piezo) is used, which slightly stretches the fiber so as to modify its refractive index. The induced laser frequency response in relation to the voltage on the piezo is 4 GHz over a 200 V range. The modulation bandwidth of the piezo is on the order of 10 kHz.
Both the Keopsys and the Nufern systems are fiber amplifiers. They are based around an Ytterbium-doped optical fiber, which is also, as in the Koheras seed laser, pumped by diode lasers to establish population inversion. The emission wavelength is controlled by the seed laser, which is injected into the active fiber of the amplifier to induce stimulated emission. Amplified in several stages, the Keopsys and Nufern amplifiers produce an output power of 1.3 W and 2.4 W, respectively.
The 1557-nm, NP Photonics, laser system includes a seed and amplifier module in one integrated system. The seed is an erbium-doped microfiber, pumped by a single mode diode laser. The fiber amplifier provides a little over 2 W of power. Most of the systems' functionality is controlled using a computer interface via RS-232, although the front panel offers some adjustment settings.

Frequency locking
The frequency of the 1083-nm seed laser is locked through a saturated absorption spectroscopy setup. The method is described extensively in the aforementioned theses. Briefly, the basic principle is that an atomic transition (in this case, the 1083-nm transition between the 2 3 S 1 and 2 3 P 2 states) is probed to well within the Doppler-broadened profile. Fig. 2.7 schematically depicts the setup. An RF-discharge is established in a helium gas cell to populate the metastable 2 3 S 1 state. A pump beam is directed through the cell to saturate the transition for some velocity class of the atoms. A probe beam, derived from the pump beam but much weaker in power, is retro-reflected back through the gas cell and its intensity is monitored on a photodiode. If the laser frequency is detuned from the atomic transition frequency, the probe beam will be absorbed  by atoms with the same velocity-class, but moving in opposite direction. If, on the other hand, the laser frequency is at the atomic transition frequency, the pump and probe beams address the same atoms (i.e., those without parallel velocity components), in which case the medium becomes more transparent for the probe beam. Thus, scanning the laser frequency across the atomic transition and monitoring the probe beam intensity, this feature appears as a bump in the Doppler broadened absorption profile, referred to as the Lamb dip. This feature is only slightly broader than the natural linewidth of the transition (mostly due to power broadening), and is used to stabilize the laser. An error signal is obtained from the saturated absorption signal using a lock-in amplifier. The laser frequency is modulated at 2 kHz, by a small amplitude. The lock-in amplifier demodulates the output of the photodiode to recover the error signal, which is passed to a PID controller which in turn is connected to the modulation input of the seed laser to lock its frequency.

Beam layout
With the fiber-based laser amplifiers, an excess of laser power has become available to the experiment, allowing for the introduction of optical fibers into the system to separate sections of the optical setup. This has several practical advantages. A fiber directly after the seed/amplifier laser system allows for easy swapping of a laser in case of a problem with either a seed or an amplifier system, without needing to realign any of the optics. It allows for easy trouble shooting since any beam misalignment is easily spotted by noting after which fiber the power has dropped. Generally speaking, large sections of the optical setup become uncoupled from one another. Fig. 2.8 displays the beam layout. The saturated absorption spectroscopy setups for locking of the laser frequencies are not explicitly shown. The Toptica laser, providing seed light for the 3 He experiments, is located on a separate optical table, along with its saturated absorption spectroscopy setup. Because the beam paths of the laser light for the experiments with 3 He are virtually identical 5 to those for 4 He, the two beams are merged into a fused optical fiber shortly after the laser amplifiers. The fused fiber is specified to have a 50/50 output ratio (although in practice it is roughly 60/40), such that light injected into either of the inputs 1 and 2 is divided evenly over output-1 and output-2. Light from one of the two outputs, say output-1, is directed to the collimation section of the experimental apparatus. Light from output-2 is coupled out to provide the MOT-, Zeeman-, spin polarization-, Doppler-and imaging-beams. Reason for the distinction between the collimation section from output-1 and the rest from output-2 is simply the relatively large laser power required for collimation and deflection. The great advantage of the fused fiber is that it ensures perfect overlap between the 3 He and 4 He laser beams, without the need of injecting two beams into the same optical fiber input coupler. In contrast, an earlier version of the section where the beams are overlapped was through the use of several beam-splitter cubes. Even though this approach generally works, it is more prone to misalignment and requires the beam profiles of the 3 He and 4 He beams to be matched. Especially given the relatively large length of propagation of the beams afterwards, a fiber-based overlap section simplifies the setup and increases its stability.
To further increase the stability and decrease the complexity of the setup, we aimed to also decouple the section of the optical table where the various beams are prepared from the table where they are injected into the apparatus. As of this writing, the imaging-, Zeeman-and MOT-beams are fiber-coupled to serve this purpose, using single mode, polarization maintaining, 10-m length fibers. The beams for spin polarization and Doppler cooling may be injected together into one fiber, as their propagation through the apparatus, as well as their polarization, is identical. As of yet, however, these are still free-space propagated. The Zeeman beam is coupled out using a fiber-coupled beam expander (Schäfter+Kirchhoff 60FC-L-4-M200-37), producing a (1/e 2 ) beam diameter of 3.6 cm. The six MOT beams are coupled out pair wise, each pair in a retroreflecting alignment. The Zeeman-slower beam and the two MOTbeams in the vertical plane are depicted in Fig. 2.9, along with the fiber-coupled imaging beam. Expanded using a telescope, the imaging beam passes through the apparatus from below and intersects the trap center. Subsequent to leaving the apparatus from the top, it passes a 2 , f = 25 cm, lens which is placed at

Detection
Detection is an essential part of the experimental procedure for any measurement. The detection methods available for metastable helium are two-fold: via impact detection and through imaging. At the probing wavelength of 1083 nm, the rather small quantum efficiency available through the use of silicon-based CCD cameras (1.5% for the Hamamatsu C4880-21 available in our lab) makes imaging with the large magnifications required to resolve a BEC, and thus with low light intensity, suboptimal. On the other hand, the high internal energy (19.8 eV) of metastable helium atoms allows for very efficient detection of neutral atoms on a microchannel plate (MCP) detector. A recent addition to the experiment is a camera based around an Indium Gallium Arsenide (InGaAs) photodetector array, the Xenics Xeva-1.7-320, with a quantum efficiency of over 70%.

Microchannel plates
Two Hamamatsu F4655 MCP detectors are present in the setup. MCPs are commonly used to detect electrons and ions, but can also be employed to detect neutral metastable helium atoms. Upon impact with one of the channels of the detector, the high internal energy of a metastable helium atom causes an electron to be released from the surface with high probability. A high voltage (∼ 2 kV) across the MCP then results in an avalanche of electrons, similar to an electron multiplier, which is detected as a current via the anode of the detector. One of the MCPs is located 8 cm above the trap, slightly off-center to provide optical access from the top of the trap chamber, which attracts ions produced during Penning-and associative ionization. The other is positioned 17 cm below the trap center, on a horizontal translation stage which allows the MCP to be moved out of the imaging path (which runs vertically through the trap chamber). It is electrically shielded by a grounded metallic grid to avoid attracting electrons or ions, and used primarily to detect neutral metastable helium atoms. While the atomic sample is trapped, ionizing collisions can be monitored in real time using the ion-sensitive MCP located above the trap. This can, for instance, be used to observe the onset of a BEC forming [15], or to provide a measure of the density at some point in time by counting the arriving ions (using a pulse discriminator). After release from the trap the atomic cloud expands and falls down, to be detected on the MCP below. From the time-offlight signal the temperature and chemical potential can be inferred, as well as the number of atoms, relying on a calibration obtained by comparing with absorption imaging.

MCP fit functions
The thermodynamic properties of the gas in the trap are recovered by application of one, or several, fit functions to the MCP signal (more specifically, the MCP located at a distance h = 17 cm below the trap center, dedicated to measuring the flux of metastable atoms). These functions were developed specifically for the geometry of this setup, in that they incorporate the location of the MCP. The models assume the cloud size in the trap is negligible compared to the widths of the distribution after expansion, and that the trap is switched off non-adiabatically (i.e., fast compared to the period of trap oscillations).
The critical temperature at which the BEC phase-transition occurs is given by: with the definition of the Bose functions g n (u) = ∞ j=1 (u j /j n ), g 3 (1) ≈ 1.202, andω = (ω x ω y ω z ) 1/3 the geometric mean of the trap frequencies. For temperatures well above the transition temperature, the gas can be described by a Maxwell-Boltzmann velocity distribution, and the normalized flux through the surface of the detector is given by [63]: where σ = kBT m t represents the ballistically expanding width of the gas, t 0 = 2h/g is the arrival time of an atom initially at rest, x 0 = h − 1 2 gt 2 and R = 7.25 mm is the radius of the MCP. This model holds for 4 He * as well as for 3 He * , so long as the temperature is much higher than T c . When this condition no longer holds, the following model is used in case of a thermal (i.e., non-condensed) Bose gas: where z = e µ/kBT is the fugacity of the gas (which reduces to z = 1 below T c ).
In this model it are the Bose functions, g n (u), which introduce the effects of quantum statistics to the density distribution. Compared to distinguishable particles, the density of a Bose gas is increased by g 3/2 (z)/z. When considering a gas of 3 He * , the g n (u) in Eq. 2.18 need only be replaced by −g n (−u) to properly take into account the effect of identical fermions.
In the case of a pure Bose-Einstein condensate the model for the flux through the MCP is quite different. Interactions between the atoms, described by the s-wave scattering length a, dominate the dynamics of the gas when the mean field energy, nŨ = n 4π 2 a m , is large compared to the trap spacing ω [71]. In the limit of strong interactions (nŨ ω), the kinetic energy, or quantum pressure, can be neglected (known as the Thomas-Fermi limit) and the density of the gas in the trap takes on a parabolic form: is the harmonic trapping potential. Right after release from the trap, the cloud is accelerated outwards due to the conversion of its mean-field energy into kinetic energy. The shape of the cloud, however, simply evolves as a rescaling of its parabolic shape. Assuming the entire condensate drops onto the MCP detector, the model for the normalized flux becomes: . (2.20) In practice, however, the size of the condensate is slightly larger than the area of the MCP. This effect will only marginally alter the chemical potential as obtained through a fit of this model. In the Thomas-Fermi limit, the following expression relates the chemical potential to the number of atoms: When the atoms are released from a magnetic trap, decaying magnetic field gradients influence the initial expansion of the cloud, leading to an overestimation of µ [15]. This problem is overcome when the atoms are released from the optical dipole trap instead (see chapter 4.4).
In general, the time-of-flight traces from the experiments involving quantum degenerate 4 He * are described by a bimodal distribution. A superposition of Eq. 2.18 and Eq. 2.20, i.e., a combination of a thermal Bose gas and a BEC, then provides an accurate model to fit to the data.
In all the above mentioned models, the parameter t can be replaced by t+∆t to account for an arbitrary time-offset introduced by delays in switching off the trap.

Simplified fit functions
When processing large amounts of time-of-flight traces, which all need to be fitted to a bimodal distribution of the form N th Φ th (t) + N c Φ c (t), the sum of Eq. 2.18 and Eq. 2.20 is inconvenient to apply directly, as it requires well chosen initial guesses of the fitting parameters to converge to the global optimum (using the Levenberg-Marquardt algorithm). For this reason, when the only parameter of interest is the relative number of atoms in the thermal gas or the BEC, simplified fit functions often prove to be more reliable.
In case of the Bose gas (Eq. 2.18), when the thermal energy is much smaller than the potential energy, i.e., k B T mgh, the model takes the shape of a simple gaussian function. For a 4 He * atom released 17 cm above the MCP detector, k B T equals the potential energy for T ≈ 0.8 mK. Thus, when performing experiments at or below the critical temperature (sub-microKelvin), this condition holds well, such that the model can be replaced by the following form: for some fit parameters c 0 and c 1 . In the case of a BEC, Eq. 2.20 can be simplified when the chemical potential is much smaller than the potential energy. Under typical conditions in our experiment this ratio is on the order of thousands, which validates using a model of the following form: (2.23)

Imaging
For imaging of the atomic cloud a Xenics Xeva-1.7-320 camera is used. Its detector is an Indium Gallium Arsenide (InGaAs) photodetector array, with 320 × 256 pixels, sized 30 µm × 30 µm. The reason for choosing this detector type over conventional silicon-based CCD cameras is the relatively high quantum efficiency at a wavelength of 1083 nm: approximately 70%, as opposed to 1.5% for a Hamamatsu C4880 camera (previously used in our lab) with a silicon-based CCD detector. The downside, however, is larger noise levels. Exact figures are dependent on detector gain, but 150 noise electrons are specified for our gain configuration (compared to 12 for the Hamamatsu camera). Similarly important for low-light imaging is how many electrons per pixel are discernible after the analog-digital (A/D) converter. The Xenics camera has a 12-bit depth A/D converter, corresponding to 4096 analog-to-digital units (ADUs). The full well capacity, i.e., the maximum number of stored electrons per pixel, of the InGaAs detector is 187500 (determined by the capacitors connected to the photodiodes). Dividing these numbers one finds that the resulting sensitivity is 45.8 electrons per ADU. By measuring the noise on dark images (by comparing single images to the average image), a level of 3.8 ADU (rms) is determined, corresponding to 174 electrons (close to the noise of 150 electrons as specified by the manufacturer). For imaging purposes, intensities of around one-tenth of the saturation intensity (0.1 × 0.167 mW/cm 2 ) are applied to the detector. Given a pixel size of 30 µm × 30 µm and an illumination time of 50 µs, the number of photons incident on a pixel is 40 × 10 3 , corresponding to 28 × 10 3 electrons. With a noise level of 174 electrons, the expected signal-to-noise ratio is approximately 160.
For the Hamamatsu camera, the noise level was established to be 12 electrons. With a pixel size of 24 µm × 24 µm the incident number of photons is 26 × 10 3 which, at a quantum efficiency of 1.5%, results in 390 electrons of signal. The signal-to-noise ratio then becomes 32, i.e., five times lower than that of the Xenics camera.

