An Analogy with Optical Lasers
In the very general picture depicted in figure 1.3, the atom laser can be thought of as an analog to the well-known optical laser. The following points briefly outline this analogy:
• Just as the lasing mode in an optical laser is a source of coherent photons, the lasing mode of an atom laser is a BEC, a source of coherent atom waves.
• In an optical laser, the photons are held within a cavity created from two optical mirrors. Similarly, for an atom laser, the BEC is constrained by either a magnetic or an optical trapping potential.
• In an optical laser, the extraction of the beam can be achieved as one of the mirrors forming the cavity has been carefully chosen to partially transmit the light. Comparatively, atom laser beams are usually extracted by transferring the atoms from their initial magnetically trapped internal state into a state that does not interact with the trapping field, leading the way for atoms to fall away under gravity. This extraction process can be applied either as short or long (quasi-continuous) pulses.
• Both optical and atom lasers are extracted from a macroscopically populated mode of the cavity (or the trap). In the case of optical lasers, this mode is highly excited and multiple wavelengths long whereas in the case of an atom laser it is the lowest energy mode (usually the ground state of the trap).
• Finally, in order to maintain the production of a laser beam, one has to continuously refill the lasing mode as it is depleted. In an optical cavity, a gain medium of atoms is pumped by some source of energy in order to sustain a population inversion that coherently amplifies the lasing mode via stimulated emission of photons. Pumping (or refilling) of an atom laser is nowadays a real challenge in the field of atom lasers. Recently, a method has been demonstrated in our group  to out-couple an atom laser beam while simultaneously and irreversibly pumping new atoms from a physically separate cloud into the trapped Bose-Einstein condensate that forms the laser source.
Definition of an ’Atom Laser’
The term ’atom laser’ has become widely used in the last decade and it is therefore important to have a more rigorous definition of what really defines an atom laser. For Wiseman , a general laser is a device containing a highly populated mode of a boson field. This so-called ‘laser mode’ is continuously replenished so that the output continues indefinitely. The output beam is formed via an out-coupling mechanism and is well approximated by a classical wave of fixed amplitude and phase. This definition leads to four properties defining the atom laser, in analogy with optical lasers :
1. The output beam must be highly directional, clearly defining a longitudinal direction along which propagation and dispersion occur and two transverse directions along which diffrac-tion may occur. The directionality condition does not necessarily imply that the laser out-put must travel in free space and wave-guides have been proved to be useful for atom lasers to support against gravity, prevent spreading due to diffraction, and also improve the spatial profile .
2. The beam must be narrow in linewidth. There is a limit on the uncertainty δ k of the longitudinal spatial frequency of the output such that δ k ≪ kAL = 2π /λAL, where kAL is the wave vector of a boson in the output which, as mentioned earlier, continuously changes during the propagation.
3. The output beam must have a well defined phase, implying that phase fluctuations in the output beam are small. This condition can be expressed quantitatively using Glauber’s normalized first order coherence function  defined by
G(1)(τ ) = Ψ†(t + τ )Ψ(t) (1.7)
or its normalized form
g(1)(τ ) = Ψ†(t + τ )Ψ(t) (1.8)
where Ψ(t) is the complex amplitude of the field. For τ = 0, |G(1)(τ )| is simply the mean intensity I of the field. However, as τ increases, |G(1)(τ )| decreases as the phase of the field gradually becomes decorrelated from its value at time t with |G(1)(τ )| → 0 when τ → ∞. By contrast, an ideal first order coherent source will have g(1)(τ ) = 1 for all τ . In practice, the phase of an atom laser, and hence G(1)(τ ), will remain constant over a long period (tcoh) before significantly decaying and this coherence time is defined by tcoh = 0 |g(1)(τ )|dτ . (1.9)
The first order temporal coherence of an atom laser was demonstrated in  where the authors showed that the linewidth of the atom laser, defined as = 1/tcoh, was Fourier limited (by the out-coupling duration and the detection resolution) giving an upper limit on the phase fluctuations of the atom laser. The first order spatial coherence of an atom laser beam was also demonstrated experimentally [56, 57] by observing high contrast in-terference patterns from two interfering atom laser beams that were phase related (created from the same initial source).
