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Theory of quantum gases
Every particle falls into one of two categories: if its spin is /2 it is called a fermion; if it is it is called a boson. This is as true of composite particles such as atoms as it is of elementary particles like electrons. The quantum mechanical properties of both classes are very diﬀerent: the many-particle wavefunction of identical fermions must be anti-symmetric under the exchange of two particles, while the bosonic wavefunctionmust be symmetric. For fermions, this leads to the Pauli exclusion principle: two fermions may never occupy the same quantum state. This can be seen in the following way. Let φ, χ be single particle wavefunctions. Two identical particles in these states are described by Ψ+1,2/− = (φ1χ2 ± φ2χ1)/√2. The +(-) makes the wavefunction symmetric (anti-symmetric) and describes bosons (fermions). The wavefunction Ψ− describing two fermions in the same state φ = χ vanishes: Ψ−1,2 = (χ1χ2 − χ2χ1)/ 2 = 0 and is thus unphysical. Two fermions can not occupy the same state. The link between the spin and the statistics of a particle is the spin-statistics theorem, which can be derived using relativistic quantum mechanics.
The statistical and the scattering properties are dominated by this (anti-) symme-trization requirement. At zero temperature, trapped fermions occupy, one by one, the lowest energy states, forming a Fermi sea, whereas (non-interacting) bosons are con-densed in the lowest state as a Bose-Einstein condensate (see figure 1). The scattering of distinguishable particles is possible in all angular momentum orders. But only the even angular momentum orders (s,d,…), which correspond to symmetric wavefunctions are permitted for identical bosons. Only odd orders (p,f,…), corresponding to anti-symmetric wavefunctions, are permitted for fermions. In the following, the properties of fermionic and bosonic gases relevant for the described experiment such as density distributions, heat capacities etc. are briefly derived. For a more detailed description see .
Properties of a classical gas
Before describing the properties of gases of identical particles, we will briefly recall the properties of a gas of distinguishable particles, for which (anti-) symmetrization of the wavefunction is not required. In the limit of low phase-space densities, a fermionic gas, a bosonic gas and a gas of distinguishable particles all behave in the same manner. This limit is called a classical gas. We consider N non-interacting particles with mass m trapped in a cylindrical harmonic potential with trapping frequencies ωrad radially and ωax = λωrad axially. The energy distribution is the Boltzmann distribution f( T, C ) = Ce−β , (1.1) where β = 1/kBT , kB the Boltzmann constant and C a normalization constant. The hamiltonian describing the gas is
→− 2 k 2 mω2
→− + rad ρ2 ; ρ x2 + y2 + λ2z2 ) 1/2 .
H ( r , k
→−)= 2 = ( 2m
The constant C is determined using the normalization N = dr dk f( = H ( r , −→) ) 3 3 →− k ,T,C and yields C = N λ/(2π)3σ3 σ3 and σk = √ with σx = kBT /mω2 2π/λdB where √ x k rad λdB = h/ 2πmkBT is the de Broglie wavelength. The distribution in momentum and in real space decouple and are both gaussian:
Nλ 2 k 2 2 2
f r , k , T exp − β −→ exp −β mωrad . (1.4)
) = (2π)3σ3 σ3 2m 2 ρ
The momentum distribution is isotropic with the RMS size σp = σk. The spatial distribution follows the anisotropy of the trap and has the RMS size σi = kBT/mωi2. For ωrad > ωax the gas is distributed in a cigar shape and has the aspect ratio λ.
A property which is very important for sympathetic cooling (see section 1.2.2) is the specific heat capacity of the gas. It is defined as CF ≡ ∂E∂T N and is a measure for the amount of energy that must be dissipated, to cool the sample by ∆T . The heat capacity of a classical gas is Ccl = 3NkB .