Absorption imaging
An absorption image of the atomic cloud is obtained by illuminating it with a low intensity, resonant, laser beam and image the shadow. The beam has a diameter of several centimeter to encompass the expanding cloud. The number of atoms and the temperature of the atomic distribution can be deduced from the amount of absorption and the shape of the feature, respectively. In the low intensity regime (I I sat ), the intensity of the beam decreases exponentially as it passes through, and is absorbed by, the atomic cloud. The probe beam intensity distribution, as it exits the cloud, can be expressed as: where the factor σ a n(x, y) is the optical density of the cloud, with n(x, y) the column density of the atoms (which is the density, integrated over the direction parallel to the probe beam) and photon absorption cross section: where ω/2π is the laser frequency, Γ/2π = 1.6 MHz is the natural linewidth of the transition and I sat = f I sat /χ is the effective saturation intensity, which is the saturation intensity I sat = ωπΓ/3λ 2 , adjusted by the temperature dependent line-shape factor χ: which is a convolution of the transition lineshape (with ∆ the detuning from atomic resonance) and the Maxwell Boltzmann velocity distribution of the trapped sample. The factor f in the effective saturation intensity depends on the populations of the atomic sublevels and the polarization of the probe light. For atoms in the m = ±1 state, illuminated with linearly polarized light, f = 17/10. For unpolarized atoms, f = 18/10. An extensive account of the effective saturation intensity is given by Tol [62]. The properties of the atomic cloud, i.e., the number of atoms, temperature and/or chemical potential, are contained in the column density n(x, y), which can be expressed in terms of the normalized intensity profile by rewriting Eq. 2.24 as: The normalized intensity profile, I out (x, y)/I in (x, y), is obtained by taking a sequence of three images of the atomic cloud. The first image, I abs (x, y), contains the shadow of the atoms due to absorption of the probe beam. The second image, I prb (x, y), is shot 300 ms later and contains only the probe beam intensity profile as the atoms are absent at this time. A third image, I bgr (x, y), with neither the probe light nor the atoms in it, is shot to correct both the absorption image and the probe image for any stray light which does not involve the probe intensity profile, as well as to correct for systematic deviations in the detector in the camera. The normalized intensity profile can then be expressed as: by subtracting and dividing the pixel values of the obtained images on a pixelby-pixel basis, and correcting the spatial coordinates for the pixel sizes and magnification of the imaging setup. Fig. 2.10 shows an example of the three camera images, as well as the normalized intensity profile.

Imaging fit functions
Taking the natural logarithm of the normalized intensity profile (Eq. 2.27), the thermodynamic properties of the atomic cloud are obtained by fitting of −σ a n(x, y). The form of n(x, y) depends on the distribution of the atoms. For a thermal cloud, the column density can be expressed as: where N th is the number of (themal) atoms and σ i the width of the gas after expansion time t: where ω i /2π is the trapping frequency in the i ∈ {x, y, z} direction, and σ i (0) = kBT mω 2 i the initial width of the gas in-situ. At time-of-flights for which t 1/ω 2 i , the expression for the width can be approximated as kBT m t.
At temperatures around and below T c , the indistinguishable character of the atoms becomes apparent, and the Maxwell-Boltzmann distribution fails to properly describe the gas. In this case, the column density is described by a thermal (i.e., non-condensed) Bose-gas: with z = e µ/kBT the fugacity of the gas (which reduces to z = 1 below T c ) and the Bose-functions defined as g n (u) = ∞ j=1 (u j /j n ). In the Thomas-Fermi approximation, the kinetic energy in the BEC is negligible compared to the mean field energy. The density in the trap is then determined by the relation between the chemical potential and the external potential, given by Eq. 2.19. In case of a harmonic trap, the density distribution takes on a parabolic shape. Integrating over one dimension, the column density of a BEC, in the Thomas-Fermi approximation, becomes: When the condensate is released from the trap, its expansion progresses as a rescaling of the parabolic shape: where the initial width of the BEC in-situ is the Thomas-Fermi radius r i (0) = 2µ mω 2 i . In the case of partially condensed clouds, the densities of the Bose gas and BEC can be added and fitted together to obtain the thermodynamic properties of both gases simultaneously.

Fringe removal algorithm
A fringe-removal algorithm can be applied to significantly improve on the quality of absorption images. One common issue with absorption imaging as explained in the previous section, is that artifacts arise on the normalized intensity profile due to mismatches of the illumination between the first two shots (I abs and I prb ), not considering the difference due to actual absorption. An example of a typical artifact can be seen in the normalized intensity picture in Fig. 2.10, as the fringe pattern in the bottom-center of the image.
The fringe-removal algorithm is based on the idea that the second shot in the sequence of the three images, that is, I prb , is not necessarily the best image to use to divide out the illumination of the probe beam in I abs . Rather, a linear combination of various I prb images can be constructed to provide an optimal reconstruction of the probe illumination profile to most closely match the illumination in the absorption image I abs . The method described here is based on Ref. [72]. The criterion as to what constitutes an optimal reconstruction is that the least-squares deviation between the pixel values of the absorption image and the probe image are minimized, excluding the area where the probe beam was absorbed by the atoms. To ensure there are indeed no atoms in the area under consideration, only the corners and/or borders of the images are used as the criterion to minimize the least-square deviation. The assumption is then that if a suitable linear combination of probe images is constructed for which the corner areas are a close match to the corresponding pixel values in the absorption image, so too is it a good match for the rest of the image. This is reasonable as long as the reproducibility of the images is high. The actual size, in pixels, of the corner areas to use for the reconstruction is not too critical, as long as it spans a reasonable part of the image and does not overlap with the atoms. Given that the images taken by the Xenics camera consist of 320 × 256 pixels, corner areas of 50 × 50 will suffice (the image is usually centered around the atomic clouds). Using all four corners of the image, the total area under consideration corresponds to n = 4 × 50 × 50 = 10000 pixels. If the number of probe images used in the reconstruction is m, and the corner pixels of image i are stored in an array, or vector, p i of length n, we are looking for a solution to: where a is the corresponding array of corner pixels in the absorption image, and the s i are the components of the solution vector s. The corrected probe image is then obtained as: Taking the p i as column vectors of a matrix P : Because a, in general, does not lie in the space spanned by the column vectors of P , this equation has no solution. The vector closest to a which does lie in the column space of P is the projection of a onto the column space of P . The projection is found by solving the normal equations: The term which solves the equation, P T P −1 P T , is referred to as the Moore-Penrose pseudo-inverse, denoted P + . This method is essentially equivalent to performing a linear least-square fit. On a desktop computer, computing P + using several hundred probe illumination images, where from each image 10000 pixels are gathered from the corner areas, takes about 10 seconds. Whereas applying P + to a, constructing the corrected image I cor and using it to divide out the probe illumination from I abs takes less than a second. Fortunately, considering large sets of images needed to be processed, P + needs to be calculated only once, as it depends solely on the probe images. Having calculated P + , it can then be rapidly applied to various images. An example of the fringe-removal procedure can be seen in Fig. 2.11. The image on the left is the same as the right-bottom image in Fig. 2.10. The image on the right displays the effect of the fringe-removal algorithm, applied to the same absorption image I abs . The procedure effectively deals with the visible fringe pattern.

Computer control
To produce ultracold gases and perform measurements on them, accurately timed sequences controlling the various components of the experimental setup are a necessity. As measurement sequences often involve repeated iterations of identically prepared atomic samples, reproducibility and reliability are similarly important. Previously, the control sequences were executed by means of a LabView program, which operated the digital and analog cards in a desktop computer. With this particular computer becoming somewhat aged and therefore being less than adequately equipped to run modern software, a replacing system has been installed. With the introduction of a newer computer system, including new digital and analog I/O cards to provide a larger degree of control and stability, the software has been rewritten to suit the needs required for present and future experiments.
The digital card (DIO-64 Viewpoint Systems) provides 64 digital output lines. A timetable 6 of TTL triggers is stored to the card and executed independently of the clock speed of the host computer. The digital outputs are updated at an effective scanrate of 1 MHz. The analog card (PCI-6713 National Instruments) has 8 analog lines, with a -10 V to +10 V span, programmable at 12-bit depth (i.e., 1 in 4096, roughly 5 mV resolution). A separate timetable is buffered onto the analog card, which is triggered by the digital card for synchronization. This triggering mechanism effectively allows a temporal resolution for the analog lines of 3 µs (to trigger subsequent sets of outputs in the analog timetable, the digital trigger bit is lowered and raised again), which is well below the shortest relevant timescales in our experiments (50 µs).
The control software which manages the digital and analog cards, represented in a block diagram in Fig. 2.12, can be distinguished into two parts, a front-end and a back-end. The back-end directly communicates with both the front-end and the two I/O cards. Its main task is to convert a timetable as specified in a file into the buffers of the I/O cards and start execution. The front-end, i.e., the user interface, allows for easy creation and editing of the timetable and controls the back-end to start and stop execution. Besides this, the front-end communicates with oscilloscopes for data acquisition and initiates devices prior to the execution of the timetable. Initiation of, and data acquisition from, the Xenics camera is managed via a separate program which runs alongside the control software, but is controlled by the front-end. The sequence script which holds the timetable which is to be loaded by the back-end into the buffers of the I/O cards, also contains auxiliary information which is used by other programs. One example is the frequency locking routine for the 1557-nm laser, for which the set-point value is stored in the script. This laser is referenced to a frequency comb system, from which the beat signal is transported over the computer network to the locking routine, where it is compared with a set-point value to provide an error signal. The sequence script is also used for automated data analysis.
A screenshot of the front-end is shown in Fig. 2.13. The aim of the software is to provide a graphical representation of the experimental sequence, which can then be manipulated, as well as display the most recently acquired data (i.e., camera images and oscilloscope traces).  The user interface communicates with software controlling the Xenics camera and several peripheral devices such as oscilloscopes and waveform generators. The sequence script can be used by other programs running at the same time, such as the 1557-nm laser locking algorithm, and it can also be used for data analysis. 3

Frequency metrology
Precision spectroscopy of simple atomic systems has refined our understanding of the fundamental laws of quantum physics. In particular, helium spectroscopy has played a crucial role in describing two-electron interactions, determining the fine-structure constant and extracting the size of the helium nucleus. Here we present a measurement of the doubly-forbidden 1557-nanometer transition connecting the two metastable states of helium (the lowest energy triplet state 2 3 S 1 and first excited singlet state 2 1 S 0 ), for which quantum electrodynamic and nuclear size effects are very strong. This transition is 14 orders of magnitude weaker than the most predominantly measured transition in helium. Ultracold, submicrokelvin, fermionic 3 He and bosonic 4 He atoms are used to obtain a precision of 8 × 10 −12 , providing a stringent test of twoelectron quantum electrodynamic theory and of nuclear few-body theory.

