In the previous section, we have introduced the relative potential U(r) without putting constraints on it. Here we will filter out a certain class of potentials.
In the framework of ultracold atomic gases, we are interested in collisions be-tween two neutral atoms. This means that the considered interaction is an induced dipole-dipole interaction and is described by a van der Waals type potential, which is attractive at the long-range and has a hard-core at r =b. For more detailed information about the atomic potentials we refer the reader to the following ref-erences [48, 145–149]. At intermediate-long distances this interaction is described by a −C6/R 6 potential, which has a length scale given by the C 6 parameter: the van der Waals length RvdW = 1 2µC6 .
This length scale has an interesting physical interpretation . At the length scale r ∼RvdW, the potential energy becomes comparable to the collision energy E ∼U(RvdW). This means that the eﬀect of the potential outside of this range vanishes and the wavefunction approximates the free space wavefunction. In other words, the van der Waals length indicates a range over which the potential mod-ifies the behavior of the wavefunction. In the following, we will only discuss the wavefunction outside the range of the interaction and condense the eﬀect of the potential into a boundary condition at r = 0.
In the above, we have written the solutions of the Schrödinger Equation in the basis of the spherical harmonics. The next step is to describe the eﬀect of the scattering potential on each of the spherical waves. We will use the fact that the potential has a finite range RvdW and start by considering an incoming wave outside of the range of the potential r ≫RvdW. Outside the range of the potential, the incoming wave is described by the r ≫RvdW solution. We will consider a virtual propagation of the wave to the scattering center and as soon as the wave starts to feel the potential, it is deformed and finally reflected within the range of the potential. The potential is norm conserving (elastic scattering), so when the wave is coming back out of the range of the potential it will again be the r ≫RvdW solution, but with an acquired phase δl due to the potential U(r) in the r < RvdW region. Let us write this in terms of the wavefunction 2k 2 Ylm(θ,φ) 1 e+i(kr−l π +2δl ) − e−i(kr−l π which at far distance will behave as 1/r sin(kr −lπ/2+δl). If we write this in terms of the incoming and outgoing waves, the wavefunction takes the form ψlm r→∞≃ slϕlm(out) − ϕlm(in), (1.10) where we have introduced sl ≡ Aout/Ain = e2iδl . This sl described the scattering of a wave in channel l due to the potential U(r). This is a trivial case of the scattering matrix diagonal (corresponding to one channel), but as we will see in Chapter 2, it becomes useful when several channels are coupled.
As a final step, let us separate the outgoing wave with no interaction from the part with the phase factor slϕlm(out) = ϕlm(out) + (sl − 1)ϕlm(out), (1.11) where (sl −1) = 2ie iδl sinδl. When the result is written into the form of the wave-function, it is given by ψlm r →∞≃ ϕlm(out) − ϕlm(in) + (sl − 1)ϕlm(out) r ≃ ϕlm + 2ieiδl sinδl ϕlm(out). (1.12) →∞
This is the result of scattering in a specific channel l. To summarize the elastic scattering let us note that the scattering potential fixes a boundary condition on the long-range result.
In the next section we will apply this on the initial condition of the problem: the plane wave.
The spherical waves are practical to introduce the eﬀect of the scattering into the wavefunction, however, the initial condition is the incoming plane wave ψ(0). In order to use the spherical waves, let us project the plane wave onto the basis of spherical waves. The projection is given by ∞ ∞ ψ(0) = e+ikz = il 4π(2l + 1) jl(kr)Yl 0(θ,φ) = cl ϕl0(r). (1.13) l=0 l=0
The fact that there are only terms with m = 0 in the projection of the plane wave, shows the cylindrical symmetry of the scattering. Here we have made, without loss of generality, the choice of having an incoming plane wave traveling along the z-axis towards the center, which means that θ is defined as the angle between r and z.