Properties of a fermionic gas
Now we will discuss the properties of N identical non-interacting fermions with mass m trapped as above in a cylindrical harmonic potential with trapping frequencies ωrad radially and ωax = λωrad axially. At zero temperature, fermions occupy one by one the N lowest lying energy states due to the Pauli principle. The energy of the highest occupied state is called the Fermi energy EF . This energy, together with the energy corresponding to the temperature kBT and the two oscillator energies ωrad and ωax are the four energy scales of the system. In the following we will always assume kBT and EF ωrad, ωax. Then the energy distribution has no structure on the scale of the energy level splitting and eﬀects due to the discreteness of the energy level spectrum are negligible. Thus we are left with two energy scales EF and T . We may define a parameter T/TF called the degeneracy parameter, with TF = EF /kB being the Fermi temperature.
In the case T/TF 1 the occupation probability of the quantum states is low and thus it is very improbable that two particles occupy the same state, independent of the particle statistic. Thus a Fermi gas will behave classically.
We are here more interested in the case T /TF < 1, in which case the gas is called degenerate. This term is used in analogy with the degenerate Bose gas, but without the notion of having all the atoms in the same energy level, a notion that is normally associated with the word degenerate. For a classical or bosonic gas at low temperature, the probability to find several particles in the lowest states is high. This is forbidden for fermions and thus the properties of the fermionic degenerate gas are nonclassical and diﬀer also from the bosonic case. The shape of the cloud is no longer gaussian. Its characteristic size is given by the Fermi radius RF which corresponds to the maximum distance from the trap center, that a particle with energy EF can reach in the radial direction
RF ≡ 2EF
mωrad2 . (1.5)
This size exceeds the size of a classical gas, which means that by cooling a gas from the classical to the degenerate regime, its size will stop shrinking when degeneracy is reached. This is the eﬀect of the Fermi pressure, a direct consequence of the Pauli exclusion principle. Its observation is one of the main results of this work. The momen-tum distribution is position dependent and the heat capacity is reduced in comparison to a classical gas.
In the following the discussed properties will be exactly derived in the case of zero temperature and finite temperature results are cited. The complete treatment and a discussion of the approximations applied may be found in the useful article by Butts and Rokhsar .
From the Pauli principle it is easy to derive the energy distribution of fermions, the Fermi-Dirac distribution f( ) = 1 , (1.6) eβ( −µ) + 1 where µ is the chemical potential (see e.g. ). The latter is determined by the normalization condition N = d f ( )g( ) , (1.7) where g( ) is the density of energy states which, for a harmonic potential, is g( ) = 2/(2λ( ωrad)3). The Fermi-Dirac distribution never exceeds 1, reflecting the Pauli exclusion principle. At zero temperature the energy distribution f( ) = 1 below the Fermi energy EF ≡ µ(T = 0, N) and 0 above. For lower degeneracies, this step function is smeared out around EF with a width of the order of EF T /TF (figure 1.1a).
For high temperatures or energies higher than the Fermi energy, f( ) can be approx-imated by the classical Boltzmann distribution, f( ) = e−β( −µ). The same is true for the bosonic energy distribution function (see 1.1.6). The three distribution functions are compared in figure 1.1b). This behavior leads to a simple detection scheme for quantum degeneracy. A classical gaussian distribution is fitted to the high energy part of the spatial or momentum distribution and to the whole distribution. For quantum degenerate gases the resulting width diﬀer, in contrast to a classical gas (see section 1.1.5 and figure 3.20).
By integrating equation (1.7) for T = 0 the Fermi energy is found to be EF = ωrad(6N λ)1/3 . (1.8)
To obtain a high Fermi energy it is necessary to use a strongly confining trap and to cool a large number of atoms. The Fermi energy corresponds to a wave number kF ≡ 2mEF = (48N λ)1/6σr−1 , (1.9) with σr = /(mωrad) the radial width of the gaussian ground state of the trap. kF is the maximum momentum reached in the Fermi gas.
