Zero-Range Three-Body Physics in Ultracold Bosonic Gases Master’s thesis in Physics KIRILL DANILOV DEPARTMENT OF PHYSICS CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2022 www.chalmers.se www.chalmers.se Master’s thesis 2022 Zero-Range Three-Body Physics in Ultracold Bosonic Gases KIRILL DANILOV Department of Physics Chalmers University of Technology Gothenburg, Sweden 2022 Zero-Range Three-Body Physics in Ultracold Bosonic Gases KIRILL DANILOV © KIRILL DANILOV, 2022. Supervisor: Johannes Hoffmann, University of Gothenburg, Department of Physics Examiner: Henrik Johannesson, University of Gothenburg, Department of Physics Master’s Thesis 2022 Department of Physics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Three body state binding energy spectrum for identical bosons. Typeset in LATEX, template by Kyriaki Antoniadou-Plytaria Printed by Chalmers Reproservice Gothenburg, Sweden 2022 iv Zero-Range Three-Body Physics in Ultracold Bosonic Gases Kirill Danilov Department of Physics Chalmers University of Technology Abstract Due to their universality, effective field theories developed in the study of nuclear physics have found a direct application in the research of many-body physics in quantum degenerate gases such as Bose-Einstein condensates. This thesis will re- produce from one such theory, the zero-range model, the discrete scaling symmetries emergent for the physical observables of three-body physics in bosonic gases within the ultracold temperature regime. The conditions of the ultracold regime allow for the treatment of bosonic gas atoms as quantum-mechanical point particles inter- acting exclusively through scattering in the s-wave channel. The Feynman rules of the zero-range model are derived and implemented to formulate the three-body scattering amplitude and the three-body bound state equation using the inclusion of an auxiliary diatomic field operator. The zero-range model has been shown to be renormalisable within the three-body sector utilizing a renormalisation group limit cycle through numerical tests of an ultraviolet cutoff momentum dependent three-body scaling term G(Λ) parametrised by a three-body parameter Λ∗. The universal properties of the Efimov effect have been confirmed for the three-body physics of a single species of bosonic gas, as well as for mixtures of species distinguished by different masses. Furthermore, results for the three-body scattering amplitude demonstrate the impact of different parameters Λ∗ and two-body scattering lengths on the three-body scattering length of the quantum degenerate gas. In conclusion, the zero-range model enables the description of few-body physics of bosonic quantum degenerate gases with respect to a finite set of parameters: the masses of gas species, the spin-statistic of gas species, the two-body scattering length and the three-body parameter, and the temperature of the gas. Keywords: Bose-Einstein gas, Efimov effect, ultracold, universality, zero-range model. v Acknowledgements I would like to acknowledge my supervisor, Professor Johannes Hofmann, for his lengthly and thorough coordination of my work, and my family for their invaluable support to the fulfillment of my education. Kirill Danilov, Gothenburg, December 2021 vii Nomenclature Below is the nomenclature of indices, constants, parameters, variables, operators and Feynman diagrams that have been used throughout this thesis. Indices i, j, n, n′, N Generic indices l Quantum mechanical angular momentum number s, I Quantum mechanical spin, nuclear spin state number f, i Final and initial states Constants kB Boltzmann’s constant ℏ Reduced Planck’s constant λ0 Bare two-body coupling constant ZΨ Dimer field renormalisation constant γ0 Bare three-body coupling constant Parameters m, mr Mass, reduced mass T Gas temperature λT Thermal de Broglie wavelength n Gas particle density lvdW, rs Van der Waals length, effective range a, a2, aAD Two-body scattering length, bare two-body scattering length, atom- dimer scattering length ix Λ∗, κ∗ Three-body parameter λ Discrete scaling symmetry parameter rm Mass ratio Λ Ultraviolet cutoff momentum Variables k, qi, pi Wavevector, momentum of quantum-mechanical point particle i ω, ωT , Etype Frequency, energy of channel T , generic energy by context δl Phase shift at angular momentum number l ri Position of QM particle i ti Time of QM particle i ψscatter, |ψ+⟩ Scattering wavefunction, outgoing scattering wavefunction state fl, Al