4. Finally, the output beam must have a well defined intensity. Like the previous condition, this can be expressed in terms of correlation functions. The second order normalized cor-relation function is given by
g(2)(τ ) = Ψ†(t + τ )Ψ†(t)Ψ(t + τ )Ψ(t) (1.10)
which is also commonly written as
g(2)(τ ) = : I(t + τ )I(t) : (1.11)
where I = Ψ†(t)Ψ(t) is the intensity of the field and :: denotes normal ordering (i.e. all creation operators to the left of all annihilation operators). For fields that are second order coherent, the intensity fluctuations are small in the sense that |g(2)(τ ) − 1|2 ≪ 1 and the only intensity noise contribution is quantum noise (also called shot noise). Second order coherence of an atom laser has been demonstrated experimentally using a Hanbury Brown-Twiss type experiment where the second order correlation function was measured to be g(2)(τ ) = 1.00 ± 0.01 .
It is important to note that the use of the term ’atom laser’ in the literature has not followed completely Wiseman’s definition. Indeed, it has been widely used for out-coupled beams which are either short or long-pulsed but not rigorously continuous since the condensate (i.e. the laser source) was not continuously replenished. In our experiment, the atomic beams are almost always long-pulsed (quasi-continuous) unless specified otherwise. The notion of short and long atomic pulses will be described in section 1.4.3. For consistency with the literature the term ’atom laser’ will be used although the condensate is never replenished and the output cannot continue infinitely.
Although all atom lasers produced to date utilize a Bose-Einstein condensate as a source (or reservoir) to populate the atomic beam, they can be distinguished by the different techniques used to out-couple the atoms. There are many ways of coherently transferring atoms from a condensate into the propagative matter-wave of the laser. These methods can be classified in two main groups namely the non-state changing and the state changing out-coupling techniques. A non-state changing out-coupling technique is a method where the internal state of the atoms is not modified during the out-coupling process. Conversely, a state changing out-coupling process couples the magnetically trapped atoms out of the condensate by transferring them into a different internal state which is magnetically un-trapped and falls under gravity. To do so, one can use either a radio-frequency (RF) (or microwave) field, or alternatively an optical Raman field. Both techniques are described in detail in the following sections.
Non-State Changing Out-coupling
Although the majority of the experimentally demonstrated out-coupling mechanisms are based on either Raman or RF transfer of atoms from a magnetically trapped into an un-trapped Zeeman state, it is interesting to mention a few examples of non-state changing out-coupling techniques.
The first atom laser beam was created when the first time-of-flight measurement of a con-densate was performed after switching off the magnetic trap. The pulse of atoms falling under gravity can be considered as forming a coherent matter-wave. This is analogous to the optical ‘cavity dumping’ where all the photons in a cavity would be extracted at once. This very simple technique is also very limited since, by definition, it is a short-pulse method, with the repetition rate being the time it takes to create a new BEC (typically a few tens of seconds) and the linewidth of the beam being very broad, on the order of the full width of the condensate (see section 1.3.2).
The second technique was a quantum tunneling effect demonstrated by Anderson and Kase-vich . In this experiment, a condensate was loaded into the top of a vertical array of optical traps created by a standing wave. Acceleration due to gravity induced tunneling between the traps and constructive interference between the lattice sites resulted in falling pulses of atoms. The size of the pulses could be adjusted by altering the depth of the optical wells. However, this outcoupling technique has important limitations. First, the atomic beam is again inherently pulsed and cannot work in a continuous regime. Second, the density of these pulses is limited as for large densities, mean-field interactions cause significant dephasing between lattice sites, thus degrading the interference process. No further consideration was given to this technique in the development of an atom laser.