Spatial and momentum distribution
To derive the spatial and momentum distributions it is convenient to apply the semiclassical, also called the Thomas-Fermi, approximation. Particles are distributed in phase-space according to the Fermi-Dirac distribution. The density of states is (2π)−3 and sums over states are replaced by integrals. The number density in phase-space is
w( r , −→; ) = 1 , (1.10)
→− k T, µ 1 3
(2π) e →− − ) + 1
β(H( r ,→−) µ
with the Hamiltonian
→− 2 k 2 mω2
→− + rad ρ2 ; ρ x2 + y2 + λ2z2 ) 1/2 . (1.11)
H ( r , k
→−)= 2 = (
The chemical potential is determined using the normalization
= d r d→− ( →− −→; ) (1.12)
N →− k w T,µ .
r , k
After determination of µ the spatial and momentum densities can be calculated by integrating over the momentum or spatial degrees of freedom, respectively:
n −→ k w →− T, µ ,
( r , T ) = ( r , k (1.13)
d→− →−; )
n k , T ) = →− −→ k T, µ . (1.14)
˜(→− d r w( r , →−; )
This method gives solutions for all temperatures, but can only be applied numeri-cally. For T = 0 it is easy to calculate the distributions analytically. For each spatial point in phase-space we can determine the local Fermi wave number using
2 kF (→−) 2
r r E , (1.15)
2m +V(→−)= r 1 mω2 ρ2
with V (→−) = r being the potential. All momentum states corresponding to the r k r position −→ are filled up to the momentum F (−→). This expresses the fact that all states up to EF are occupied according to the definition of the Fermi energy. EF is constant over the sample since any position dependence would provoke a flow of atoms in phase-space and the sample is already defined to be in equilibrium. Equation (1.15) shows that position and momentum distributions do not decouple. At the outer edges of the cloud, the atoms have a lower momentum then in the middle, in contrast to a classical distribution. The density at each point is the number of states that fit in the Fermi sphere with radius kF in k space, which is the volume of the Fermi sphere multiplied by the density of states (2π)−3
4 F (−→) 3 1 2 3/2
r , T πk r m =0) = 3(2π)3 = 6π2 2 (EF − V (r)) (1.16) n(→− N λ 8 ρ2 3/2 = 1 − (1.17)
RF3 π2 RF2
for ρ < RF and 0 else. Here the definition of the Fermi radius (1.5) has been used. The distribution of the cloud is a cigar shaped ellipsoid with length 2RF /λ and diameter 2 RF (for ωrad > ωax). The same aspect ratio λ is also obtained for a classical gas.
Measuring the spatial and momentum distribution
In the experiment, the density distribution is probed by sending a probe beam through the gas cloud and measuring the absorption. The measured quantity is the optical density
Dopt = −ln I(y, z) = σ0 dx n(x, y, z) = σ0n2D(y, z) , (1.18)
where I0(y, z) is the initial probe beam intensity, I(y, z) the intensity after passage through the cloud and σ0 the absorption cross section. This quantity is proportional to the column density n2D(y, z). Normally it is not possible to resolve the radial direction and a second integration is necessary, leaving only n1D(z) = dy n2D(y, z). These distributions take the form
3N λ y2 + λ2z2 2
n2D(y, z) = 1 − , (1.19)
48Nλ λ2z2 5/2
n1D(z) = 1 − . (1.20)
These distributions are non-gaussian, but the diﬀerence from the classical gaussian distribution is always small and diminishes with more integrations, as demonstrated in figure 1.2.
By suddenly switching oﬀ the trap and allowing an expansion of the gas, it is also possible to observe the momentum distribution of the cloud. It can be derived, for T = 0 analogous to equation (1.17) and is with formulas analogous to (1.19) and (1.20) for the integrated distributions. The distributions have the same functional form because position and momentum enter both quadratically in the Hamiltonian. However, the momentum distribution is isotropic, similar to that of a classical gas, unlike the momentum distribution of a BEC. Again the diﬀerence between the classical and the quantum degenerate solution is small.
Finite temperature distributions
To study the dependence of the position and momentum distributions on degeneracy, one must solve equation (1.7) and calculate (1.13) or (1.14) numerically. For T /TF 1 the classical gaussian distributions are obtained. As T /TF → 0+, where T /TF < 1, the distributions approach the parabolic distributions (1.17) and (1.21). In the degenerate but finite temperature regime, the center of the distribution is well approximated by the T = 0 solution, whereas the wings are better approximated by a gaussian (figure 1.3a).