Partial wave amplitude, radial scattering wavefunction C Modified three-body scattering amplitude function G(Λ) Three-body scaling term at ultraviolet cutoff Λ Operators H, H Hamiltonian, Hamiltonian density V Interaction potential U(t, t0) Quantum-mechanical unitary time-evolution operator S, T Matrices for quantum-mechanical state evolution ψ, ψ† Atomic field annihilation/creation operators Ψ,Ψ† Dimer field annihilation/creation operators T{} Time-ordering operator Feynman Diagrams FN fields Generic Feynman diagram at N loop level, the field operators in- volved listed in subscript An, An,l=0, A+ n-body scattering amplitude, n-body scattering amplitude pro- jected on total angular momentum l = 0, high-momenta contri- butions to the scattering amplitude Dfield Propagator for a quantum field operator x Contents Nomenclature ix 1 Introduction 1 2 Background 3 2.1 Degenerate Quantum Gases: Macroscopic Level . . . . . . . . . . . . 3 2.2 Energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Two-body Scattering: Microscopic Level . . . . . . . . . . . . . . . . 6 2.4 Low-energy limit for two-body scattering . . . . . . . . . . . . . . . . 9 2.5 Universality and the Efimov Effect . . . . . . . . . . . . . . . . . . . 11 3 Zero-Range Model: Effective Quantum Field Theory 13 3.1 Two-Body Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 The S-matrix in the Interaction Picture . . . . . . . . . . . . 14 3.1.2 The Scattering Amplitude: Green’s Function and Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.3 Renormalisation of the Two-Body Coupling Coefficient . . . . 20 3.2 Three-Body Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.1 The Exact Dimer Propagator . . . . . . . . . . . . . . . . . . 22 3.2.2 The Atom-Dimer Scattering Amplitude . . . . . . . . . . . . . 23 3.2.3 Renormalisation of Three-Body Term G(Λ) . . . . . . . . . . 25 3.3 Observables of the Three-Body System . . . . . . . . . . . . . . . . . 27 4 Results 29 4.1 Results for Scattering Amplitudes of Identical Bosons in the Zero- Range Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.1 Renormalisation of the Three-Body Scattering Amplitude . . . 29 4.1.2 Results for the Atom-Dimer Scattering Length . . . . . . . . . 31 4.2 Results for Three-Body Bound State Energy in the Zero-Range Model 31 4.2.1 Results for Bound State Energy of Identical Bosons . . . . . . 31 4.2.2 Results for Bound State Energy of Distinct Bosons . . . . . . 38 5 Conclusion 41 Bibliography 43 A Appendix: Feynman Rules I xi Contents A.1 Zero-range model, two-body sector . . . . . . . . . . . . . . . . . . . I A.2 Zero-range model, three-body sector . . . . . . . . . . . . . . . . . . . II B Appendix: Numerical Methods III B.1 Gauss-Legendre Quadrature . . . . . . . . . . . . . . . . . . . . . . . III B.2 Methodology for Trimer Binding Energies . . . . . . . . . . . . . . . VI B.3 Methodology for Scattering Amplitudes . . . . . . . . . . . . . . . . . VI B.3.1 Methodology for Three-Body Renormalisation . . . . . . . . . VIII xii 1 Introduction In theoretical physics, it is typically challenging to formulate a detailed and accurate understanding of any complex system without applying some sensible intuition to the problem. Non-relativistic effective field theories, utilized frequently in the study of strongly-interacting particles such as the constituent subatomic particles of nuclei, resort to the separation of energy scales in a system in order to reduce the amount of unnecessary detail of the problem. Through understanding that the relevant phenomena for the desired observables at a given energy scale will not be affected by conditions that occur as a result of higher-energy interactions, it is possible to greatly simplify the theory modelling a physical interaction down to the dependence on a handful of manipulable or observable parameters, as long as the low-energy conditions are met. A prediction made in 1971 by Vitaly Efimov stated that subatomic particles could, if their scattering was of sufficiently low kinetic energies, form weakly bound three- body states [1]. This prediction would start the development of effective field theories in order to understand the conditions under which these bound three-body states could form and what the recombination rates into other configurations would be [2]. Despite the steady progress of the mathematical science applied to these theories, in particular the introduction of renormalisation group transformations