To implement what we have derived in Section 1.1.5, let us do the replacement ϕl0 → ψl0 and write the resulting scattering wavefunction ψ, ψ = clψl0(r) = clϕl0 + cl 2i eiδl sinδl ϕl(out)0 l=0 l=0 l=0 = ψ(0) + fk(θ) eikr (1.14)
Here fk(θ) is the scattering amplitude given by 1 ∞ 0(θ)eiδl sinδl. (1.15) fk(θ) ≡ 4π(2l + 1)Yl k l=0
The scattering amplitude fk(θ) is a measure for the strength of the scattering. For a more detailed description and higher-order corrections we refer the reader to [145, 151].
The two parts in Equation (1.14) are the incoming wave ψ(0) =e+ikz and the outgoing scattered wave ψsc =fk(θ) eikr /r. This result is the well-known two-particle scattering wavefunction in the long-range. In Chapter 2 we will use a similar method to described the scattering of three particles.
Scattering Cross Section
In order to characterize the scattering, an often used property is the scattering cross section. The total scattering cross section is defined by the amount of initial plane wave ψ(0) scattered into the scattered wave ψsc. In order to calculate the total scattering cross section, we have to consider the diﬀerential cross section. This is given by the ratio of the current density of the scattered wave through a surface element on a sphere and the current density of the incoming wave (see Figure 1.3).
As a final step, we can use the result from Equation (1.15) and plug it into Equation (1.19). The spherical functions Yl0(θ) are orthonormal and satisfy Yl0(π− θ) = (−1)lYl0(θ). This allows us to write the total scattering cross section in the following forms: for identical bosons : σk = 8π (2l + 1) sin2 δl(k), even for identical fermions : σk = 8π (2l + 1) sin2 δl(k), (1.20) l odd for distinguishable particles : σk = 4π (2l + 1) sin2 δl(k).
The Unitary Limit
The Equations (1.20) are bounded by the maximum value of sin2 δl(k), which is 1.
The scattering cross section for a specific partial wave (σk = l σl) is bounded by For identical bosons : σl ≤ 8π (2l + 1) k2
For identical fermions : σl ≤ 8π (2l + 1) (1.21) k2 For distinguishable particles : σl ≤ 4π (2l + 1). k2 These inequalities give the maximum value the scattering cross section can attain. The limit in which the maximum possible cross section is obtained, is the so called Unitary Limit.
Low-Temperature Limit: Bosons versus Fermions
Let us reconsider Figure 1.2 in the limit of low energy. In the figure, the eﬀective potentials for l = 0, l = 1 and l = 2 are shown. The purple line shows the energy of an incoming wave ψ(0), given by 2k2/(2µ). For l >0, we can associate a length scale rl to the rotational barrier by comparing the energy of the incoming wave to the rotational energy. Following from the Schrödinger Equation, we find krl = l(l + 1).
The point r =rl corresponds to the classical turning point, where the kinetic energy is 0 and we neglect the interaction potential U(r) (which is true for rl ≫RvdW). Comparing the two length scales gives RvdW ≪ rl = l(l + 1) . (1.23)
The first inequality in this equation defines the zero-temperature limit. Following, this limit can be expressed as kth →0. So if this limit is reached, the waves in the l >0 channels will avoid the scattering center. This means that for l >0, the phase shift δl vanishes i.e. limk→0 δl = 0.
When considering bosons, the scattering cross section can, in the limit of cold- collisions, be written as 8π 2 (1.24) σk k≃→0 sin δ0 (k). k2
From here on, we will, unless otherwise specified, assume s-wave scattering of identical bosons.
In the previous section, we have seen that the limit of cold-collisions greatly simpli-fies the problem, because we only need to consider s-wave scattering. This same limit allows us to further simplify the problem. In order to do so, let us define a which is the zero-temperature limit of fk , a ≡ −klim0 fk = −klim0 δ0(k) (1.25)
The length scale that is defined here is the scattering length a, which is a measure of the strength of the scattering. When this result is plugged into Equation (1.24), the zero-energy limit of the collisional cross section is found σk k ≃0 8πa2. (1.26)
To physically understand the scattering length, we will consider the long-range behavior of the wavefunction (see Equation (1.5) for the definition of uk0(r)) and extend that to the short range region: vk0 (r) ≡ rlim uk0(r) →∞ (1.27) = C sin (kr + δ0(k)).