To characterize the eﬀects of the Fermi pressure quantitatively we fit gaussian dis-tributions to the spatial distributions n1D(T /TF ) and trace σ2/RF2 over T /TF , where σ is the root mean square size of the gaussian fits (figure 1.3b). For a classical gas this treatment results in a straight line, with slope 0.5, which has a y-intercept of zero. Taking the correct fermionic distributions, one obtains asymptotically the same behavior in the high temperature regime. For T = 0 the curve turns upwards to inter-cept the y axis at a constant value of 0.15 instead of 0, which is a result of the Fermi pressure. Inbetween these limits one may interpolate. This has also been applied to experimental data (see figure 3.22 in section 3.3.4). In addition to the gaussian fit to the fermionic distribution for the determination of σ, a good method of measuring the temperature and the atom number must be used in order to determine T /TF and RF experimentally.
The heat capacity CF ≡ ∂E∂T N can be calculated using the expression E = d ( )g( ) (1.22) for the total energy. For the high temperature region the classical result CCl = 3N kB is obtained, whereas for low temperatures the heat capacity is smaller, CF = π 2N kB(T /TF ). This is due to the fact that at high degeneracies most atoms are al-ready at their T = 0 position. Only in a region of size kBT around EF will the atoms still change their energy levels. For a power law density of states the fraction of atoms in this region is proportional to dependence of the heat capacity cases.
T /TF , leading to the T /TF suppression in CF . The on degeneracy is an interpolation between these two
The eﬀect of the discreteness of the harmonic oscillator states
If the temperature is smaller than the level spacing and the number of atoms stays small enough, the parabolic density distribution of the degenerate Fermi gas is slightly modified and shows a modulation pattern. This comes from the population of discrete energy shells in the Fermi sea. It also shows up in the heat capacity, which is strongly modulated in dependence of the particle number .
The eﬀect of interactions on the degenerate Fermi gas
Until now we have considered a gas of noninteracting fermions. In our system this accurately models the experiment at low temperature, when all fermions are in the same internal state (see section 1.3), since then elastic collisions are suppressed. With respect to the phenomenon of Cooper pairing it is interesting to consider an interacting Fermi gas ( ,). Mediated through an attractive interaction, the fermions pair, forming bosonic quasiparticles, which may then undergo a BEC transition. This will be discussed in section (1.1.14). Here we are interested in the shape change of a degenerate Fermi gas at zero temperature due to interactions. In experiments this interaction can be due to the magnetic dipole-dipole interaction or the s-wave interaction between fermions in diﬀerent internal states. This case is particularly interesting in lithium, since the s-wave scattering length may be arbitrarily tuned using a Feshbach resonance.
We consider fermions in two diﬀerent spins states | ↑ and | ↓ . The | ↑ state r atoms, experiences a mean field potential gn↓(→−) from the interaction with the | ↓ r g is the coupling constant which is linked which have the density distribution n↓(→−). to the scattering length a↑↓ through g = 4π 2a↑↓/m. The mean field potential must be added to the external potential in equation 1.16, resulting in 1 2m 3/2 n↑ = (EF ↑ − Vext − gn↓) since the situation is symmetric. This coupled set of equations must be solved numer-ically by iteration. It can be simplified by assuming N↑ = N↓. Then EF ↑ = EF ↓ and n↑ = n↓ = n and we obtain a single equation (6π2n)2/3 + Vext + gn = EF . (1.26)
The result of this calculation is shown in figure 1.4. The shape of the integrated distribution is compared for negative (a) and positive (b) values of the interaction strength. Experimentally the interaction strength can be arbitrarily tuned using a Feshbach resonance (see section 1.3.3). A trap with frequencies ωax = 2π × 70 s−1 and ωrad = 2π ×5000 s−1 was used in the calculation and 105 atoms at T = 0 were assumed to be in each state. For negative values the mean field potential confines the atoms more strongly than in the case of an ideal gas, leading to higher densities and a more peaked distribution. If the scattering length approaches a↑↓ = −2100 a0, a very small change in a↑↓ provokes a strong change in the size. For slightly smaller values the gas becomes unstable: the strong mean field potential leads to an increase in density which in turn increases the mean field potential, without the increase in kinetic energy being able to counterbalance the collapse.