by Wilson [3], there was little fortune in providing the experimental results to confirm or reject Efimov’s theories within the scope of nuclear physics itself. In parallel to these developments, the science of ultra-cold states and materials made some of its most significant breakthroughs when the Bose-Einstein condensate was created for the first time in laboratory conditions in 1995, and when its half- integer counterpart, the degenerate Fermi gas was first contained in optical trapping in 2001 [4]. The study of these quantum degenerate gases utilized the possibility to trap ultracold with optical tweezer technology and to tune the strength of the two-body interactions between gas atoms with the use of magnetic fields, allowing experimentalists to create strongly interacting gases. Studies on degenerate quantum gases have included, for example, 40K, 6Li, 7Li, 87Rb and 23Na [5], [6]. More recently Bose-Einstein condensates have also been realised for 39K and metastable 4He, and dipolar degenerate Fermi gas was realised from 167Er [7], [8], [9]. Although different from nuclear particles, in 2005 quantum degenerate gas atoms were found to be a valid experimental proving ground for the study of many-body physics [10], and the predictions of Efimov physics due to the universal properties of the effective field theories that governed them both in the ultracold regime [11]. This thesis documents a work that aims to detail the effective field theories developed from nuclear physics and apply these in the study of strongly interacting quantum 1 1. Introduction degenerate gases in the ultracold temperature regime. In chapter 2, we cover the physical arguments that motivate the implementation of an effective field theory derived from nuclear physics as a model for quantum degenerate gases. In chapter 3 we will derive the effective field theory, known as the zero-range model, used in conjunction with numerical methods discussed in the appendix B.1 to demonstrate the physical observables of three-body physics that are then presented in chapter 4 along with a discussion of how these three-body observables may be reflected in the experimental studies of quantum degenerate gases. The thesis is closed by concluding remarks in chapter 5. 2 2 Background In this chapter, we introduce the foundation for the effective field theories used for describing degenerate quantum gases in the ultra-cold regime. Section 2.1 covers the distinction between degenerate quantum gases and classical gases at ultracold temperatures, i.e. below 1 µK. Section 2.2 discusses the various scales of energy inherent to the interactions of atoms in a gas, defines the natural energy scale and establishes that in the degenerate limit the atoms can be treated as quantum mechanical point particles. Sections 2.3 and 2.4 discuss quantum-mechanical two- body scattering, introduce the s-wave scattering length and demonstrate that in the ultra-cold regime, scattering is dominated by the s-wave channel. Section 2.5 describes universal properties of degenerate quantum gases that are expected to arise as the result of the Efimov effect. 2.1 Degenerate Quantum Gases: Macroscopic Level Atoms in the gas phase exhibit particular behaviour as the quantum-physical realm is reached. At high temperatures, dilute gases are modelled as point-like particles that follows the Maxwell-Boltzmann distribution, for particles with momentum p, f(p) = 1 (2πmkBT ) 3 2 e − p·p 2mkBT , (2.1) where m is the mass of the gas particles, kB is Boltzmann’s constant and T is the gas temperature. To procure the quantum-mechanical limit, we set the wavevector k = p 2πℏ and renormalise equation 2.1: (mkBT2πℏ2 ) 3 2 ∫ dk3e − 4π2ℏ2k·k 2mkBT = 1 Reading off the forefactor gives a characteristic length scale, the thermal de Broglie wavelength: λT = √ 2πℏ2 mkBT . (2.2) The thermal de Broglie wavelength can be thought of as the average size of the atoms as quantum-mechanical wave packets. When it is comparable to the other length scales, such as the interparticle separation, it is not possible to disregard the emergent quantum physics. In order to make an estimate of the temperature at which the quantum effects become relevant, we make the following comparison 3 2. Background Temperature (K) 273 1 1 × 10−3 1 × 10−6 λT (µm) 1.1 × 10−5 1.9 × 10−4 5.9 × 10−3 1.9 × 10−1 Table 2.1: Thermal de Broglie wavelength at 300 K, 1 K, 1 mK and 1 µK. for 87Rb, with mass m = 1.44 × 10−25 kg. between the