In Figure 1.4, the long-range behavior extended to the short-range is depicted. The red dot in the figure corresponds to the point r =rP ≡ − δ0/k. In the limit of cold collisions k →0 this point corresponds to the scattering length a (see Equa-tion (1.25)).
We can also define a in a diﬀerent manner, which, in the limit of cold-collisions, is equivalent to the above definition. In order to do so let us look at the expansion of vk0(r) (Equation (1.27)) in terms of k up to first order in k vk0(r) ≃ kr cosδ0 + sinδ0.
This definition shows that δ0(k) needs to be defined up to a factor of π and it is suﬃcient to consider δ0(k) to be between −π/2 and π/2. This statement can be graphically interpreted by considering Figure 1.4 and imagining that π was added to δ0. This would shift the wavefunction and change the slope direction of the first intersection, but it would not change the position of the intersection.
Using the previous results, we are in a position to write down the wavefunction vk0. Here we have chosen the normalization vk0(0) =C.
This result includes all the information about the short-range in a single condition close to zero. In the rest of this thesis, we will be interested in the long-range behavior of the two-particle system and use the boundary condition to include the short-range physics.
Feshbach Resonances – Tuning the Scattering Length
In this section, we will consider a simple model to understand the properties and the physical origin of Feshbach resonances. For a more elaborated discussion, we refer the reader to a review on Feshbach resonances .
In Figure 1.5a we have plotted two molecular potentials: Vbg(r) (the entrance or open channel) and Vc(r) (the closed channel). Let us consider to be in the limit of ultracold collisions, then the collisional energy E ≈0. When there is a bound state in the closed channel, with energy Ec, that is close to the asymptotic potential energy in the open channel (by definition E = 0) the scattering in the open channel is modified. Due to the diﬀerent magnetic moments of the two channels, the relative oﬀset of the closed channel can be modified by changing the magnetic field, which will change the value of Ec. Around the point Ec = 0 the scattering length can be tuned from −0 to +0, through ±∞. This resonant behavior is called a Feshbach-resonance and is described by the following equation .
Table of contents :
1. Two-Particle Problem
1.1.1. Center of Mass (CoM) Motion
1.1.2. Radial Schrödinger Equation
1.1.3. Scattering Potential
1.1.4. Spherical Waves
1.1.5. Elastic Scattering
1.1.6. Scattering Amplitude
1.1.7. Scattering Cross Section
1.1.8. The Unitary Limit
1.1.9. Low-Temperature Limit: Bosons versus Fermions
1.1.10. Scattering Length
1.2. Feshbach Resonances – Tuning the Scattering Length
1.2.1. Two-Channel Model
1.2.2. Determining the Position and Width
2. Three-Particle Scattering
2.1. Elastic Scattering
2.1.1. Three-Particle Hamiltonian
2.1.2. Hyperangular Problem
2.1.3. Scattering Regimes
2.2. Unitary Interactions – Efimov’s Ansatz
2.2.1. Hyperspherical Waves
2.2.2. Short-Distance Scattering – R<Rm
220.127.116.11. Elastic Scattering
18.104.22.168. Efimov Bound States
22.214.171.124. Zero-Range Model
2.2.3. Long-Distance Scattering
126.96.36.199. Long-Range Wavefunction
188.8.131.52. Coupling of the Long-Range to the Short-Range