For positive scattering length the size of the cloud increases. If a↑↓ > 4000 a0 it is energetically favorable to introduce a boundary and form two distinct phases. One state forms a core surrounded by a mixed phase consisting of both states. For high enough mean field the | ↑ and | ↓ states separate completely. A pure phase of one state forms a core surrounded by a pure phase of the other state. This should be observable experimentally. The experimental signal, distributions integrated in two dimensions, is shown in figure (1.4b). At first glance it might be astonishing that the outer component in the doubly integrated distribution has a completely flat profile in the region of the central component. This can be understood analytically by calculating the doubly integrated density of a hollow sphere, for which the central part of the profile is also flat.
The same calculation has been performed for a trap with frequencies ωax = 2π × 1000 s−1, ωrad = 2π × 2000 s−1, corresponding to our crossed dipole trap. The collapse appears at a↑↓ < −1800 a0 and the phase separation occurs at about a↑↓ > 4000 a0.
Until now the calculation has been performed at zero temperature. This does not correspond to the actual experiment where typically degeneracies of T /TF = 0.2 are reached. To obtain an estimate for the uncertainty of the result, the same calculation can be performed for a classical gas by solving n(→−) = exp( − ( ext + (→−))) with = dr3 n (−→) (1.27) r C β V gn r N r .
The results are qualitatively the same but the collapse and phase separation appear at more extreme values of a↑↓. For T /TF = 1 they occur at a↑↓ < −10000 a0 and a↑↓ > 40000 a0 respectively when the other parameters are held fix. These values are extreme and would in an experiment probably be accompanied by strong losses.
The Pauli exclusion principle leads to a suppression of scattering of light or particles from a degenerate Fermi gas. Consider a scattering process that produces a fermion with less than the Fermi energy. For a zero temperature Fermi gas, all states with this energy are occupied. Thus the scattering process is forbidden by the quantum statistics. This eﬀect is called Pauli blocking [66, 65]. The degree of suppression of collisions depends on the degeneracy of the Fermi gas and on the energy of the incoming particle or photon, called the test particle in the following. If the energy of the test particle is much greater than the Fermi energy, also the majority of scattered fermions will have an energy above EF , where the occupation of states is low. In this case no modification of the scattering probability is observed, independent of the degeneracy of the Fermi gas. If the energy of the test particle is below the Fermi energy only atoms at the outer edge of the Fermi sphere can participate in the scattering, since these atoms can obtain an energy higher than EF during the scattering process. For a test particle with zero energy, the scattering rate decreases in proportion to T 3 when compared to the classically expected rate (see figure 1.10a). This eﬀect can be used to measure the degeneracy of the Fermi gas. It is detailed by G. Ferrari in , employing a static integration of the Boltzmann collision integral in phase-space. In the experiment the rethermalization of a cold test cloud of impurity atoms in the Fermi gas is observed (see section 3.3.5). This is a dynamic process that has been simulated by solving the coupled energetic Boltzmann equations of the two clouds, as detailed in the section on evaporative cooling (1.2.3). The inhibition of elastic scattering also becomes important at the end of sympathetic cooling as it slows down the cooling process. This is especially true if both components participating in sympathetic cooling are fermionic as is the case in .