Introduction
Because the ionization energy of helium is lower than the energy for any doubly-excited state in helium, at least one of the two electrons always populates the ground (1s) orbital. Also, the chance of radiative decay from doubly excited states is much lower than the chance of ionization [73]. This makes the level structure of helium below the ionization limit relatively simple, as the spatial wavefunction of any state can be expressed as: The plus (minus) sign denotes the symmetric (anti-symmetric) spatial wavefunction. Historically, the symmetric spatial states were referred to as parahelium, the anti-symmetric as orthohelium. Since the total wavefunction of the two-electron system is necessarily anti-symmetric, it follows that the spinwavefunctions of the states have opposite symmetry compared to those of the spatial-wavefunctions: spin singlets constitute parahelium, spin triplets orthohelium. Consequently, a transition from singlet to triplet requires a symmetry change, which electric-dipole interactions do not induce. Because of this, transitions between singlet and triplet states in helium are forbidden. The restriction is somewhat lifted however due to triplet-singlet mixing in states with higher angular momentum (l > 0), where spin-orbit and spin-spin interactions play a role. Eigenstates of the Hamiltonian become of the form a |2 3 P 1 − b |2 1 P 1 , where in the case of P states the ratio b/a is of order 10 −4 [74]. This chapter describes a measurement of a transition between the two metastable states in helium: the n=2 triplet S and the n=2 singlet S (see Fig. 3.1). It is forbidden in the electric dipole approximation by two selection rules: spin and parity (besides connecting the triplet to the singlet state, this transition connects an S to an S state, i.e., ∆l = 0). With an Einstein A-coefficient calculated to be 9.1×10 −8 [75] (a rate of approximately 1 in 4 months), this magnetic dipole transition is 14 orders of magnitude weaker than typical electric dipole transitions found in helium. The metastable triplet state, denoted 2 3 S 1 , has a lifetime of approximately 7.9 × 10 3 s [19,76]. The linewidth of the transition is 8 Hz, determined by the 20 ms natural lifetime of the metastable singlet state, 2 1 S 0 , which decays to the ground state in a two-photon process.
From a theoretical point of view, this transition is interesting because quantum electrodynamics (QED) contributions to the ionization energies of the two metastable states are large and calculated to very high accuracy [47,53].
The narrow linewidth, in combination with the transition being exceptionally weak, is the reason that this line had not yet been observed before. It requires high laser power at a narrow laser linewidth, interacting with a helium atom for seconds. The former is achieved by referencing of the laser to a frequency comb, the latter by trapping the atoms at sub-microkelvin temperatures.  A focused 1557-nm laser also constitutes a trap for ultracold atoms in the 2 3 S 1 state because it is red detuned from the 2 3 S 1 → 2 3 P J transitions. As the 1557-nm laser light is blue detuned from the 2 1 S 0 → 2 1 P 1 transition, atoms in the 2 1 S 0 state are anti-trapped.

Apparatus and dipole trap
The experiment described here was performed using an apparatus designed for the production of quantum degenerate gases of helium [15,18]. Briefly, the metastable 2 3 S 1 state, denoted He * , is populated by electron impact in an electric discharge. The atomic beam is collimated, slowed and trapped using standard laser cooling and trapping techniques on the 2 3 S 1 → 2 3 P 2 transition at 1083 nm. The atoms, optically pumped to m J = +1, are then transferred to a Ioffe-Pritchard type magnetic trap. 4 He * atoms are evaporatively cooled towards Bose-Einstein condensation by stimulating RF transitions to untrapped states. For 3 He * , elastic collisions are suppressed at sub-mK temperatures due to the Pauli exclusion principle. These atoms can, however, be cooled down to quantum degeneracy through collisions with 4 He * atoms. A crossed-beam optical dipole trap at 1557 nm is overlapped with the magnetic trap. The intensity of the 1557 nm light induces an AC Stark shift on the 2 3 S manifold, predominantly due to the red detuning relative to the 1083 nm 2 3 S 1 → 2 3 P 2 transition. The AC stark shift induced by the dipole trap is proportional to the local laser light intensity, which gives rise to a trap profile as shown in Fig. 2.4. At the same time, atoms residing in the 2 1 S 0 state experience a repulsive force due to the intense 1557 nm light, because this light is blue detuned from the dominant electric dipole transition at 2059 nm (see Fig. 3.1). Thus, the crossed-optical dipole potential inverts for an atom excited from the 2 3 S 1 to the 2 1 S 0 state, resulting in the atom rapidly leaving the interaction region.
To efficiently load atoms from the magnetic trap into the crossed-beam optical dipole trap, the dimple at the intersection of the beams is overlapped with the magnetic-trap center. Once switched on, the magnetic trapping fields can be ramped down, leaving the atoms trapped at the intersection of the dipole trap beams, provided the temperature of the ensemble is lower than the depth of the dimple.
A commercial telecom fiber laser (NPPhotonics Scorpio) produces the 1557nm laser light. It is frequency tunable by means of a temperature control and a piezo input for fast modulation. As the 1557-nm trap laser is far red detuned from the nearest electric dipole transition (2 3 S 1 → 2 3 P 2 at 1083 nm), the scattering rate (roughly 1/300 s −1 , see chapter 2.2.2) can be neglected. Instead, the lifetime of atoms in the trap is limited by two-body decay (see chapter 4). The depth of the trap at the largest laser power used for this experiment is approximately 10 µK, which translates into an AC stark shift of the 2 3 S 1 manifold of 200 kHz.
A separate beam for the purpose of spectroscopy is derived from the same laser as the dipole trap, but switched and frequency shifted by a 40 MHz acousto-optic modulator (AOM). Overlapped with the returning dipole trap beam (see Fig. 3.2), it intersects the trapped cloud of atoms. Aligning the spectroscopy beam (that is, to overlap it with the atomic cloud) can be efficiently done by blocking the returning trap beam and using the spectroscopy beam for trapping instead. In this case the incoming trap beam is responsible for trapping of the sample in the radial direction and the spectroscopy beam holds the atoms in the axial direction, effectively producing a similar crossedtrap setup albeit with a larger dimple due to the larger waist (approximately 280 µm) of the spectroscopy beam. The overlap of the spectroscopy beam with the dipole trap center can then be optimized by imaging the trapped atoms from a direction perpendicular to the plane of the crossed beams (i.e., from above).

Laser lock to frequency comb
Determining the absolute frequency of the spectroscopy light, or that of any laser in general, requires some source to reference to. A widely used and commercially available source is that of a rubidium atomic clock, locked to GPS for enhanced long term stability, reaching accuracies below 1 × 10 −12 . These signals, however, are in the RF domain, roughly six orders of magnitude lower in frequency than the optical domain. Advances in the field of mode-locked lasers have facilitated the development of a device which directly bridges the gap between these domains: the frequency comb [77], a femtosecond modelocked laser emitting a train of ultrashort, coherent, pulses. It renders its name from the comb like structure of the emitted spectrum, when regarded in the frequency domain, with the teeth of the comb spaced equidistantly. Any mode of the comb can be written in terms of two radio frequencies, the repetition rate (f rep ) of the pulses (or equivalently, the mode spacing) and the carrier envelope offset frequency (f ceo ), related to the phase and group velocities within the laser cavity. The spectrum is graphically represented in Fig. 3.3. The frequency of the n th comb mode, with n being an integer, can be written as: i.e., every comb mode is an integer multiple of the repetition rate, incremented by the carrier envelope offset frequency. Given that f rep and f ceo are both in the radio domain, n is of the order of a million for an optical frequency emitted by the frequency comb laser. Without going into detail as to how f rep , and especially f ceo , are measured, it suffices to say that both can be locked to a stable external source such as the rubidium atomic clock (for more information, see for instance Ref. [78]). These parameters thus define the absolute frequency f (n) of each of the comb modes, and the stability to which they can be locked is transferred to the optical spectrum. The comb can then be used as a frequency ruler for other laser sources. When the light of the comb is spatially overlapped with light of a CW laser, a beat frequency is produced. In fact, every mode in the spectrum of the comb gives rise to a unique beat frequency with the CW laser. The n th comb mode producing the lowest frequency beat is the one for which f (n) matches the CW-laser frequency most closely. Determining the absolute frequency of the CW laser then reduces to resolving the modenumber n.
At the time of writing, the laboratory of the Ultrafast Physics and Frequency-Comb Laser Metrology group, led by Prof. dr. K.S.E. Eikema, hosts three frequency comb lasers. Two of these are home-build and based on mode-locked

Frequency lock
The 1557-nm fiber laser which is used for trapping and spectroscopy is tunable over the range 1557.183 nm -1557.425 nm (approximately 30 GHz) by adjusting the operating temperature of the active fiber. For fast tuning over a range of 150 MHz, a piezoelectric transducer is used. The laser has a specified short term linewidth of 5 kHz. To frequency lock this laser, a small fraction of the light is split off and send via an optical fiber to the lab where the frequency combs are based. There it is coupled out and overlapped with the light of the frequency comb laser to produce a beat signal, measured on a fast AC-coupled photodiode (see Fig. 3.4). The comb used for this purpose is the Menlo Systems erbium doped fiber comb (referred to as the Erbium comb hereafter). A frequency counter (Agilent 53132A) measures the beat frequency, which is collected by a PC through a GPIB connection. In order to lock the 1557nm laser, the beat frequency is regulated through a Proportional-Integral (PI)

Absolute frequency determination
The absolute frequency of the laser, which is referenced to the comb, can be written as f (n)+f beat , where f (n) is the frequency of the comb mode (Eq. 3.2) nearest to the laser frequency, and f beat is the corresponding beat frequency between these two frequencies. The exact comb modenumber, n, needs to be resolved in order to determine the absolute frequency of the 1557-nm laser. When the free-running frequency of the laser is stable enough, the absolute frequency can be determined by measuring the beat frequency for various settings of the repetition rate of the comb. Stability of the laser is required to prevent the frequency from drifting while the frequency comb is adjusted to the new repetition rate. In case of the 1557-nm laser, the free-running frequency stability was not sufficient to use this method. We did, however, employ a similar method by leaving the 1557-nm laser locked to the Erbium comb and making use of a Ti:sapphire comb for an alternative beat measurement. To do so, a Periodically Poled Lithium Niobate (PPLN) crystal was used to frequency double the 1557-nm light to 779-nm, well within the wavelength range of the Ti:sapphire comb. Measuring the beat frequency between the Ti:sapphire comb and the frequency doubled light, f Ti:S beat , and at the same time having the beat frequency with the Erbium comb, f Er beat , locked at a certain value, results in one equation and two unknowns (the modenumbers): The constraint that the modenumbers must be integers results in only a discrete set of possible solutions for n and m. The actual modenumbers are obtained by rewriting the equation to m as a function of n, and finding the m(n) which is closest to an integer as n is varied in integer valued steps. Fig. 3.6 shows the distance to integer values of m(n) as n is varied between 770704 and 770778 (corresponding to the modenumbers for a wavelength range of 1557.183 nm -1557.425 nm, i.e., the wavelength range of the 1557-nm laser system). The point marked by a circle depicts the situation where m(n) is closest to an integer. In this particular situation n = 770765 and m = 2586769.99989. In fact, even when adding another 4000 modenumbers n to the procedure, effectively searching over a range of 1 THz (or 8 nm around 1557 nm), these modenumbers still emerge to be the best estimates for the solution to Eq. 3.3. As a check the procedure was repeated for different values of f rep and f ceo of the Ti:sapphire comb, but the same modenumber n for the Erbium comb was obtained.
A different way of determining which of the comb modes is closest to the 1557-nm laser frequency, without the need for a second frequency comb, is by simply measuring the light with a wavelength meter. For this purpose we inserted the frequency-doubled light into a wavelength meter (Burleigh WA-VIS). Having first applied the above method allowed a recalibration of the Burleigh if necessary. As it turned out, the Burleigh was accurate to within 500 MHz, which is sufficient given that the light is frequency-doubled.

Power meter calibration
The measured transition frequency will be shifted due to the AC Stark effect, the amount of which is linearly proportional to the laser-light intensity of the trap-and spectroscopy beams. When the transition frequency is measured as a function of laser intensity, the AC Stark shift can be accounted for by extrapolating to zero intensity. In order to apply such an extrapolation, an accurate determination of the applied laser intensities needs to be known; not so much on an absolute scale, as the extrapolation goes to zero intensity, but on a relative scale. To this end, over the course of the measurements described in this chapter, two power meters have been used. A Thorlabs thermopile power meter (model D3MM in conjunction with a PM100 console), and an Ophir thermopile power meter (3A-P-V1, with a AN/2E console). To test and calibrate the linearity of the power meters a self consistent procedure was applied, loosely based on the method described in Ref [79].
Two laser beams of equal power are overlapped and applied to the power meter. The beams can be individually shuttered, such that the light of either of the beams, or both at the same time, can be measured. If the power meter has a linear response, the sum of the readings of the individual beams equals the reading of the beams combined. Assuming the power meter has a quadratic response, and possibly an offset, the response function becomes R(P ) = aP 2 + bP + c, where P is the actual power (ideally a = c = 0 and b = 1). In general, the combined power x = R(2P ) is not equal to the sum of the constituents, y = 2R(P ). Measuring y as a function of x then allows the shape of the response function to be derived by applying a least-square fit to the data. An absolute power calibration can not be obtained in this manner, as P can be arbitrarily scaled. This scale factor, however, can be chosen to demand a specific corrected power to represent a desired value, which is relevant when dealing with multiple ranges (scale settings) on a single power meter. The Ophir, for example, may read 305 mW at the high end of the 300 mW range, but reads 297 mW on the 1 W range, while applying the same laser power. The above procedure for linearization is applied for several range settings of the power meters. The corrected ranges are then glued together by proper adjustment of the scale factor.