2.3. Finite-a – Hyperspherical Channels
2.3.1. Long-Distance Scattering
184.108.40.206. Long-Range Wavefunction
220.127.116.11. Coupling of the Long-Range to the Short-Range
18.104.22.168. Effective Two-Channel System
2.4. Inelastic Three-Particle Processus
2.4.1. Elastic versus Inelastic Scattering
22.214.171.124. Elastic → Inelastic Scattering
126.96.36.199. Inelastic Zero-Range Model (ZRM)
188.8.131.52. Resonant Interactions: Efimov Physics
184.108.40.206. Finite Interactions
2.4.4. Flux and Recombination
2.4.5. Temperature Average
2.4.6. Optical Resonator Analogy
2.4.7. Oscillations of L3(T) at Unitarity
2.4.8. Numerical Analysis of L3(T, a)
2.5. Three-Particle Losses on the Positive-a Side
2.5.1. Weakly Bound Dimer
2.5.2. Weakly Bound Dimers and the Efimov Channel
2.5.3. Atom-Dimer Decay with Chemical Equilibrium
3. Three-Particle Recombination in a Harmonic Trap
3.1. Three-Particle Losses in a Trap
3.1.1. Trapping Potential
3.1.2. Weakly and Deeply Bound Dimers in a Trap
3.1.3. Number Decay
3.2. Heating Effects
3.2.1. Weakly-Interaction Limit
3.2.2. Extending the Model to Include Strong Interactions
3.3.1. A Simple Evaporation Model
3.3.2. More Advanced Model of Evaporation Effects
3.3.3. “Magic” η
4. The Road to Strongly Interacting Bose Gases
4.1. Experimental sequence
4.1.2. Laser System
4.1.3. Zeeman slower
4.1.4. Magneto-Optical Trap (MOT)
4.1.5. Optical Pumping
4.1.6. Magnetic Trapping and Evaporation
4.1.7. Optical Dipole Trap (ODT)
4.1.8. Radio-Frequency (RF) Transitions
4.2. Feshbach Resonance in 7Li
5. Lifetime of the Resonant Bose Gas
5.1. Recombination Rate Measurements and Assumptions
5.1.1. Quasi-Thermal Equilibrium Condition
5.1.2. Separation of Time Scales
5.1.3. Starting Point for the Measurements
5.1.4. Number Calibration
220.127.116.11. Pressure calibration
18.104.22.168. Recombination and Temperature calibration
5.1.5. Constant Temperature
5.1.6. Data Analysis
5.2. Results – Unitary Interactions
5.2.1. Temperature Dependence of L3 at Unitarity
5.2.2. Reanalysis using the Advanced Evaporation Model
5.3. Results – Finite Interactions
5.3.1. Saturation of L3 for Resonant Interactions
5.3.2. Comparison with Previous Data – 133Cs
22.214.171.124. The First Efimov Resonance
126.96.36.199. Resonance Position
5.3.3. Temperature Behavior of L3 – 39K
188.8.131.52. Validating the 1/T 2 Law for L3(T)
184.108.40.206. Excess Heat Measurements
Appendix A. Technical Details – Theory
A.1. Jacobian and Hyperspherical Coordinates
A.1.1. Jacobian Coordinates
A.1.2. Jacobian → Hyperspherical Coordinates
A.1.3. Jacobian → Hyperspherical Hamiltonian
A.1.4. Hyperradial and Hyperangular Schrödinger Equations
A.2. Incoming and Outgoing waves
A.3. Saddle Point Method
A.4. Efimov’s Ansatz
A.5. The s-matrix at Unitarity
Appendix B. Peer-reviewed papers
B.1. Dynamics and Thermodynamics of the Low-Temperature Stronglyn Interacting Bose Gas
B.2. Lifetime of the Bose Gas with Resonant Interactions
B.3. -enhanced sub-Doppler cooling of lithium atoms in D1 gray molasses
Appendix C. Data Loss Measurements
C.1. Unitary Interactions
C.2. Finite-a Interactions
Appendix D. Efimov resonances
D.1.1. Universality of the Efimov resonances
D.2.1. L3 vs. a
D.4.1. Efimov resonance
D.4.2. Universality of the Efimov resonances in 39K
D.5. How to determine the Efimov parameters