A further unconventional form of an atom laser was produced in , again using an optical method. In this experiment, a single spin state condensate was produced in a strongly focused optical trap by applying a magnetic field gradient stronger than the transverse optical confining force. All atoms were removed by this field gradient, except those in the magnetically insensitive (mF = 0) state which stayed confined in the optical dipole trap. Here mF is the projection of the total angular momentum F of the hyperfine atomic state that is considered. An atom laser was finally produced by smoothly lowering the optical trapping potential so that atoms from the condensate could fall under gravity. The benefits of this scheme are that the outcoupling is not limited by the stability of magnetic fields. It is dependent only on the laser intensity fluctuations but stabilizing laser intensities is technically easier than eliminating stray magnetic fields and producing a highly stable magnetic trap. However, magnetic fields are still required to produce cold atoms that will continuously replenish the condensate. Moreover, the use of strong magnetic field gradients and shallow optical potentials could make it difficult to extend this method in the case of a pumped atom laser.
State Changing Out-coupling
In order to illustrate the state changing out-coupling process, the atomic spectroscopy of the 87Rb ground state is presented in figure 1.4. It has two distinct hyperfine states of total angular momentum F = 1 and F = 2 characterized by Landé factors of gF = −1/2 and gF = 1/2 respectively. In most experiments, and in particular in ours, the BEC is trapped in a magnetic potential. In the bias field at the minimum of the trap, a hyperfine level is split into 2F + 1 non-degenerate Zeeman states. For weak magnetic fields, in which the Zeeman effect is small and may be treated as a perturbation, the splitting between two Zeeman states characterized by their projection mF and m′F can be approximated as h¯δZeeman = ΔmF gF µB|B| (1.12) where ΔmF = m′F − mF , µB is the Bohr magneton and |B| is the magnetic field amplitude. Zeeman states experience differently the magnetic trapping potential. In the case of 87Rb, the |F = 2, mF = 2 , |F = 2, mF = 1 and |F = 1, mF = −1 states are trapped and can poten-tially form the condensate. Any mF = 0 state is un-trapped and insensitive to the magnetic field (under the first order Zeeman effect), whereas the |F = 2, mF = −2 , |F = 2, mF = −1 and |F = 1, mF = 1 states are anti-trapped and quickly expelled from the magnetic field region. A state changing process will coherently couple two Zeeman states that can be either inside a given hyperfine state (this is the case of RF and the Raman out-coupling used in our experiment) or from two distinct hyperfine states (which is the case of microwave or sometimes Raman out-coupling).
Radio-Frequency Output Coupling
Mewes et al.  demonstrated the first pulsed atom laser output coupler based on the appli-cation of pulsed radio-frequency (RF) fields, before continuous output coupling was achieved by Bloch et al. .
Radio-frequency output coupling consists of applying a monochromatic RF magnetic field B of given amplitude BRF and frequency ωRF in order to induce controlled spin flips between adja-cent Zeeman states. The aim is to resonantly and coherently transfer the atoms from their initial magnetically trapped state into a state which is insensitive to magnetic fields. Consequently, the atoms are no longer trapped and will be extracted from the BEC due to gravity, as shown in figure 1.5a. Experimentally, a BEC is produced in the F = 1 manifold and the magnetically trapped |F = 1, mF = −1 > atoms are transferred to the |F = 1, mF = 0 state (in which they no longer interact with the magnetic potential). The process is resonant provided the frequency of the field satisfies ωRF ∼ δZeeman = µBB/(2h¯) (figure 1.5b). Because the first order Zeeman shift symmetrically splits adjacent Zeeman states, all 2F + 1 sublevels are coupled, producing a multi-state atom laser beam. For instance in the F = 2 manifold, the condensate atoms initially in the |F = 2, mF = 2 trapped state are transferred, via the |F = 2, mF = 1 state, into the mF = 0 un-trapped state but can also populate the anti-trapped |F = 2, mF = −1, −2 states.
Microwave out-coupling is a similar technique with the difference that it couples two Zeeman levels of different hyperfine states, producing a single-state atom laser.