Sequence of a transition measurement
At the start of a measurement of the transition frequency the power of the trap and spectroscopy beams is measured on a thermopile power meter. Subsequently an automated procedure is started to perform trap-loss measurements for a set of, typically, 40 laser frequencies to scan across the atomic transition. Each of these start by laser cooling and magnetic trapping of metastable helium atoms, which are then evaporatively cooled to quantum degeneracy and loaded into the optical dipole trap. After 500 ms of rethermalization time, the spectroscopy beam, set to one of the predefined setpoint frequencies, is applied to the atoms for a fixed duration. The duration is chosen so as to be long enough to excite approximately half of the sample to the 2 1 S 0 state when the laser frequency is at resonance with the atomic transition, which, depending on the laser power, ranges between 0.5 and 6 seconds. The beat frequency of the 1557-laser with the frequency comb is recorded to provide an estimate of the average laser frequency during excitation. The number of atoms left unexcited, i.e., still populating the 2 3 S 1 state, is determined by dropping these onto an MCP detector: when the optical dipole trap is switched off, the cloud of atoms expands and falls down due to gravity. The high internal energy of  the metastable atoms (approximately 20 eV) allows for efficient detection on an MCP, which is located 17 cm below the trap center. The MCP signal is displayed on an oscilloscope and downloaded to a PC for analysis.

Analysis
From each of the runs (that is, each instance of loading atoms into the dipole trap, shining in the excitation laser and recording the remaining atoms on the MCP), the number of atoms needs to be extracted from the MCP signal.
To convert from the MCP signal to atom number, a model is required which describes how a certain atomic distribution expands during time of flight and intersects with the MCP. In the case of 3 He * , the fermionic nature of the cloud is, near quantum degeneracy, described by the Fermi-Dirac distribution. For bosonic 4 He * , the cloud shows a bimodal distribution, described by a superposition of a thermal Bose gas and a Bose-Einstein condensate (see Fig. 3.7).  The models which incorporate ballistic expansion and the intersection with the MCP are, for both isotopes, detailed in chapter 2.4.1. As the parameter of relevance is the relative number of atoms from shot to shot (as opposed to the absolute number), the models can be simplified for ease of fitting to the data. Especially for 4 He, the term in the model describing the BEC fraction can be approximated by an inverse parabola and the thermal fraction by a Gaussian, decreasing the complexity of the fit function. Given the number of data sets which needed to be analyzed (several thousands), the reliability of an automated fitting routine was essential. Most important in successful fitting of the bimodal distribution for 4 He are the initial guesses. To this end, an algorithm is applied to the data prior to the actual fitting procedure to provide these starting parameters.
The frequency of the 1557-nm laser is continuously locked by referencing to the frequency comb, the beat frequency is sampled during the excitation time every 30 ms (effectively corresponding to the integration time in the feedback loop of the frequency lock). Plotting the number of atoms remaining in the trap after the excitation time as a function of the average of the recorded beat frequencies during that time, we obtain a graph showing a dip in the signal where the resonance is (expressed in terms of the beat frequency). Fig. 3.8 shows an example of such a measurement. The observed lineshape of the transition is dominated by the laser frequency lineshape, which is to say, other broadening effects of the 8 Hz natural linewidth of the transition (such as Doppler broadening) are mostly unresolved. Assuming the fractional atom loss to be proportional to the resonant laser intensity, so long as the trap is not depleted, the transition lineshape can be approximated by a Gaussian. A nonlinear regression then yields the line center and its uncertainty, both of which can be readily converted to the actual absolute laser frequency.

Systematics
Several systematic effects cause the measured transition frequency to be shifted from the atomic resonance frequency. Most notably these are due to the presence of external fields, inducing differential Zeeman and Stark shifts of the atomic energy eigenstates, but also due to the recoil the atom undergoes during excitation when the photon is absorbed and due to inter-atomic interactions.

Zeeman shift
The Zeeman shift poses the largest of the systematic perturbations to the transition frequency. Any atomic energy level corresponding to a state with a nonzero magnetic moment (be it spin, orbital or nuclear), is perturbed in the presence of a magnetic field. In particular, the frequency of the transition between two states is shifted by the differential Zeeman shift of the two states involved. In case of the 2 3 S 1 (m J = +1) → 2 1 S 0 (m J = 0) transition, for 4 He, the singlet state has no magnetic moment, and thus the differential Zeeman shift of the transition frequency is determined only by the shift of the 2 3 S 1 (m J = +1) state. Considering the shift rate of g s µ B /h ≈ 2.8 MHz/G, where h is the Planck constant, g s is the electron g-factor, and µ B is the Bohr magneton, the earth magnetic field of approximately 0.5 Gauss introduces a 1.4 MHz perturbation to the transition frequency. In the case of 3 He, both the initial and final states have a nuclear spin contribution. The initial 2 3 S 1 , F = 3/2 (m F = +3/2) state shifts by (g s µ B − 1 2 g i µ N )/h, where g i is the nuclear g-factor and µ N is the nuclear magneton. The final state, 2 1 S 0 , F = 1/2 (m F = +1/2), shifts by − 1 2 g i µ N /h. As such, the differential shift rate is g s µ B /h, equal to that of 4 He. To measure the perturbation to the transition frequency in 3 He and 4 He due the magnetic field, then, amounts to determining the Zeeman shift of the 2 3 S 1 (m J = +1) state in 4 He. This is done by inducing RF transitions to the 2 3 S 1 , m J = 0 and m J = −1 magnetic substates: the frequency at which most atoms are transferred equals the Zeeman shift. In order to avoid introducing unforeseen additional shifts, the experimental sequence which is ran to perform this measurement is identical to the sequence needed to run the 1557-nm transition measurement, up to the point where the atoms would be dropped from the trap (with the exception that the excitation laser is several MHz detuned as to not excite the transition). Making use of the antenna which also serves for evaporative cooling, a 40-µs RF pulse is applied to drive transitions between the magnetic substates in the 2 3 S 1 manifold. Prior to the pulse the RF generator (Agilent 33220A) is enabled to produce the desired frequency (set by an analog control voltage), and the output measured on a frequency counter (Agilent 53132A). After the spin populations have been redistributed due to the RF pulse, a magnetic field gradient is applied in which the atoms experience a force, the direction of which is dependent on their magnetic moments. The atoms are released from the dipole trap and the spin-state populations are separated during time-offlight. Subsequently, an absorption image is taken from which the relative populations of the magnetic substates are obtained. Fig. 3.9 shows a selection of absorption images as a function of applied RF frequency. In the graph below the images, the corresponding fraction of the atoms which is populating the m J = 0, −1 states is plotted. A fit of a Gaussian to the ratios as function of RF reveals the frequency for which most of the atoms are transferred with an uncertainty of, on average, 0.5 kHz.
Several such Zeeman shift measurements were performed during a day, intermittently with 2 3 S 1 → 2 1 S 0 transition measurements in order to compensate for slow changes in the magnetic field. The standard deviation of the value of the Zeeman shift over the course of a day was, on average, 2 kHz which corresponds to a stability of the magnetic field of ±0.7 mG.

Recoil
Due to momentum conservation, when an atom is excited from the 2 3 S 1 state to the 2 1 S 0 state, the momentum of the atom is increased by the momentum of the absorbed photon: ∆p = hf /c, where h is the Planck constant, f is the photon frequency and c is the speed of light. Consequently, the atom gains a kinetic energy of ∆E = ∆p 2 2m = 1 2m hf c 2 . As this energy is acquired from the photon, the transition frequency is lower than the photon frequency by f recoil = ∆E/h, such that: f photon = f transition + f recoil as required for energy conservation. For a 4 He atom excited by a 1557-nm photon, the recoil shift is 20.6 kHz and for 3 He it is 27.3 kHz, with negligible uncertainty.

Mean field
An additional correction to the transition frequency for 4 He is the cold collision frequency shift [80]. Due to the extremely low temperatures and diluteness of the gas, interactions in the condensate can be modeled by a Dirac-delta type potential, with a strength given by the s-wave scattering length. Each atom experiences an interaction potential due to the mean field of the other atoms, proportional to the density of the gas. Originating from the difference in meanfield energy between the ground state and the excited state, this frequency shift is proportional to the density of trapped atoms and to the difference in swave scattering lengths for the triplet-triplet and the triplet-singlet molecular potentials, calculated as: The triplet-triplet scattering length a 3−3 = 7.512(5) nm [81]; however, the value of the triplet-singlet scattering length is presently unknown. By measuring the 2 3 S 1 → 2 1 S 0 transition frequency with a high atomic density and with a low atomic density (reduced by a factor of 5) and taking the difference, we find a mean-field shift of 0.07±1.08 kHz which is consistent with no mean-field shift to within 1.1 kHz.

AC Stark shift
One systematic effect which is frequently encountered in laser spectroscopy is the Stark shift. Since our trap and anti-trap rely on the AC Stark shift, this systematic effect is of particular relevance. One might suggest to measure the applied laser power and deduce the magnitude of the shift ab initio. This would, however, prove to be too inaccurate. As the AC Stark shift is of the order of 100 kHz in our measurements, reducing the systematic uncertainty to below the 1 kHz level would require knowledge of the laser intensity accurate to 1%. Because the laser intensity depends on factors such as the overlap between the two trap beams and the spectroscopy beam, their waists at the intersection and powers, this would be unfeasible. Another difficulty with this method would be that, especially in the case of 3 He, the atoms are not necessarily in the ground state of the dipole trap. The calculated AC Stark shift of the two trap beams designates the (maximum) depth of the trap, whereas an atom populating an arbitrary trap state |n has an additional potential energy E n corresponding to that state. This makes the shift effectively dependent on the distribution of the atoms in the trap, and is further enhanced due to the anti-trapping potential of the excited state.
To correct for the AC Stark effect, the transition frequency measurement is repeated for a range of laser powers to allow for an extrapolation to zero power. An example of such an extrapolation for 4 He is plotted in Fig. 3.10. This method has the advantage of being independent of many of the previously mentioned factors, such as beam geometries and absolute laser powers. It relies on the measured shift to be linear in laser power, which is valid to a large extent. Several issues were encountered, however, while employing this method. The first was that at the initial stages of the experiment, an AOM was used to tune the overall laser power. As it turned out, the beam profile suffered from thermal lensing of the AOM at relatively high laser powers ( 1 W), which introduced a very nonlinear response in the AC Stark shift measured as a function of power. Because of this, the AOM, as an optical attenuator, was replaced by a waveplate in combination with a Glan-laser calcite polarizer. Another issue was that the intensity at the trap center is dependent on the stability of the overlap between the two trap beams with respect to one another and to the spectroscopy beam. To avoid this problem we aimed to gather the required measurement sequences to allow for a single  He as a function of total laser power (red circles). The measured AC Stark shift increases significantly for higher laser powers when the 3 He * is sympathetically cooled with a 4 He * BEC in the dipole trap (blue squares). The black line through the circles is a fit to the data up to 300 mW, the dashed extrapolation indicates the deviation at higher laser powers. The blue line is a fit through the squares.
extrapolation as quickly as possible, preferably within one day, to minimize the effect of slow beam misalignment on the power to intensity linearity. The most pronounced effect causing nonlinearity in the extrapolation originates from the dependence of the trap geometry on laser power, as discussed above.
In the case of 4 He this effect is suppressed as the condensed atoms populate the ground state of the dipole trap, irrespectively of the depth. For 3 He, however, any state of the dipole potential has a maximum occupancy of one, causing the atoms to be distributed over a multitude of energy levels, such that the measured shift need not necessarily have a linear dependence on laser power. At laser powers 1 below 300 mW, the trap can be considered to be saturated, which means that if the temperature would be zero, all states in the trap are populated and the Fermi energy equals the trap depth. When the optical dipole trap is switched on, a selection of the density of states in the magnetic trap is transfered into the dipole trap. At the lower laser powers, only the least energetic atoms are transfered. At laser powers from around 300 mW and upwards (see Fig. 3.11(a)), the whole thermal distribution can be transfered into the dipole trap. This effect caused a nonlinearity in the AC Stark shift curve, shown in Fig. 3.11(b) (red circles). When a BEC of 4 He * is additionally loaded into the optical dipole trap, it acts as a cold reservoir to the 3 He * atoms. Kept in thermal equilibrium with 4 He * , especially at high laser powers, an increased effective AC Stark shift is measured for 3 He, as shown by the blue squares in Fig. 3.11(b). To reduce the dependency on this effect, we restrict the maximum power used for an extrapolation to 300 mW such that the measured AC Stark shift remains linear in power.

Combined systematics
To deduce the unperturbed 2 3 S 1 → 2 1 S 0 transition frequency, each point on the AC Stark shift curve is adjusted with the best estimate for the Zeeman shift during the time at which the point is obtained. The uncertainty is added in quadrature. The frequency coordinate for each point is corrected for the power meter nonlinearity. A linear regression to the AC Stark shift curve is applied to obtain the intersect with the frequency axis. The uncertainty of the power meter calibration is estimated by applying the same regression, but for a set of different power meter calibration parameters (adjusted by the uncertainty in those parameters). The spread in the results of these regressions is added in quadrature. Finally, the recoil shift is subtracted, the uncertainty due to the frequency comb is taken into account and, in the case of 4 He, an uncertainty due to the mean field effect is added. The average contribution of the uncertainties is listed in Table 3.1. Over the course of several months, 20 independent extrapolations were obtained (as shown in Fig. 3.12) to deduce an absolute frequency of the 2 3 S 1 → 2 1 S 0 transition for 1 The quoted laser power is that of the trap beam together with the spectroscopy beam. It is convenient to refer to the combination of powers, rather than the powers of the individual beams, as that is the relevant parameter in the AC Stark shift extrapolation. The power ratio between the trap-and spectroscopy beam is 1:2, respectively. Due to the different beam waists at the trap center, and the double pass of the trap beam, the ratio of the contribution to the AC Stark shift is roughly 10:1.