values for the wavelength λT of 87Rb at temperatures ranging from temperature at 300 K to near absolute 0 at 10−1 K. For any dilute gas the volume per particle exceeds the volume of tha gas, therefore the interatomic separation n− 1 3 is much greater than the radii of the atoms. An assessment of the interatomic separation of degenerate quantum gases can be made from the example of a 87Rb Bose-Einstein condensate, which was observed to have density n = 2.5 × 1018 m−3 [12]. Therefore: n− 1 3 ≈ 0.7 × 10−6 m = 0.7 µm. (2.3) Comparing with table (2.1), at T = 300 K the thermal de Broglie wavelength is much smaller than the interatomic separation, so there are no discernible quan- tum mechanical effects and atoms may be regarded as classical particles. If the temperature is lowered, the thermal de Broglie wavelength increases and quantum mechanical effects become significant to the statistical properties of the gas. At temperatures below T = 10−6 K, the statistics of the atoms are not the classical Maxwell-Boltzmann distribution, but instead either the Bose-Einstein or the Fermi- Dirac distributions, fBE(E) = 1 e E kBT − 1 , fFD(E) = 1 e E kBT + 1 . The macroscopic properties of the degenerate quantum gas and the nature of its degenerate phases are depend on the Bose - or Fermi statistics of the atoms. The spin statistic is dependent on a particle’s nuclear composition and electron number, where each fermion contributes 1 2 spin. Bosonic isotopes will have full integer spin while fermionic isotopes will have half-integer spin. 2.2 Energy scales The internal energies of atoms includes excitation energies of the electron into greater orbitals, the interaction between electron, the nuclear level splitting known as the hyperfine energy, and the magnetic-spin interaction of the electron in the presence of a magnetic field. For many-body systems, the relevant energies are the kinetic energy of the gas and the interaction energy between atoms, which consists of van der Waals forces and the short-range contact potentials. To demonstrate the energy scale of electronic excitation, we use the alkali metals as an example. The first electronic excitation of alkali atoms scales as ERyd/N 2 where ERyd = 13.6 eV is the excitation of hydrogen. For the lowest value such as 13.6 62 eV in the case of 133Cs, the corresponding temperature scale is Telec ≈ 103 K. 4 2. Background The hyperfine splitting energy EI = 2 2I+1Ehf and magnetic energy Em = µB = 2µBB contribute to the internal energy of the atom due to the interaction of the nuclear spin , with quantum number I, with the electron spin s, and the interaction of the electron spin with any applied magnetic field. The temperature scales for experiments at a magnetic field of 2×10−2 T are Thf ≈ 10−1 K and Tm ≈ 10−2 K [6]. The potential energy of the many-body system is determined by interactions of the gas atoms through short-range contact potentials Vshort with long-range van der Waals tails. The van der Waals contribution is caused by the polarizability and motion of electron clouds in the atoms [13]. At distances larger than the radius of the atoms, the van der Waals long-range tail has the form VvdW(r) ∼ −C6 r6 . V (r) = Vshort(r) − C6 r6 . (2.4) The van der Waals length lvdW follows from dimensional analysis, where [E] is the dimension of energy and [L] is the dimension of length: [C6] = [E][L]6, [ℏ 2 m ] = [E][L]2, [mC6 ℏ2 ] = [L]4, Hence, lvdW := (mC6 ℏ2 ) 1 4 . (2.5) The corresponding energy scale is EvdW = ℏ2 ml2vdW = ℏ3 √ m3C6 . (2.6) The van der Waals energy is the natural interaction energy scale of the system. Using results for 87Rb atom [14]: EvdW[87Rb] = 6.4 × 10−9 eV, lvdW[87Rb] = 8.6 × 10−9 m. In terms of temperature, the scale is TvdW ≈ 10−4 K. The particle density of the gas sets an additional characteristic scale, the Fermi energy EF. For transitioning to the ultra-cold phase necessary to obtain a degenerate quantum gas, the thermal energy kBT = 2π2ℏ2 mλ2 T must be similar to the Fermi energy, so that the thermal wavelength λT will exceed the interatomic separation of the atoms n− 1 3 : EF = ℏ2(3π) 2 3n 2 3 2m ∼ kBT . (2.7) In a condensate of 87Rb the Fermi energy in terms of temperature is TF ≈ 10−8 K. The energy scales of the atomic structure ∆E1, EI and Em exceed the energy scales of the many-body system EvdW and EF. The various energy scales are summarised in table 2.2. Experiments typically contain and manipulate the gas using lasers for optical