Raman Output Coupling
The outcoupling mechanism used in most of the experiments described in the thesis is also a state changing technique but utilizes an optical Raman transition. It was originally suggested as a mechanism for atom laser outcoupling by Moy et al.  and first demonstrated experi-mentally by Hagley et al.  who produced a quasi-continuous multi-state atomic beam. The scheme (figure 1.6a) is based on the absorption and stimulated emission of photons from two optical laser beams, again transferring the atoms from a magnetically trapped to an un-trapped state. The process is coherent and inherently controllable. However, the main difference with the previously described RF coupling technique is that the atom laser is produced with an ini-tial momentum kick in any chosen direction defined by the incoming directions of the two laser beams. Experimentally, this two photon transition is applied on the F = 1 manifold, as shown in figure 1.6b. An atom in the trapped mF = −1 state coherently absorbs a photon from a beam with frequency ω1 and is stimulated to emit into the other beam with a frequency ω2 thus changing its internal state to an un-trapped magnetic level (mF = 0).
The process is reversible and un-trapped atoms can be coherently transferred back into the condensate. In order for this scheme to be efficient, one has to create the following conditions :
1. The two laser beam polarizations must be chosen to allow optical transitions between the Zeeman sublevels. In figure 1.6b, one laser beam is thus linearly (π ) polarized whereas the other one has a σ − circular polarization where the polarization is defined in terms of absorption.
2. The frequency difference between the two lasers (δ = ω2 −ω1) has to be adjusted to satisfy energy and momentum conservation. An atom, initially at rest, gains a momentum kick p = h¯(k2 − k1) following momentum conservation upon absorption and emission of each photon. Here k1,2 are the wavevectors of the lasers of the two laser beams. When the beams make an angle θ and are produced from the same laser source of wavevector k (which is always the case in our experiments), one can write : p = 2hksin¯ θ . (1.13)
Table of contents :
CHAPTER 1: EXPERIMENTAL AND THEORETICAL BACKGROUND OF ATOM LASERS
1.1 GENERAL OVERVIEW OF ATOM LASERS
1.1.2 An Analogy with Optical Lasers
1.1.3 Definition of an ’Atom Laser’
1.2 ATOM LASER OUT-COUPLING TECHNIQUES
1.2.1 Non-State Changing Out-coupling
1.2.2 State Changing Out-coupling
1.2.3 Radio-Frequency Output Coupling
1.2.4 Raman Output Coupling
1.3 RESONANT WIDTH OF THE CONDENSATE
1.3.1 Gravitational Sag
1.3.2 Resonant Frequency Width
1.4 RABI-FREQUENCY AND THE DIFFERENT OUT-COUPLING REGIMES
1.4.1 Output Coupling Strength
1.4.2 Pulsed and Quasi-Continuous Output Coupling
1.4.3 Out-coupling regimes
CHAPTER 2: RUBIDIUM CONDENSATE AND RAMAN BEAMS
2.1 EXPERIMENTAL SETUP TO PRODUCE BEC
2.1.1 Atomic structure of 87Rb
2.1.2 Laser system
2.1.3 Vacuum system
2.1.4 2D MOT
2.1.5 3D MOT
2.1.6 Transfer to a magnetic trap
2.1.7 Transport by a Translation Stage
2.1.8 QUIC Trap
18.104.22.168 Transfer to the QUIC Trap
22.214.171.124 Optical Imaging
126.96.36.199 Trap Frequencies
2.2 EXPERIMENTAL SETUP TO PRODUCE RAMAN BEAMS