Ionization energies
Calculations of the level energies in helium, or helium-like ions in general, have a long history in theoretical atomic physics. For hydrogen-like ions, the non-relativistic Schrödinger equation, as well as the relativistic Dirac equation, can be solved exactly. Exact analytic solutions to the two-electron problem for helium on the other hand, have so far not been possible, but various approximation methods have been developed to push the degree of accuracy to practically arbitrary precision. QED-and relativistic effects are then included perturbatively, expanded in powers of the fine-structure constant α as E(α) = E (2) + E (4) + E (5) + E (6) + E (7) + ..., where E (n) ≡ mα n E (n) is a contribution of order α n [53]. Each E (n) is the expectation value of some effective Hamiltonian H (n) , and is in turn expanded in powers of the electron-to-nucleon mass ratio m/M . The term E (2) is the eigenvalue of the non-relativistic Hamiltonian. The next term, E (4) , comprises relativistic corrections which are given by the Breit-Pauli Hamiltonian. The subsequent terms with n ≥ 5 are QED corrections, which have recently been calculated complete up to orders mα 6 and m 2 /M α 5 . The largest source of uncertainty to the ionization energies of the low lying S-states is due to terms of order mα 7 . Table 3.2 lists the contributions to the ionization energies of the (n=2) S states due to the various terms in powers of α, these values are collected from Drake [47]. The energy contribution as a consequence of the finite-nuclear size is added for completeness, even though it is not calculable through QED theory, but rather relies on precision measurements of the nuclear size via electron scattering experiments. The contribution of the QED terms (n ≥ 5) combined is of the order of 10 9 Hz, on a total ionization energy of 10 15 Hz. The accuracy of the highest order terms is of the order of 10 6 Hz, presenting a relative uncertainty of two parts per billion. These ionization energies agree with calculations by Pachucki et al. [53]. Taking the difference between the ionization energies, the transition frequency is 192 510 703.4(2.5) MHz, where the uncertainty is the maximum of the theoretical uncertainties of both ionization energies since both uncertainties are expected to be correlated (private communication, K. Pachucki and V. . The present experimental result from our work (192 510 702 145.6(1.8) kHz) is in agreement with this number, but more accurate by three orders of magnitude. This presents a significant challenge for groups involved in atomic structure theory.

Isotope shifts and nuclear charge radii
Isotope shift measurements, combined with high-precision QED theory, provide a method to isolate contributions due to finite nuclear size effects. The difference in nuclear charge radii between 3 He and 4 He is determined by comparing experiment and theory. Because the 4 He nuclear charge radius is one of the most precisely known of all nuclei [58], 1.681(4) fm, a value of the 3 He nuclear charge radius with similar precision can be deduced. In calculating the isotope shift, QED theory is more precise than our measurement as massindependent terms cancel. From Table 2  The latter values were calculated applying the methodology of Ref. [53]. All these values exclude hyperfine structure effects and contributions due to finite nuclear size effects, i.e., assuming point charge radii. Taking the difference between the isotope shifts of the 2 1 S 0 and 2 3 S 1 ionization energies, the calculated isotope shift for the atomic transition by Drake is 8 034 148.5(7) kHz and by Yerokhin and Pachucki is 8 034 148.75 (69) kHz. To calculate the nuclear shift, the average of the theoretical isotope shifts will be used: 8 034 148.63 (70) kHz. The measured transition frequency for 3 He is between specific hyperfine states, i.e., 2 3 S 1 , F =3/2, and 2 1 S 0 , F =1/2. The hyperfine structure of 3 He [85] shifts the 2 3 S 1 F =3/2 state by -2 246 587.3 kHz and the 2 1 S 0 F =1/2 state by 60.6 kHz (due to hyperfine-induced singlettriplet mixing). Taking these hyperfine effects into account, theory predicts an isotope shift measurement of δf QED = 5 787 500.7(0.7) kHz. Subtracting the measured transition frequencies for 3 Table 2) for the measured transition, a value for ∆r 2 c of 1.019(11) fm 2 is deduced. ∆r 2 c represents a more universal parameter than the value of the isotope shift as it is obtained from various branches of physics. Besides through spectroscopic means, it can be determined from nuclear theory and from electron-scattering experiments. Nuclear few-body theory provides ∆r 2 c = 1.16±0.12 fm 2 [57,86], whereas ∆r 2 c = 1.01±0.13 fm 2 is found through electron-scattering experiments [58,87]. Comparing the values of ∆r 2 c , we find our result to be in good agreement but an order of magnitude more precise. An independent spectroscopic measurement in helium on the 2 3 S 1 → 2 3 P 0 transition [88] gives ∆r 2 c = 1.059(3) fm 2 , obtained using the QED calculations from Ref. [89]. Although the measurement precision of the isotope shift for this transition is comparable to our precision, the smaller uncertainty in ∆r 2 c is due to a larger sensitivity to differential nuclear charge effects. Presently, the accuracy to which the 4 He charge radius is known sets a lower limit on the uncertainty of the 3 He charge radius determined from helium spectroscopy. Our measurement presents a value for the 3 He nuclear charge radius of 1.961(4) fm.
Equation 29 from Ref. [57] relates the nuclear charge radius to an effective rms radius of the nucleus (corresponding to the distribution of point-like protons and neutrons in the nucleus), to the mean-charge radius of the proton and to the mean-charge radius of the neutron. The effective rms radius is taken from Tables 7 and 8 of Ref. [86]. The results given in these Tables correspond to state-of-the-art calculations of nucleon-nucleon and three-nucleon forces and give values of r p ( 3 He) = 1.777±0.014 fm and r p ( 4 He) = 1.434±0.040 fm.
Our result for ∆r 2 c disagrees with a very recent determination of the 3 He charge radius by Cancio Pastor et al. [90], obtained via spectroscopy on the 2 3 S -2 3 P transitions in 4 He and 3 He, by 4σ. This discrepancy may either be due to unforeseen systematic effects or to yet to be discovered physics of the electron-nucleon interaction. Additional measurements are underway, aiming to push the uncertainty in the isotope shift of the 2 3 S 1 -2 1 S 0 transition down by an order of magnitude.
We have experimentally studied the magnetic-field dependence of the decay of a Bose-Einstein condensate of metastable 4 He atoms confined in an optical dipole trap, for atoms in the m = +1 and m = −1 magnetic substates, and up to 450 G. Our measurements confirm long-standing calculations of the two-body loss rate coefficient that show an increase above 50 G. We demonstrate that for m = −1 atoms, decay is due to three-body recombination only, with a three-body loss rate coefficient of 6.5(0.4) stat (0.6) sys × 10 −27 cm 6 s −1 , which is interesting in the context of universal few-body theory. We have also searched for a recently predicted d-wave Feshbach resonance, but did not observe it.

Introduction
The realization of a BEC [13,14] and a degenerate Fermi gas [18] of helium in the metastable 2 3 S 1 state (He * , radiative lifetime of 8000 s) has opened interesting possibilities for research [12]. Prominent examples are measurements of higher-order coherence in atomic matter waves, including the direct comparison between the bosonic and fermionic Hanbury Brown-Twiss effect [37], direct measurement of third-order coherence [38], and production of squeezed states by four-wave mixing in colliding BECs [91]. These experiments take advantage of the 19.8 eV internal energy of He * atoms, which allows for single atom detection with high spatial and temporal resolution by using microchannel plate detectors. Also, ultracold trapped He * allows for precise spectroscopy of The realization of ultracold and dense samples of He * atoms is quite remarkable since the large internal energy allows for detrimental Penning (and associative) ionization loss processes due to collisions between two He * atoms. The corresponding inelastic rate constant, ∼ 1 × 10 −10 cm 3 s −1 [12,20], would limit evaporative cooling of He * atoms and inhibit the possibility of achieving BEC. However, in a gas of spin-polarized atoms, Penning ionization is forbidden by spin conservation and leads to a suppression of inelastic collision rates [20]. Shlyapnikov et al. [21] considered the magnetically trappable 2 3 S 1 , m = +1 state (where m is the magnetic quantum number) of 4 He * and found Penning ionization to be suppressed by four orders of magnitude, indicating the possibility of BEC for 4 He * . The strongest inelastic two-body processes for m = +1 were found to be spin-relaxation (SR) and relaxation-induced Penning ionization (RIPI), both induced by the spin-dipole interaction. At zero magnetic field a rate constant of 2 × 10 −14 cm 3 s −1 , dominated by RIPI, was calculated [21], which was confirmed by other calculations [92][93][94]. Several experiments measured losses in magnetic traps in agreement with this loss rate [14,15,22,95,96]. For the high densities typically present in a BEC, trap loss caused by three-body recombination competes with two-body loss. A three-body loss rate of 2 × 10 −27 cm 6 s −1 was calculated by Fedichev et al. [97].
Recently, trapping of 4 He * [98-100] and 3 He * [100] in optical dipole traps (ODT) has been demonstrated. This has opened new possibilities over magnetic trapping. Most notably, it allows for trapping of the m = 0 and m = −1 spin states [98] and the application of Feshbach resonances to control the scattering properties by a magnetic field. Two-body loss rates of low density m = 0, ±1 spin mixtures in an ODT for a small, fixed magnetic field have recently been measured [98]. This study confirmed strong Penning ionization (loss rates on the order of 10 −10 cm 3 s −1 ) for those spin mixtures. In this chapter we present trap loss measurements in 4 He * for single-spin m = +1 and m = −1 clouds, which are expected to show suppression of Penning ionization, for fields up to 450 G. In particular, we have investigated the prediction of a strong magnetic-field dependence of the two-body loss rate for atoms in the m = +1 state [21,92,93], which had not yet been experimentally tested. We have also searched for a d-wave Feshbach resonance, which was recently predicted [101].