trap- ping and cooling of the sample. As the first order approximation of the restoring force in optical trapping is linear, the potential of such optical wells is modelled as harmonic. If the gas is subject to confinement by optical trapping, the lowest scale 5 2. Background ∆E1 First electronic excitation ERyd/N 2 103 K EI Hyperfine splitting 2 2I+1Ehf 10−1 K Em Magnetic energy B ∼ 10−2 T, µB 10−2 K EvdW van der Waals energy ℏ3√ m3C6 10−4 K EF Fermi energy ℏ2(3π) 2 3 n 2 3 2m 10−8 K Ewell Confining optical well ℏωwell 10−10 K Table 2.2: Energy scales relevant to a quantum gas in descending order. of energy are the energy differences of the harmonic potential. The energy levels of a harmonic potential are spaced as ℏωwell where a typical frequency is ωwell = 2π10 Hz [6]. The corresponding temperature scale is Twell ≈ 10−10 K. When the gas is at a temperature where kBT is comparable or below EvdW, any atomic interactions will be unable to cause state transitions of the atoms between spin, hyperfine or electronic levels. At even larger temperatures below the Fermi energy, the gas enters the phase of a degenerate quantum gas. Therefore, the gas is modeled as a large collection of massive quantum-mechanical point particles occu- pying states of constant spin configurations I and s, interacting through two-body potentials given in equation (2.4). Single component gases will see all particles in- teract with each other equally, while gases consisting of mixtures of species - for instance fermions distinguished by spin - can have distinct interactions. Generally let us denote these conditions on atomic species by a tensor ynj where n and j denote two atoms with some interaction relationship. The Hamiltonian in a coordi- nate system with the origin at the bottom of the trapping potential and interaction potential V (r) from equation (2.4) is: Htotal = N∑ n=1 [ −ℏ2∇2 n 2m + 1 2mω 2 well|rn|2 ] + 1 2 N∑ n=1 N∑ j ̸=n ynjV (rn − rj). (2.8) The physical behaviour of the gas determined by the Hamiltonian in equation (2.8) is still rather complicated. For any given gas species it would be necessary to determine the interaction due to relative positions of molar amounts of atoms to describe the system, and solve a Schrödinger equation of order 1023 interacting particles. However it will be demonstrated in the following sections that ultra-cold degenerate quantum gases are universal: there exists another energy scale, dependant only on the s-wave scattering length a, at which all quantum degenerate gases obtain common features. 2.3 Two-body Scattering: Microscopic Level The interaction between two atoms in a quantum gas modeled by equation (2.8) is determined by quantum mechanical scattering processes derived from the two- body interaction potentials of equation (2.4). Given that the potential is time- independent, the relative two-body scattering wavefunction ψscatter in the centre-of- 6 2. Background mass frame solves a time-independent Schrödinger equation [15]:[ −ℏ2∇2 2mrel + V (r) − E ] ψscatter(r) = 0. (2.9) where the relative distance r = ri − rj between two interacting atoms i, j while the mass mrel is the reduced mass mimj mi+mj . This allows the treatment of the problem as a one-body interaction of a quantum-mechanical wave with a potential. The solution to the scattering wavefunction has asymptotically the form of a sum of a plane wave and a spherical wave modified by the effect of the potential V (r) in the form of the scattering amplitude fk(θ): ψscatter(r) ∼ eikz + fk(θ) eikr r . (2.10) The probability flux j = − iℏ m [ψ∗∇ψ − ψ∇ψ∗] is determined from the wavefunc- tion ψscatter(r). The differential cross-section is defined as the ratio of the radially scattered flux into an area r2dΩ to the incident flux of the plane wave: dσ dΩ = |fk(θ)|2. (2.11) The elastic scattering cross-section, a measure of the probability of particle scatter- ing, is found as the integral of the differential cross-section over solid angle dΩ and relates to the scattering wave amplitude in general as σ = ∫ |fk(θ)|2dΩ = ∫ ∫ |fk(θ)|2 sin (θ)dθdφ, (2.12) where integration occurs for the solid angle dθ in the interval 0 to π. However, in the case of identical particles there is interference between scattering amplitudes, and the integral need only evaluate angle θ from 0 to π 2 : σ = ∫ |fk(θ) ± fk(π − θ)|2dΩ, (2.13) where the constructive or destructive interference is distinguished by the scattering wavefunction being particle-exchange symmetric for identical bosons and antisym- metric for identical fermions. If the potential V (r) vanishes for large r, particles scattered by the potential will be observed as free particle waves far from the potential. Moving into the interaction picture, we let V (r, t′) be defined as V (r, t′) = e iH0t′ ℏ V (r)e −iH0t′ ℏ , (2.14) The time-evolution of states is performed by the unitary time-evolution operator U(t, t0), which satisfies the following nested integral equation: U(t, t0) = U(t0, t0) − ∫ t t0 i ℏ V (r, t′)U(t′, t0)dt′ = U(t0, t0) − ∫ t t0 i ℏ V (r, t′)[U(t0, t0) − ∫ t′ t0 i ℏ V (r, t′′)U(t′′, t0)dt′′]dt′... 