2.2.1 Optical setup
2.2.2 Adjusting the polarization of each of the beams
CHAPTER 3: DIVERGENCE OF AN ATOM LASER
3.1 M2 QUALITY FACTOR
3.2 OUT-COUPLING FROM THE CENTER OF THE BEC
3.3 REDUCING THE DIVERGENCE OF THE ATOM LASER
3.4 THEORETICAL MODEL OF THE EXPERIMENT
3.4.1 The model
3.4.2 Data analysis
3.5 DEPENDENCE ON TRAPPING FREQUENCIES
CHAPTER 4: COHERENT ATOM BEAM SPLITTING
4.1 OVERVIEW ON BRAGG DIFFRACTION
4.2 DIFFRACTION FROM A SINGLE LASER BEAM
4.3 A VELOCITY RESONANT PROCESS
4.3.1 Theoretical Model
4.3.2 Experimental measurement
4.4 BRAGG DIFFRACTION EFFICIENCY
4.4.2 Theoretical Model
CHAPTER 5: RF OUT-COUPLING FROM TWO- AND MULTI-LEVEL SYSTEMS
5.1 THEORETICAL MODEL
5.1.1 Time-dependent Gross-Pitaevskii Equations (GPE)
188.8.131.52 GPE for a condensate
184.108.40.206 GPE for a multi-level system
220.127.116.11 Dimensionality reduction
18.104.22.168 Initial conditions
5.1.2 Numerical method
22.214.171.124 The grid
126.96.36.199 Results of the simulations
5.2 EXPERIMENTAL COMPARISON OF THE MODEL
5.2.1 Bound state of an atom laser
5.2.2 Spatial structure of an atom laser
5.3 COMPARISON OF TWO- AND MULTI-STATE SYSTEMS
5.3.1 Flux of the atom laser
5.3.2 Population dynamics
188.8.131.52 Five-state system
184.108.40.206 Three- and two-state systems
5.3.3 Spatial dynamics
5.3.4 Density fluctuations
220.127.116.11 Five-state system
18.104.22.168 Three- and two-state systems
5.3.5 Flux and fluctuations trade-off
CHAPTER 6: HELIUM BEC: EXPERIMENTAL SETUP
6.1 THE METASTABLE HELIUM ATOM 4He∗
6.1.1 The metastable 23S1 triplet state
6.1.2 Penning collisions
6.2 EXPERIMENTAL SETUP
6.2.1 Vacuum system
6.2.2 Optical Setup
6.2.3 The source of atoms
6.2.5 The Zeeman Slower
6.2.6 Channel Electron Multiplier (Channeltron)
6.2.7 The MOT
6.2.8 Magnetic trap and evaporative cooling
CHAPTER 7: OPTICAL TRAPPING OF 4HE ATOMS
7.1 OPTICAL DIPOLE POTENTIALS
7.1.1 Oscillator Model
22.214.171.124 Interaction with a light field
126.96.36.199 Atomic Polarizability
188.8.131.52 Dipole Potential and Scattering Rate
7.1.2 Dressed State Picture
184.108.40.206 Two-Level Atom
220.127.116.11 Multi-Level Atom
7.2 RED-DETUNED DIPOLE TRAP FOR HE∗
7.2.1 Single Gaussian Beam
7.2.2 Crossed Dipole Trap
7.2.3 Experimental Layout
18.104.22.168 Light Source
22.214.171.124 Output Beam Waist
126.96.36.199 AOM efficiency
188.8.131.52 Lens focussing
184.108.40.206 Loading an optical dipole trap
7.3 INELASTIC COLLISION RATES IN A GAS OF SPIN-POLARIZED METASTABLE HELIUM ATOMS
7.3.1 Spin-dipole Hamiltonian
7.3.2 Spin relaxation
7.3.3 Spin relaxation towards Sf = 2
7.3.4 Spin relaxation towards Sf = 0
7.3.5 Inelastic collision rates and magnetic field dependence
7.3.6 Future experiment
CHAPTER 8: NOVEL ATOM TRAP FOR HE ATOMS IN OPTICAL LATTICES
8.1 PERIODIC LATTICE POTENTIALS
8.1.2 Quantum Phase Transition from a Superfluid to a Mott Insulator
220.127.116.11 Bose-Hubbard Model
18.104.22.168 Superfluid-Mott Insulator quantum phase transition
8.1.3 New insight with metastable helium atoms
8.2 NOVEL ATOM TRAP
8.2.1 Experimental challenge
8.2.2 Optical lattice requirements
8.2.3 Coil and beam geometry
8.2.4 Trap simulations
8.2.5 Electric circuitry
22.214.171.124 Wiring circuit
126.96.36.199 Water cooling