Collisional properties of spin-polarized ultracold 4 He *
The collisional properties of an ultracold atomic gas, dominated by s-wave collisions, are intimately linked to underlying two-body potentials. The electron spin of a He * atom with s = 1 gives rise to three distinct Born-Oppenheimer (BO) potentials: singlet (S = 0) 1 Σ + g , triplet (S = 1) 3 Σ + u and quintet (S = 2) 5 Σ + g , where the total electronic spin is given as S = s 1 + s 2 . Since 4 He * has no nuclear spin the total atomic spin is equal to the electron spin, s, with projection m. The Hamiltonian of the BO potentials is spherically symmetric and therefore conserves the total electron spin projection, M S . For the interaction between two atoms in either the m = +1 or m = −1 state the total spin projection is |M S | = 2, and therefore scattering is only given by the 5 Σ + g potential. The energy of the least bound state of the 5 Σ + g potential was measured to be h × 91.35(6) MHz [81], from which a precise quintet scattering length of 142.0(0.1) a 0 (where a 0 is the Bohr radius) has been derived. The absence of hyperfine coupling between the different BO potentials excludes the possibility of s-wave Feshbach resonances with spin-stretched 4 He * atoms.
Penning ionization (PI) and associative ionization (AI) play an important role in ultracold He * gases. For an unpolarized sample, the following two-body loss processes limit the stability of the trapped sample: He * + He * → He + He + + e − (PI) He + 2 + e − (AI) (4.1) In the following, we will refer to both processes in Eq. 4.1 as PI. PI is spin-forbidden for scattering in the 5 Σ + g potential, since the total spin of the final PI state cannot exceed 1. Therefore, a spin-polarized sample (i.e., atoms prepared in either the m = +1 or m = −1 state) is stable. However, weak higher-order interactions can couple the different BO potentials which will induce loss processes. Shlyapnikov et al. [21] identified the spin-dipole interaction as the most important higher-order interaction that leads to a weak coupling of the S = 2, M S = 2 state to the S = 2, M S = 0, 1 and S = 0, M S = 0 states. For scattering between identical particles in the absence of nuclear spin, the condition S + = even (where is the total angular momentum) is required, which excludes coupling to the triplet 3 Σ + u potential. For spin-dipole interactions only M S +M is conserved, where M is the projection of , and ∆ = 0, 2 so that the final channel after the spin-dipole interaction is characterized by = 2 (i.e., d waves). Furthermore, the spin-dipole interaction allows for coupling between the quintet scattering state and the singlet molecular state with = 2 and therefore d-wave Feshbach resonances are possible. The authors of Ref. [101] predicted the existence of a d-wave Feshbach resonance, either in a pure m = −1 sample below 470 G or in a pure m = +1 sample below 90 G, by varying the singlet potential within the theoretical bounds. In the latter case the molecular state would be a shape resonance.
The spin-dipole interaction induces two inelastic, two-body processes: SR, where the energy gain or loss is determined by the Zeeman energy, and RIPI, which is due to coupling of the S = 2, M S = 2 state to the S = 0, M S = 0 state, which is a strongly Penning ionizing state. At zero magnetic field the loss rate due to both SR and RIPI is independent of m and therefore is equal for m = +1 and m = −1 states. However, at magnetic fields for which 2µ B B k B T , both processes are energetically not allowed for m = −1 states. This has the consequence that in a 1 µK cloud of m = −1 He * atoms SR and RIPI can be neglected for magnetic fields larger than 10 mG. The much weaker direct dipole-exchange mechanism and spin-orbit coupling might still be possible, but are estimated to have rates 10 −16 cm 3 s −1 [21,92] and therefore do not play any role here. For m = +1 atoms trapped in a magnetic trap, SR leads to a transfer to untrapped m = 0 and m = −1 states. In an optical trap SR induces two different loss mechanisms. For sufficiently high magnetic fields (i.e., B > 0.1 G for a trap depth of 10 µK) the high gain in kinetic energy of the m = 0 and m = −1 reaction products induces instant trap loss of those atoms. For B < 0.1 G, the large PI loss rate constant for collisions between m = +1 and m = −1 atoms and/or between m = 0 atoms will also remove those atoms from the trap.
The magnetic-field dependence of SR and RIPI rate constants for m = +1 was investigated in several papers, using perturbative methods [21,92] and close-coupling calculations [93], all showing similar behavior that for small magnetic fields RIPI dominates with a rate of ∼ 2 × 10 −14 cm 3 s −1 , whereas for large magnetic fields SR becomes the dominant loss mechanism with a maximum rate of ∼ 3 × 10 −13 cm 3 s −1 at a field of 700 G (see Fig. 4.1). Around 100 G a crossover between the two loss processes occurs due to the strong magnetic-field dependence of SR [92,93]. An increase in trap loss is therefore expected for B > 50 G. The calculation for SR is very sensitive to the 5 Σ + g potential and to a lesser degree to the 1 Σ + g potential. The calculations of the authors of Refs. [21,92,93] were based on the 5 Σ + g potential of Stärck and Meyer [102] and on the 1 Σ + g potential of Müller et al. [103], which was modified to have the same long-range potential as the 5 Σ + g potential. More accurate calculations of the 5 Σ + g potential have been performed since then [104], which, within the theoretical bounds, are in good agreement with the latest experimental value of the quintet scattering length [81]. The difference between the rate coefficients obtained by these different 5 Σ + g potentials is, at most, a factor of 2 [105].
For large enough densities, trap loss caused by three-body recombination (TBR), described by: He * + He * + He * → He * 2 + He * (TBR), (4.2) will compete with two-body loss processes. He * 2 in Eq. 4.2 will undergo fast PI. The energy gain in TBR is given by the binding energy of the least bound state of the 5 Σ + g potential, which in temperature units is 4 mK, leading to loss in both magnetic and optical dipole traps. TBR depends strongly on the scattering length, a. In particular, when a is much larger than the van der Waals length, r vdW , the rate coefficient for TBR, L 3 , is given by: where, for a > 0, C(a) is an oscillating function between 0 and 70 with an unknown phase [106] and assuming that three atoms are lost from the trap [107]. For 4 He * , a/r vdW ≈ 4 1 , and universal few-body physics related to a large scattering length [106] can be expected. Since the scattering length for m = +1 and m = −1 atoms is equal and magnetic-field independent, so is the threebody loss rate coefficient.

Trap loss equation
To study the different loss processes one has to monitor the time-evolution of the density of a trapped atomic gas, which can be described as: The first term takes into account one-body loss due to collisions with background gas, scattering by off-resonant light in a dipole trap, and on-resonance excitation by stray laser cooling light, which causes exponential decay with a time constant τ . L 2 and L 3 are the rate coefficients for two-and three-body loss, respectively, and are defined such that they explicitly include the loss of two and three atoms per loss event. The constants in front of L 2 and L 3 are κ 2 = 1/2! and κ 3 = 1/3! for a BEC (where we neglect quantum depletion [95]), while κ 2 = κ 3 = 1 for a thermal gas [109].
Since we measure the atom number, N , we have to integrate Eq. 4.4, which for a BEC in the Thomas-Fermi regime gives: where: with harmonic oscillator length a ho = /(mω) and the geometric mean of the trap frequenciesω = 2π(ν ax ν 2 rad ) 1/3 [110]. Analytical solutions of Eq. 4.5 can be found for pure two-or three-body loss, but in general one has to solve Eq. 4.5 numerically.

Experimental setup
Our experimental setup and cooling procedure has been outlined in chapter 2. In short, we use a liquid-nitrogen cooled dc-discharge source to produce a beam of metastable helium atoms that is collimated, slowed and loaded into a magneto-optical trap in 2 s. The atomic gas is optically pumped into the m = +1 state, after which it is loaded into a cloverleaf magnetic trap. After 2.5 s of one-dimensional (1D)-Doppler cooling and 5 s of forced evaporative RF cooling, BEC is realized. We transfer up to 10 6 atoms into a crossed ODT at 1557 nm, which is formed by two beams that are focused to a waist of 85 µm at the intersection and cross under an angle of 19 • in the horizontal plane. The power used for our ODT is between 100 and 500 mW. A small, uniform magnetic field is applied to ensure that atoms stay in the m = +1 state after they have been transferred into the ODT. To prepare a spin-polarized sample in the m = −1 state, a small magnetic field sweep from 1 to 2 G in 50 ms is applied while the atoms are in an RF field at a fixed frequency to transfer atoms from the m = +1 state to the m = −1 state with nearly 100% efficiency (see Fig. 4.2).
Once atoms are trapped in the ODT, the axial compensation coils of the cloverleaf magnetic trap are used to create the required magnetic field. The magnetic field is calibrated by performing spin flips between m = +1 and m = −1 as described above, recording the RF resonance frequency at different currents applied through the coils. Because the coils are not in a geometrically ideal Helmholtz configuration, the field creates an anti-trapping potential for m = −1 and a trapping potential for m = +1, with a curvature of about 0.1 G/cm 2 for a field of 1 G. This curvature primarily affects the axial trap frequency. The decrease (m = −1) and increase (m = +1) of the axial trap frequency is 19% at 450 G for the loss rate measurements presented in this chapter and is corrected for in the analysis. Furthermore, the curvature effectively leads to a decrease of the trap depth for atoms in the m = −1 state, so that for a particular ODT power there is a maximum magnetic field beyond which all Bose-condensed m = −1 atoms escape from the trap. The inhomogeneity of the magnetic field across the BEC in the ODT is on the order of 1 mG and plays no role in this study.
The number of trapped 4 He * atoms is measured by turning off the ODT, which causes the atoms to fall and be detected by a microchannel plate (MCP) detector, which is located 17 cm below the trap center and gives rise to a timeof-flight of approximately 186 ms. From a bimodal fit to the MCP signal, the BEC and thermal fraction are extracted as well as the temperature and the  chemical potential, µ, of the trapped gas. We only use absorption imaging for setting up the transfer scheme between m = +1 and m = −1 states by using Stern-Gerlach separation, as shown in Fig. 4.2(a), and to measure trap frequencies by recording induced trap oscillations. The BEC part of the signal from the MCP detector (V MCP ) relates to the number of condensed atoms as N c = αV MCP , where α is a conversion factor dependent on several parameters, such as the applied potential difference across the MCP detector and on its detection efficiency. In the Thomas- atoms to the power of 2/5. The MCP signal gives a relative measure for the number of condensed atoms, V MCP , and therefore this signal is corrected by a factor α to represent the number of condensed atoms, N c = αV MCP . The value of α is determined such that the MCP signal (circles), for all data acquired, has a slope that equals the theoretical slope of 1 2 (15aω 3 2 M 1/2 ) 2/5 (line).
Fermi limit, the relation between µ and the number of condensed atoms is 2µ = (15aω 3 2 M 1/2 ) 2/5 N 2/5 c [110], where M is the mass of a helium atom, and is used to determine α (see Fig. 4.3). Since the scattering length is known [81] and the value for the average trap frequency,ω = 2π(ν ax ν 2 rad ) 1/3 , where ν ax = 55.3(0.3) Hz and ν rad = 363.4(2.1) Hz, was measured, the theoretical slope of a µ versus N 2/5 c plot is known. A value of α was determined, for the entire data set, such that the slope of µ versus (αV MCP ) 2/5 equals the theoretical slope.
The determination of α for atoms in an ODT is much more reliable than in a magnetic trap. In the case of atoms in a magnetic trap, the velocity distribution may be distorted during trap switch-off as a result of magneticfield gradients on the initial expansion of the atomic cloud. These gradients can lead to an overestimation of µ [15]. A ballistic expansion from the ODT, however, is not hindered by these effects.
The experimental procedure to study magnetic-field dependent trap loss is as follows. Atoms are confined in the ODT and are in the m = +1 state. Next, a small magnetic field of 0.5 G is applied to ensure a quantization axis. Atoms are then transferred to the m = −1 state using the spin-flip procedure illustrated in Fig. 4.2. The reason for performing this initial spin-flip is that absorption imaging showed that, unlike m = −1 atoms, hot atoms in the m = +1 state would remain trapped in the wings of the crossed ODT due to the additional trapping force resulting from the residual magnetic-field curvature. After 500 ms of rethermalization time, either another spin-flip procedure is performed in which case trap loss experiments for m = +1 atoms are investigated, or, atoms remain in the m = −1 state. The magnetic field is then ramped up to a certain value in 100 ms and the atoms remain in this field for a variable time (ranging from 10 ms to 50 s), after which the magnetic field is turned off. Finally, the small 0.5 G field and the ODT are both turned off and the number of metastables is monitored by the MCP detector.

Two-and three-body loss
We have performed two types of measurements: (1) we compare the total magnetic-field dependent loss rate for atoms in the m = +1 state versus the m = −1 state, and (2) we monitor the lifetime of the BEC for various magnetic fields for atoms in both the m = +1 state and the m = −1 state. The first measurement illustrates the differences between atoms in the m = +1 and m = −1 states, in particular, that m = −1 atoms have no magnetic-field dependent loss processes, as shown in Fig. 4.4. The experimental procedure is to measure the number of atoms remaining for a fixed hold time of either 10 ms (representing the initial atom number N 0 ) or 2 s (representing the final atom number N 1 ), in the presence of various magnetic fields between 10 and 450 G for atoms in both the m = +1 and m = −1 states. The remaining fraction, N 1 /N 0 , gives insight into the magnetic-field dependence of the loss processes. At fields <100 G the total loss for m = +1 and m = −1 samples are equal since three-body loss is the dominating mechanism and the two-body loss rate remains small at these fields. At approximately 100 G SR begins to contribute significantly to the total loss and a clear difference between the m = +1 and m = −1 states becomes evident. Atoms in the m = −1 state cannot undergo two-body loss processes because it is energetically not allowed, and since threebody loss processes are magnetic-field independent, the remaining fraction for the ODT as a function of magnetic field. N 0 and N 1 represent the number of condensed atoms remaining after exposed to a magnetic field for 10 ms and 2 s, respectively. Two-body loss processes are energetically not allowed for atoms in the m = −1 state and therefore the total loss rate is dominated by the three-body loss rate which is independent of magnetic field.
an m = −1 sample remains constant as a function of magnetic field. Due to the curvature of the magnetic field, as discussed in section 4.4, there is a maximum magnetic field for which the depth of the ODT (∼1 µK) is sufficient to confine m = −1 atoms, which was 210 G in this case. The second measurement is to monitor the time evolution of the number of condensed atoms and solve Eq. 4.5 to determine τ , L 2 and L 3 as a function of magnetic field. Since τ characterizes background collisions it is independent of magnetic field and insensitive to whether atoms are in the m = +1 or m = −1 state. By looking at trap loss for atoms in a 10 G field at long (20 to 40 s) hold times a value of τ ∼ 25 s was determined. As it can be difficult to distinguish between two-and three-body loss, we first extract L 3 by looking at trap loss for atoms in the m = −1 state (for which two-body loss is absent) and use this value to determine L 2 as a function of magnetic field for atoms in the m = +1 state. This is valid since three-body loss rates are equal for both spin states. We have shown that three-body loss is indeed independent of magnetic field (see Fig. 4.4 for m = −1 atoms) and we have determined a value of L 3 = 6.5(0.4) stat (0.6) sys × 10 −27 cm 6 s −1 . The systematic uncertainty is due to the propagation in uncertainties of a, τ , ν ax , ν rad and the conversion factor α. The present value of L 3 is in fairly good agreement with previous experimental results, but more accurate. Previously, our group has measured the three-body loss rate to be 9(3) × 10 −27 cm 6 s −1 [12,15], the experiment of Seidelin et al. [96] determined 0.8 +1. 4 −0.5 × 10 −27 cm 6 s −1 (corrected [95] for the current value of a), and an upper limit of 1.7(1) × 10 −26 cm 6 s −1 was given by Pereira Dos Santos et al. [14]. Our value is larger than the only published theoretical value of L 3 of 2 × 10 −27 cm 6 s −1 [92,97]. However, this calculation was based on a scattering-length-independent prefactor in front of the a 4 scaling (see Eq. 4.3), with C = 3.9, and on an older value for the scattering length, 190a 0 .  Having determined the one-body lifetime τ and the three-body loss rate L 3 we are now in the position to extract the two-body loss rates L 2 by numerically integrating Eq. 4.5 for m = +1 atoms. of a trapped sample of m = +1 atoms in a low (squares) and a high (circles) magnetic field. For m = +1 atoms in low magnetic fields, the theory predicts that two-body losses are small and remain relatively constant up to 50 G (see Fig. 4.1). We indeed observe that for small magnetic fields (B < 75 G) our m = +1 data are fully dominated by three-body loss. For large magnetic fields (B > 75 G) the inclusion of two-body loss is required to fit the data.
We show our experimental results for the two-body loss rate L 2 as a function of magnetic field in Fig. 4.6, together with the theory [93]. Our low field data are in good agreement with previous experimental results at zero magnetic field: 2(1)×10 −14 cm 3 s −1 [12,15], 0.4 +0.7 −0.3 × 10 −14 cm 3 s −1 [96] (corrected [95] for the current value of a), and an upper limit of 8.4(1.2)×10 −14 cm 3 s −1 [14]. Our result shows good agreement between the experiment and theory at magnetic fields up to 250 G, but deviates, up to approximately a factor of 1.3, at higher magnetic fields. The theoretical evaluation is strongly dependent on the 5 Σ + g potential. The long-range part of the potential is well known; however, the short-range part of the potential is not well known and therefore our data can be used to correct the short-range part of the potential [21,105].