7 2. Background By definition, U(t0, t0) = 1. We collect the integrals as time-ordered terms of an exponential series using the time-ordering operator T{}. U(t, t0) = 1 − ∫ t t0 i ℏ V (r, t′)dt′ + ∫ t t0 ∫ t t0 (−i)2 ℏ2 1 2!V (r, t′)V (r, t′′)dt′dt′′... = T{ [ e − i ℏ ∫ t t0 dt′V (r,t′) ] } The probability of an outgoing (+) scattered state with momentum ki and energy Ei, |ψ+ ki(r, t)⟩ = e−iEit ℏ |ψ+ ki(r)⟩ being measured as a free plane wave with momentum kf and energy Ef at t → ∞ is given by the scattering matrix S [16], S = lim t→∞ [U(t,−∞)] (2.15) = lim t→∞ lim ϵ→0 T{ [ e − i ℏ ∫ t−iϵ −∞+iϵ dt′V (r,t′) ] } (2.16) Evaluating on the incoming and outgoing states gives the elements: Sfi = lim t→∞ [ lim ϵ→0 ∫ dr′3ψ∗ kf (r′)U(t,−∞)ψ+ ki(r ′) ] = lim t→∞ [lim ϵ→0 ∫ dr′3e−ikf·r′ e i(Ef−Ei)t ℏ eϵtψ+ ki(r ′)] = δ3(kf − ki) − 2πiδ(Ef − Ei)Tfi. The term Tfi are the elements of the T -matrix that represents the energy-conserving state transitions caused by the potential. In the momentum-space basis, the el- ements of T are related to the scattering amplitude fk(θ), where θ is the angle between the initial and final momentum: T = 4πℏ2 m fk(θ). (2.17) Therefore the differential cross-section is also expressible in terms of the T -matrix: dσ dΩ = m2 16π2ℏ4 |T |2. And the total cross-section can be found as: σ = ∫ m2 16π2ℏ4 |T |2dΩ. (2.18) The scattering amplitude fk(θ) can be expanded into angular momentum compo- nents labeled by l. The partial wave expansion of the scattering amplitude is defined as: fk(θ) = ∞∑ l=0 (2l + 1)fl(k)Pl(cos θ). (2.19) Figure 2.1 indicates the plane wave expansion of the scattering wavefunction. The incident wavefunction contributions are the same as that of the free plane wave, but every outgoing l-wave is modified by a complex number Sl. The numbers Sl are the eigenvalues of the S-matrix in the basis of spherical harmonic functions. If the 8 2. Background l l Figure 2.1: a) The plane wave expansion of eikz propagating in free space is asymptotically a sum of outgoing and incoming l-waves. b) Comparison of wave undergoing scattering (solid line) off potential V to plane wave in free space (dashed line). The incoming l-waves are not modified by the interaction while the outgoing l-waves are modified by a complex number Sl ≡ 1 + 2ikfl(k). scattering is elastic, |Sl|2 = 1. Expanding ψscatter in the angular momentum basis gives a sum of radial wavefunctions Al(r), ψscatter(r) = ∞∑ l=0 il(2l + 1)Pl(cos θ)Al(r) (2.20) that has an asymptotic form for r → ∞: ψscatter(r) ∼ 1 2ik ∞∑ l=0 (2l + 1)Pl(cos θ)((−1)l+1 e −ikr r + Sl eikr r ). (2.21) Setting the partial wave expansion of (2.19) into equation (2.10) and comparing with equation (2.21), the partial wave amplitudes are related to Sl: fl(k) = Sl − 1 2ik . (2.22) Sl are represented as phase factors ei2δl where δl are real-valued phase shifts of the outgoing l-waves. That gives an expression for each l partial wave amplitude fl(k): fl(k) = 1 k cot δl − ik . 2.4 Low-energy limit for two-body scattering For illustration, we use the hard sphere potential as an approximation of the inter- action potential. Let: V (r < lvdW) = ∞, V (r > lvdW) = 0. 9 2. Background The phase shifts are resolved depending on the solutions for the radial wavefunctions Al(r): ℏ2(∂2 r + 2 r ∂r) 2mrel − V (r) + ℏ2(k2 − l(l+1) r2 ) 2mrel  rAl(r) = 0. (2.23) Outside the hard sphere, the radial component of the scattering wavefunction Al is written in terms of the spherical functions [17]: Al(r) ≈ eiδl((kr)l cos(δl) (2l + 1)!! + (2l − 1)!! sin (δl) (rk)l+1 ). (2.24) Therefore at Al(r = lvdW) = 0, tan (δl) ≈ − (klvdW)2l+1 (2l − 1)!!