To observe the eﬀect of Pauli blocking on scattered light, a two-level cycling tran-sition must be used. After the absorption process, the atom undergoes spontaneous emission to its initial state. Atoms in the Fermi sea are in the same state and thus Pauli blocking can occur. Pauli blocking can increase the lifetime of the excited state and thus lead to a narrowing of the linewidth. This eﬀect is analogous to the enhancement of the excited state lifetime in cavity experiments , but the enhancement comes from the reduction of atom, not photon, final states. A second eﬀect is the reduction of the scattering rate. Both eﬀects depend on the recoil transmitted to the atom during the scattering process. Since the occupation in a degenerate Fermi gas is highest for low energies, scattering at small angles, corresponding to small momentum transfer, is altered most severely. These eﬀects also depend also strongly on the degeneracy of the Fermi gas. Several articles have been published on this subject: see [67, 68, 69] and references therein.
Table of contents :
1.1 Theory of quantumgases
1.1.1 Properties of a classical gas
1.1.2 Properties of a fermionic gas
1.1.3 The effect of interactions on the degenerate Fermi gas
1.1.4 Pauli block ing
1.1.5 Detection of a degenerate Fermi gas
1.1.6 The degenerate Bose gas
1.1.7 The effect of interactions and the Gross-Pitaevskii equation
1.1.8 BEC with attractive interactions
1.1.9 One-dimensional degenerate gases
1.1.10 The 1D condensate
1.1.11 The bright soliton
1.1.12 The effects of Bose statistics on the thermal cloud
1.1.14 The BCS transition
1.2 Evaporative cooling
1.2.1 Sympathetic cooling
1.2.2 The Limits of sympathetic cooling
1.2.3 The Boltzmann equation
1.2.4 Simulation of rethermalization including Pauli blocking
1.2.5 Simulation of sympathetic cooling
1.3.1 Energy dependence of the cross section
1.3.2 Mean field potential
1.3.3 Resonance enhanced scattering
2 The experimental setup
2.1 Overview of the experiment
2.2 Properties of lithium
2.2.1 Basic properties
2.2.2 Confinement in amagnetic trap
2.2.3 Elastic scattering cross sections
2.3 Other strategies to a degenerate Fermi gas
2.4 The vacuumsystem
2.4.1 The atomic beamsource
2.4.2 Oven bak e-out procedure
2.4.3 Themain chamber
2.5 The Zeeman slower
2.6 The laser system
2.7 Themagnetic trap
2.7.1 Theory ofmagnetic trapping
2.7.2 Design parameters of themagnetic trap
2.7.3 Realization of themagnetic trap
2.8 The optical dipole trap
2.8.1 Principle of an optical dipole trap
2.8.2 Trapping of an alkali atom in a dipole trap
2.8.3 Setup of the optical trap
2.9 The radio frequency system
2.10 Detection of the atoms
2.10.1 Principle of absorption imaging
2.10.2 Imaging optics
2.10.3 Probe beampreparation
2.10.4 Two isotope imaging
2.10.5 Detection parameters
2.10.6 Using absorption images
2.11 Experiment control and data acquisition
3 Experimental results
3.1 On the road to evaporative cooling
3.1.1 The two isotopeMOT
3.1.2 Optical pumping
3.1.3 Transfer and capture in the Ioffe trap
3.1.4 First trials of evaporative cooling
3.1.5 Doppler cooling in the Ioffe trap
3.2.1 Measurement of the trap oscillation frequencies
3.2.3 Measurements of other parameters
3.3 Experiments in the higher HF states
3.3.1 Evaporative cooling of 7Li
3.3.2 Sympathetic cooling of 6Li by 7Li
3.3.3 Detection of Fermi degeneracy
3.3.4 Detection of Fermi pressure
3.4 Experiments in the lower HF states
3.4.1 State transfer
3.4.2 Evaporative cooling of 7Li
3.4.3 Sympathetic cooling of 7Li by 6Li
3.4.4 A stable lithiumcondensate
3.4.5 The Fermi sea
3.5 Loss rates
3.6 Experiments with the optical trap
3.6.1 Adiabatic transfer
3.6.2 Detection of the 7Li |F = 1,mF = 1 Feshbach resonance
3.6.3 Condensation in the 7Li |F = 1,mF = 1 state
3.6.4 A condensate with tunable scattering length
3.6.5 The bright soliton
Conclusion and outlook