Feshbach resonance
We have searched for the narrow d-wave Feshbach resonance, caused by a = 2 singlet molecular state, with a predicted width of 20 mG [101]. In general, at a Feshbach resonance loss processes are enhanced, leading to loss resonances as a function of magnetic field. The 20 mG width of this resonance is broad enough to be observed as resonant loss in our BEC (see Ref. [111] where even much narrower resonances down to 2 × 10 −4 mG were observed, although with a 10 times higher density).
We have scanned magnetic fields up to 560 G for m = −1 (using both the compensation and pinch coils to reach fields above 450 G, and a much deeper ODT trap) and 120 G for m = +1, fully covering the predicted range. We have used small magnetic field sweeps of 1-2 G in a few seconds, but did not observe any resonant enhanced loss. One possible explanation is that the finite lifetime of the singlet molecular state, due to Penning ionization, leads to a broadening of the Feshbach resonance [101]. We expect the lifetime of the singlet molecular state to be much shorter than that of the quintet molecular state, which is 1µs [81].

Conclusion and Outlook
We have investigated the stability of ultracold spin-polarized gases of 4 He * in both m = +1 and m = −1 states. We have experimentally confirmed the longstanding theoretical prediction of a magnetic-field dependent two-body loss rate, which limits the stability of atoms in the m = +1 state at magnetic fields above 100 G. For m = −1 atoms two-body loss is energetically not possible (when 2µ B B k B T ), and therefore only three-body loss limits the lifetime of a dense sample, which is independent of magnetic field. We have also searched for a d-wave Feshbach resonance but did not observe one within the range of magnetic fields predicted by theory. We expect the short lifetime of the corresponding molecular state of the 1 Σ + g potential to be responsible for the smearing out of the resonance.
Our measured L 3 coefficient is interesting in the context of universal fewbody physics [106], as the interactions in a spin-polarized 4 He * gas are governed by a large two-body scattering length with a/r vdW ≈ 4. We note that recent theoretical few-body studies have focused on ground state 4 He and alkali atoms with tunable scattering length around a Feshbach resonance. Even though spin-polarized 4 He * atoms do not provide a tunable scattering length, it does provide an interesting benchmark system because of very accurate knowledge of the two-body potential [104] and can be used to study effects beyond universal theory (i.e., Refs. [112][113][114]).
Future experiments will advance in the direction of ultracold 3 He * -4 He * mixtures [18]. 3 He * has nuclear spin, but because of its inverted hyperfine splitting, the lowest spin state is spin-stretched. Therefore, a mixture of 3 He * -4 He * prepared in the lowest spin channel is also stable against Penning ionization as well as SR and RIPI, which are energetically not allowed. This stability provides an ideal starting point to prepare an ultracold mixture in the dipole trap at large magnetic fields to explore the recently-predicted 800-G broad interspecies Feshbach resonance [101].

Nonexponential one-body loss in a BEC
We have studied the decay of a Bose-Einstein condensate of metastable helium atoms in an optical dipole trap. In the regime where twoand three-body losses can be neglected we show that the Bose-Einstein condensate and the thermal cloud show fundamentally different decay characteristics. The total number of atoms decays exponentially with time constant τ ; however, the thermal cloud decays exponentially with time constant 4 3 τ and the condensate decays much faster, and nonexponentially. We show that this behavior, which should be present for all BECs in thermal equilibrium with a considerable thermal fraction, is due to a transfer of atoms from the condensate to the thermal cloud during its decay.

Introduction
Atomic gases can be cooled and trapped to ultracold temperatures and densities where Bose-Einstein condensation and Fermi degeneracy can be reached. These trapped gases decay in several ways due to one-body, two-body, and three-body collisions. Two-and three-body losses are density dependent and have been extensively studied [12,[115][116][117][118][119] and applied in work on atom-atom correlations [116] and on universal few-body physics [32,107,120].
One-body loss is generally considered trivial as it usually results from collisions between trapped atoms and room-temperature atoms from the background gas in the ultrahigh vacuum chamber that contains the trapped atoms. A background pressure of ∼10 −11 mbar typically leads to a lifetime of atoms in the trap of ∼100 s. Other causes of one-body loss are scattering by offresonant light in a dipole trap and on-resonance excitation by stray laser cooling light. All these effects cause exponential decay of the trapped cloud with a time constant τ , typically in the range of 1-100 s. Experiments in ultracold atomic physics measure this time constant by monitoring the number of trapped atoms as a function of time, a procedure commonly performed using absorption imaging of the cloud on a CCD camera. Due to the high densities in a condensate, two-and three-body losses are most important in the first instances of the decay. In studies of decay, it is generally assumed that the decay becomes exponential for long enough times, where two-and three-body losses have become negligible due to the low density that is then reached.
Zin et al. [121], however, showed theoretically that the decay of a BEC is expected to be nonexponential when it is in thermal equilibrium with a substantial thermal cloud. The origin of this effect stems from the transfer of atoms from the condensate to the thermal cloud during the decay. This atomic transfer occurs only when thermalization is fast compared to the change of thermodynamic variables during the decay of the trapped cloud. When twobody and three-body decay can be neglected the model predicts that for an overall exponential decay time τ , the condensate decays faster and nonexponentially, while the thermal cloud decays exponentially with a larger time constant, 4 3 τ . To our knowledge, this has not been demonstrated experimentally.
Enhancement in the decay of a BEC in the presence of a thermal cloud was observed by Tychkov et al. [15]. In that experiment the decay of a large (> 10 6 atoms) condensate of helium atoms in the metastable 2 3 S 1 state ( 4 He * , lifetime 8000 s) was monitored with and without a thermal cloud containing approximately the same number of atoms. That study, performed in a magnetic trap with atoms in the m = +1 magnetic substate, revealed that the condensate in the presence of a thermal cloud decayed much faster with the cloud than without. However, in that study the decay was studied in the presence of large two-and three-body losses possibly obscuring an effect of atomic transfer; the enhanced decay of the condensate could also be understood from two-and three-body inelastic collisions between condensate and thermal atoms. Furthermore, two-and three-body loss rate constants were not known accurately and both processes were expected to contribute about equally to the decay [12,15]. Therefore the rate constants had to be determined from a fit.
We have accurately determined these two-and three-body loss rate constants over a range of magnetic field values (see chapter 4). We transferred 4 He * atoms from a magnetic trap into an optical dipole trap and measured, both for atoms in the m = +1 and m = −1 state, the two-and three-body loss rate constants as a function of an applied magnetic field and with a (quasi-)pure BEC. In this chapter we extend the previous experiment to partially condensed clouds with a large thermal fraction in order to investigate atomic transfer. To reduce two-and three-body losses we study long time scales and work at small magnetic fields using m = −1 atoms that, in contrast to m = +1 atoms, only show three-body losses.

Theory of trap loss
The time evolution of the density of a trapped atomic gas can be described as The first term on the right takes into account one-body loss, which causes exponential decay with a time constant τ . L 2 and L 3 are the rate coefficients for two-and three-body loss, respectively, and are defined such that they explicitly include the loss of two and three atoms per loss event. The constants in front of L 2 and L 3 are κ 2 = 1/2! and κ 3 = 1/3! for a BEC, while κ 2 = κ 3 = 1 for a thermal gas [109]. The rate coefficients are obtained by measuring the number of trapped atoms after a variable hold time. The analysis of this data requires integration of Eq. 5.1 over space. Because a BEC and a thermal cloud have different density distributions, extracting two-and three-body loss rate coefficients from partially condensed samples becomes very difficult, and experiments usually focus on either a pure BEC or a thermal sample.

Atomic transfer model
In Sec. II of their paper, Zin et al. [121] have derived a simple model for the decay of the thermal and BEC components of a partially condensed cloud in thermal equilibrium below the temperature threshold for Bose-Einstein condensation. The essential assumption of the model is that thermal equilibrium holds during one-body decay of a condensate. Although two-and three-body losses may be incorporated in the model, this complicates the discussion and therefore we here ensure those to be negligible compared to the one-body losses by using low enough densities. If an atom is removed from the thermal cloud by, for instance, a collision with a background atom, then there is a place free in the otherwise saturated thermal distribution, which can be filled by a BEC atom, thus maintaining the size of the thermal cloud at the expense of the BEC. This transfer of atoms from the condensate to the thermal cloud enhances the decay of the condensate and increases the lifetime of the thermal cloud.
For the whole cloud, containing N atoms, decay is exponential with time constant τ . The model then predicts that the thermal cloud will also decay exponentially, however, with a larger time constant, while the condensate is expected to decay nonexponentially [121]: 3) Here N C and N T are the number of condensed atoms and number of thermal atoms, respectively, and N = N C + N T . The equations show that the thermal cloud is expected to decay exponentially with a time constant τ = 4 3 τ , independent of N C , which is assumed to be nonzero. For the BEC a nonexponential decay is expected if an appreciable amount of thermal atoms is present. The model is valid up to the point that there are no BEC atoms left or thermal equilibrium cannot be assumed anymore. Starting with a partially condensed cloud finally leads to the complete depletion of the BEC; for the pure thermal cloud, decay then proceeds with time constant τ = τ . Equations 5.3 and 5.4 are valid when the energy of a condensate atom (which is, apart from a small mean-field contribution, equal to the ground-state energy 0 of the harmonic trap potential) can be neglected compared to the average energy of a thermal atom (E T /N T ) [121]: In order to observe atomic transfer experimentally, inelastic collisions within the cloud should not play a role, but still a high enough elastic collision rate is necessary to reach fast thermalization. The thermalization rate is given by γ th = γ coll /2.7 [122], with collision rate γ coll = n avv σ el . Here, n av is the average density,v = 16k B T /πm is the mean relative thermal velocity, and σ el = 8πa 2 the elastic cross section at ultralow temperatures, where a=142.0(0.1)a 0 for 4 He * [81]. The large scattering length ensures fast thermalization down to a density of n av = 10 11 cm −3 , which is reached after 40 s in our dipole trap (see section. 5.3).