(2l + 1)!! (2.25) In the low energy limit the most dominant scattering is from the l = 0 s-waves as tan (δ0) reaches 0 with k → 0 slowest: fk(θ) = 1 k ∞∑ l=0 (2l + 1)eiδl(k) sin (δl(k))Pl(cos(θ)) = 1 k cot δ0(k) − ik + O(k2). The term k cot δ0(k) can be opened using the effective range expansion as a function of k2 by using equation (2.25): k cot δ0(k) ≈ −k 1 klvdW + O(k2), := −1 a + rsk 2 + O(k4). (2.26) At those energies where k2 is much smaller than effective range of scattering rs, it is possible to approximate k → 0 and then the scattering amplitude simplifies to f0 = −a = −lvdW. (2.27) The T -matrix, and differential and total scattering cross-section become for identical bosons T = −8πℏ2a m , dσ dΩ = 4|a|2, σ = 8π|a|2, and for distinguishable bosons T = −4πℏ2a m , dσ dΩ = |a|2, σ = 4π|a|2. The fact that at low energies the cross-section only depends on a justifies an effective field theory for the system that depends on this variable as a parameter. This is because the T -matrix results in the effective field theory at that energy scale match exactly the effective-range expansion result - which in this case is only a dependence on a. To complete the argument, we step away from the lvdW-dependent hard sphere potential representation and allow the potential to have an arbitrary, a-dependent shape that nonetheless appears as a simple delta-function in the low-energy limit. 10 2. Background 2.5 Universality and the Efimov Effect Ordinarily the scattering length a is of similar scale as the natural length scale lvdW, but it can be manipulated with a magnetic by a mechanism known as a Feshbach resonance. Near resonance, the scattering length takes the form a(B) = abg(1 − ∆i B −Bi ), (2.28) where the bare scattering length abg is of order lvdW, the parameter Bi is the Fesh- bach resonance point and B = ∆i + Bi is the magnetic field strength for which the scattering length vanishes. Because a can be tuned to any size, the atomic interac- tions that depend on s-wave scattering can become very strongly repulsive or very strongly attractive, depending on the choice of magnetic field. For values of |a| that greatly exceed lvdW, the dominating effects of two-body scattering depend only on the scattering length a, atomic mass m, the spin statistic of the atomic species and the gas temperature T . For three-body scattering, a new parameter κ∗ is necessary to account for the Efimov effect. The dependence of systems on these parameters to the exclusion of particular details of the interaction potentials or specific atomic properties is called universality. The primary parameter for universality can be considered the ratio of a to the natural length scale lvdW, R = lvdW |a| , (2.29) and corrections to universality are of order O(R). Full universality is achieved in the following two limits that set R = 0: lvdW → 0, Unitary / Resonant limit: a → ± ∞. The Efimov effect is the name given to the realisation of a geometrically spaced energy spectrum of bound three-body states, known as trimers, which occurs if certain conditions for the masses, spin statistics and scattering lengths of the atoms are met [18]. When the conditions for the Efimov effect are met, the aforementioned parameter κ∗ enters the theory in the form of the universal Efimov spectrum of the (n)th trimer branch in ascending order of depth, satisfied for large |a|, that follows equation 2.30: − E (n) T = (e 2π s0 )n−n∗ ℏ2κ2 ∗ m , (2.30) where the constant s0 depends on the mass ratios of the atomic species and their spin statistics. It will be shown in section 3.2.3 that this constant is the solution of a transcendental equation that determines the renormalization limit cycle of three- body scattering [19]. The parameter κ∗ is a wavevector value that can be found from an arbitrarily shallow trimer in the spectrum for n = n∗, in the resonant limit. It is convenient to express this parameter as a term Λ∗ found from the renormalization group limit cycle relationship: s0 ln (|a|Λ∗) = [s0 ln (|a|κ∗) + 0.97] mod π (2.31) 11 2. Background The result of the Efimov effect is a dependency of three-body observables such as three-body differential cross-sections on the scale of the system, and discrete scaling symmetries for these energy and length scales. Given a discrete scaling factor λ = e π s0 , the following discrete scaling symmetries apply to the system: Λ∗ → Λ∗ a → λ−1a lim |a|→∞ E → lim |a|→∞ λ2E. (2.32) These scales affect the binding energies of the trimers in the resonant limit and certain characteristic lengths associated with them. Particularly the two-body scat- tering lengths associated to the thresholds of the bound three-body state system to the three atom system as well as the atom and two-body state system will be subject to this scaling as well. Such emergent universal bound state spectra - and their potential effects on scattering amplitudes - are also realisable in other systems of strongly interacting quantum degenerate gases. 