Experiment
We have studied one-body loss in a BEC of 4 He * atoms with a considerable thermal fraction. The experimental setup and measurement procedure have been described in the previous chapters; here we summarize only the most essential parts. A BEC of about 10 6 atoms is prepared in a crossed optical dipole trap at a wavelength of 1557 nm, and all atoms are transferred from the m = +1 magnetic substate to the m = −1 magnetic substate by an RF sweep. We measure the remaining number of trapped atoms after a variable hold time by turning off the trap, causing the atoms to fall and be detected by a microchannel plate (MCP) detector, which is located 17 cm below the trap center and gives rise to a time-of-flight of approximately 186 ms. From a bimodal fit to the MCP signal, the BEC and thermal fraction are extracted as well as the temperature and the chemical potential of the trapped. In Fig. 5.1 we show typical time evolutions of the partially condensed cloud, where the total, thermal and BEC atom number are logarithmically plotted, normalized to the total number of atoms at t = 0. Figure 5.1(a) shows the decay for a sample with an initial BEC fraction of 80% (with a thermal fraction at a temperature of 0.22(3) µK), while in Fig. 5.1(b) the BEC fraction is 50% (at a temperature of 0.36(5) µK). The experiments are performed in a dipole trap with a geometrical mean of the trapping frequency of 2π × 194(1) Hz and 2π × 245(1) Hz, respectively. Equation 5.5 is easily fulfilled throughout the decay since 0 N T /E T < 0.03. We also observe that the temperature changes very little during the decay (see the insets in Fig. 5.1). At short hold times, the loss is dominated by three-body recombination with a rate constant of L 3 = 6.5(0.4) stat (0.6) sys × 10 −27 cm 6 s −1 (see chapter 4). Two-body loss by Penning ionization is strongly suppressed for the spin-stretched states m = +1 and m = −1 [21], while for m = −1 the spin-dipole interaction also does not contribute to two-body losses because the atom is in the lowest spin state. Figure 5.1 clearly shows that, for hold times longer than 10 s, the loss of the total atom number becomes exponential, indicating the regime of one-body loss. In both measurements the initial density of the BEC is 3 × 10 13 cm −3 . The thermal cloud initially has a density n av ≈ 2 × 10 12 cm −3 , which after 40 s has decreased to n av ≈ 1 × 10 11 cm −3 . The thermalization rate decreases in that time from γ th ≈ 50 s −1 to γ th ≈ 3 s −1 , still much larger than the onebody decay rate of approximately 0.05 s −1 , ensuring sufficiently rapid thermal equilibration.
We observe a faster decay of the BEC fraction than the thermal fraction, eventually leading to a full depletion of the condensate, as expected from the atomic transfer model. Inspection of the decay of the thermal cloud for times longer than 10 s clearly shows that the thermal cloud decays with a larger time constant than the whole cloud. Fitting an exponential decay function to the data at longer hold times yields τ = 18.0(8) s, τ = 30.8(2.5) s, and τ /τ = 1.7(2) for the data of , where we give 1σ uncertainties. We attribute the relatively small lifetime to off-resonant scattering of the dipole trap light or resonant excitation by stray light, which could also explain the difference in the obtained lifetimes (our background pressure would limit the lifetime to ∼100 s). We conclude that the thermal fraction decays significantly slower than the BEC fraction and that the measured lifetime ratio is in reasonable agreement with the theoretical prediction of τ /τ = 4/3.
To compare the measurements of the BEC fraction with the atomic transfer model we first fit an exponential decay N (t) = N 0 e −t/τ to the total atom number for long hold times to obtain the overall lifetime τ as well as N 0 . Here N 0 is the apparent atom number at t=0 in the absence of three-body loss. In the second step, we fit an exponential decay N T (t) = N 0 (1 − f c )e −3t/4τ to the thermal fraction, with only the apparent BEC fraction f c as a fit parameter. Finally, with the obtained parameters N 0 , τ , and f c , we numerically solve Eq. 5.4 to obtain N C (t). The fit results are shown in Fig. 5.1 as solid lines. The 1σ errors in the fit parameters are reflected in the band around the theoretical curve for N C . We observe that the atomic transfer model describes the time evolution of the thermal and BEC fraction very well.

Conclusions
Our data provide direct verification of the atomic transfer model. The relatively short lifetime τ ≈ 20 s, together with a large scattering length, provide optimal conditions to see this effect. Also, at the relatively small numbers of trapped atoms needed to demonstrate atomic transfer, MCP detection allows better fitting of the BEC fraction compared to absorption imaging. Atomic transfer is most directly visible in the one-body loss regime, but is important in the two-and three-body regime as well, as was theoretically discussed in Ref. [121]. As a final remark we note that one should take care using a BEC for loss measurements. The thermal fraction may be small in the initial stage of BEC decay, but can dramatically increase after long hold times affecting the decay of the condensate, which will finally become nonexponential.

Summary
In a more elaborate form, the title of this thesis reads, "Precisely measuring the difference in frequency between two atomic states of helium, which is cooled down to near absolute zero".
This research combines a number of recently developed techniques of experimental physics: the cooling of atoms to absolute zero and a method to determine the frequency (color) of laser light extremely accurately. Through this combination of new techniques, we have measured an extremely weak effect in helium, which had never before been observed. To indicate the orders of magnitude: helium is cooled to less than a millionth of a degree above absolute zero. About nine orders of magnitude (a billion times) colder than room temperature. The effect which is then measured, a transition between two states of the helium atom, is fourteen orders of magnitude weaker than 'normal' transitions in helium. The frequency difference between these states is measured to a fractional accuracy of 8 × 10 −12 , similar to measuring the circumference of the earth, accurate to a hair's thickness.

Bose-Einstein condensation
First, a brief introduction of the basis of this study: Bose-Einstein condensation. This is a phase transition to a form of matter in which all atoms occupy exactly the same state, which occurs at extremely low temperatures. Similar to light, atoms exhibit a particle-wave duality. This means that, depending on the experiment, they seem to be behaving as waves, or as particles. Under normal circumstances, the particle nature prevails and an atom appears to be a little ball. At extremely low temperatures, however, the wave nature of matter becomes apparent. Similar to light-and sound waves, matter waves can interfere constructively and destructively.
The wavelength, associated with the wave nature, is inversely proportional to the velocity of the atom. Which is to say, the lower the velocity with which the atoms are moving about (i.e., the lower the temperature), the greater the associated wavelength. In an experimental setup, the atoms are confined to a small space, for example, by making use of magnetic forces. This imposes a constraint to the wavelength: it can not exceed the dimensions of the trap.

Summary
For a given geometry of the trap in which the atoms are confined, there is a maximum wavelength, corresponding to the lowest possible energy that an atom can have. This state of minimal energy is often referred to as the ground state.
When the temperature of the gas is extremely low (less than a millionth of a Kelvin), and the density is relatively high, such that the wavelength of the atoms becomes comparable to the average distance between the atoms, a phase transition occurs from a gaseous form to a Bose-Einstein condensate (BEC). In a BEC, all the atoms occupy the lowest possible energy state the trap allows: the ground state.
In the late eighties, a technique was developed that allowed the movement of atoms to be manipulated using laser light. In combination with magnetic fields, it became possible to slow atoms down (in a technique called laser cooling), and to capture them in a magneto-optical trap, in which temperatures below a milli-Kelvin were achieved. Although these temperatures were unprecedentedly low, it had yet to be a thousand times colder to reach the phase transition to BEC. To continue cooling further, techniques were developed to confine atoms in a purely magnetic trap under high vacuum conditions (necessary to minimize heating due to collisions with background gas). In the mid-nineties, the first BEC was realized in a rubidium gas by the group of Cornell and Wieman. Almost simultaneously, the group of Ketterle succeeded in producing a BEC in a cloud of sodium atoms. For these achievements, Cornell, Wieman and Ketterle received the Nobel Prize in physics in 2001.

Metastable helium
Bose-Einstein condensation of helium has been realized as well. Special to helium is that it first needs to be excited to a long-lived (metastable) state. Helium has two electrons that are both in close orbit around the nucleus. For the technique of laser cooling to be used, interaction between laser light and the electrons in the helium atom is necessary. That is to say, the laser light must have a frequency that corresponds to the frequency difference between two states of vibration of the atom. The problem with helium in the ground state, however, is that the lowest frequency required to transfer the atom to an excited state lies in the extreme ultraviolet, which is especially difficult to generate. If, however, one of the two electrons of helium is already in an excited state, the step to the next state of vibration is easily reached using laser light in the infrared.
In general, the lifetime of excited states is much too short (order of nanoseconds) to run an experiment with. Just as an atom can change its state by absorption of laser light, tuned to the the proper color, it can also decay from an excited state to a less energetic state under emission of a photon (a quantum of light). The condition that must be met for this process to occur, is that both the energy and the angular momentum of the photon equal the difference in energy and angular momentum between the two involved states of the atom. In other words, energy and angular momentum are conserved. Some states cannot easily spontaneously decay (or be excited), because the difference in angular momentum cannot easily be carried away (or absorbed) by a photon. A state of vibration that does not directly decay to a lower energetic state is called metastable. The first excited state in helium is such a metastable state, having a lifetime of more than two hours, which is sufficiently long for this form of metastable helium to be used during an experiment.

Spectroscopy
The research described in chapter 3 can be regarded as a test of atomic physics. The theory that currently describes the structure of atoms most accurately is the theory of quantum electrodynamics (QED). This theory is extremely precise, but requires progressively more computation to be even more so. To explore the boundaries of this theory, measurements are required with similarly high precision. A method that can be used for this purpose is spectroscopy of atomic transitions, i.e., determining the frequency difference between two states of an atom, as accurately as possible. In practice, this means one needs to determine the exact frequency (color) of laser light that is absorbed by the atom.
The transition that is measured in this study, connects the first two excited states of helium. This transition is extremely weak, and had therefore previously never been observed. The practical difficulty of exciting this transition is that a relatively large amount of light is required, at exactly the correct frequency, over a long period of time (order of one second) on one and the same atom, in order to attain a significant chance that the atom reacts. To apply laser light to an atom for a prolonged period of time, the atom needs to be pinpointed at a fixed location. For this reason, the atoms are confined in a trap and cooled down to near absolute zero. The other ingredients are stabilization of the laser, and the determination of the absolute frequency of the laser light. To determine the laser frequency with 11 digits of precision, a device was used which did not exist until a little over ten years ago, the optical frequency comb laser. This is a system that defines frequencies in the optical spectrum, with the accuracy of an atomic clock. The combination of these techniques makes this measurement a thousand times more accurate than QED theory can predict at this time. We have measured the atomic transition in the usual helium-4, as well as in the more rare isotope helium-3. Combining these results, the difference in size of the nuclei of the isotopes has been determined.

Interactions between atoms
Chapters 4 and 5 describe research that addresses the interactions between the ultra-cold metastable helium atoms. As a metastable helium atom is populating an excited state, it possesses a certain amount of internal energy. In case of helium, this energy is so high that a collisional process, during which the energy is released, can be regarded as a kind of nano-explosion. Therefore, the lifetime of a trapped sample of ultra-cold metastable helium is directly dependent on the frequency with which collisions take place. The various collisional processes are characterized by the number of metastable helium atoms which partake in a collision. During an interaction with background gas (that is gas which, despite the ultra-high vacuum, is still present in the setup), the gas particle is ionized and the helium atom is launched from the trap. What can also happen (but is highly suppressed), is that two trapped metastable helium atoms collide with each other and are both expelled from the trap. The latter process is known as a two-body interaction, and prevails as a loss process at high density of the metastable helium cloud, in spite of the suppression.
Chapter 4 focuses mainly on two-body interactions in a Bose-Einstein condensate. The frequency with which collisions between the atoms take place, depending on the size of an externally applied magnetic field, is examined. These measurements confirm the theory, which predicts that the two-body interaction rate increases significantly above a certain magnetic field strength.
In chapter 5, the loss process due to one-body interactions (collisions with background gas) is examined. Not only the Bose-Einstein condensate is of interest here, but also the ultra-cold metastable helium gas in the trap that is not Bose-Einstein condensed. Hence, there are three systems which play a role here: the BEC, the non-condensed gas, and the background gas in the setup. Investigated here, is how the lifetime of a BEC, determined by one-body interactions, is influenced by the presence of the non-condensed metastable helium gas in the trap. Because the BEC and the non-condensed gas are continuously in equilibrium with each other, the loss processes in either system are not independent. For example, when an atom from the non-condensed helium gas is expelled from the trap due to a collision with background gas, this may be followed by a BEC atom transferring to the non-condensed gas in order to restore equilibrium. In this way, losses in the non-condensed cloud indirectly contribute to the losses in the BEC. This effect, previously predicted theoretically, has been experimentally demonstrated with this study. waarmee de atomen bewegen (ofwel, hoe lager de temperatuur), hoe groter de geassocieerde golflengte. In een experimentele opstelling worden de atomen in hun bewegingsruimte beperkt tot een kleine ruimte, bijvoorbeeld gebruikmakend van magnetische krachten, waardoor er een limiet wordt opgelegd aan de golflengte: deze kan niet groter zijn dan de afmeting van de ruimte. Voor een gegeven geometrie van de val waarin de atomen opgesloten zitten, bestaat er dus een maximale golflengte, overeenkomstig met de laagst mogelijke energie die een atoom kan hebben. Deze laagst energetische toestand wordt ook wel de grondtoestand genoemd.
Half way through my promotion period Juliette Simonet, a PhD student from the cold atoms group of Michèle Leduc at ENS, joined our group for several months. With her inexhaustible enthusiasm and hard work, she provided a great contribution to the development of the experiment. Shortly after, Maarten Hoogerland joined our team for a sabbatical from Auckland, New Zealand. It was great having an experienced scientist around the lab. He contributed a great deal to the initial measurements. During the final year I had the pleasure to be working together with Joe Borbely. I'm very grateful for the amount of effort he put into our measurements. I remember coming into the lab early in the morning and turning on the setup, only to find it was still cooling down from the last measurement of the night. A big thanks to all of you.