12 3 Zero-Range Model: Effective Quantum Field Theory In this chapter, an effective field theory known as the zero-range model will be constructed to describe the three-body interactions of degenerate quantum gases. Section 3.1 provides the derivation and renormalisation of the theory using the Feyn- man diagram formalism in the two-body sector for identical bosons. The solution of two-body scattering is directly implemented for the three-body sector in section 3.2 with the introduction of a two-particle field operator - the dimer - reducing the treatment with Feynman diagrams down to atom-dimer scattering as a two-body problem. Physical observables, such as the energy spectra for the three-particle bound states - known as trimers - and the atom-dimer scattering length, are de- scribed in section 3.3. 3.1 Two-Body Interactions An effective field theory for ultracold atoms is built by replacing the detailed atom interaction potential V (r) with an effective 2-body interaction (ψ†ψ)2 H = ∫ d3r [ 1 2m∇ψ† · ∇ψ + λ0 4 (ψ†ψ)2 ] , (3.1) for the case of two identical bosons, where λ0 is a bare coupling coefficient of the theory. We have chosen units such that ℏ = 1. As a result, wavevectors denoted q, p, k are equivalent to momenta and frequencies ωq, ωp, ω are equivalent to energies. The dimensions of the fields ψ must satisfy the condition that the Hamiltonian has the dimension of energy. In the first term the dimension of 1 m is [EL2] and the Laplacian operator has dimension [L−2]. This in turn means the fields ψ and ψ† must have dimension [L− 3 2 ] in order for the integration over dr3 to yield a first term with dimension energy. The interaction term (ψ†ψ)2 has dimension [L−6] and to compensate this the bare coupling coefficient must have dimension [EL3]. Forthwith the exponent of the negative power of length for a field operator On(ψ, ψ†,∇ψ,∇ψ†) will be called the engineering dimension dn. O0 = ∇ψ† · ∇ψ, d0 = 5. O1 = (ψ†ψ)2, d1 = 6. 13 3. Zero-Range Model: Effective Quantum Field Theory The field operators are Fourier transforms of the ladder operators a,a† of the quantum-mechanical harmonic oscillator: ψ(r) = 1 (2π)3 ∫ ake −ik·rd3k ψ†(r) = 1 (2π)3 ∫ a† ke ik·rd3k and therefore act on spaces of coherent states, which are superpositions of harmonic oscillator eigenstates [20]. To describe particle scattering, the time-independent operators ψ†, ψ are transformed into the interaction picture. ψ → ψ(r, t) = eiH0tψe−iH0t (3.2) H → H0,I +HI = ∫ dr3 1 2m∇ψ†(r, t) · ∇ψ(r, t) + λ 4 (ψ†(r, t)ψ(r, t))2 (3.3) Because H0 = ∫ d3r 1 2m∇ψ† · ∇ψ is time-independent, the unitary time-evolution operators e±iH0t commute with H0, therefore H0 = H0,I . 3.1.1 The S-matrix in the Interaction Picture The S-matrix unitary operator will determine the evolution of the incoming states into the outgoing states: Sfi = ⟨ψ−(r, t)|ψ+(r, t)⟩ (3.4) So in the basis of coherent states: Sfi = ⟨f|eiH0te−iH(t−t′)e−iH0t′|i⟩ (3.5) The operator S(t, t′) = eiH0t−iH(t−t′)e−iH0t′ may be differentiated in final time t: d dt S(t, t′) = −ieiH0tV e−iH0teiH0t−iH(t−t′)e−iH0t′ = −iVI(t)S(t, t′) (3.6) where VI is the interaction potential operator in the interaction picture which in our field theory matches HI = H − H0 presented in equation (3.3). S(t′, t′) = 1 is the identity of a state at a given time, so the integral equation of S can be formed as S(t, t′) = 1 − i ∫ t t′ HI(t′′)S(t′′, t′)dt′′ = 1 − i ∫ t t′ HI(t′′)[1 − i ∫ t′′ t′ HI(t′′′)S(t′′′, t′′)]dt′′dt′′′ = 1 − i ∫ t t′ HI(t′′)[1 − i ∫ t′′ t′ HI(t′′′)[1 − i ∫ t′′′ t′ HI(t′′′)S(t′′′′, t′′′)]]dt′′dt′′′dt′′′′... And so on indefinitely. This can be expressed as an infinite sum of interaction Hamiltonians, known as the Dyson series, and evaluated at infinitely distant points from the interaction: S(∞,−∞) = 1 − i ∫ t t′ HI(t′′)dt′′ + (−i)2 ∫ t t′ ∫ t′′ t′ HI(t′′)HI(t′′′)dt′′dt′′′ + ... 14 3. Zero-Range Model: Effective Quantum Field Theory Let time-ordering operator T{} be defined using the Heaviside step-function, which ensures only the correct time order term is respected in the result: T{f(r1, t1)} = f(r1, t1) T{f1(r1, t1)f2(r2, t2)} = Θ(t1 − t2)f1(r1, t1)f2(r2, t2) + Θ(t2 − t1)f2(r2, t2)f1(r2, t1) T{ N∏ n=1 fn(rn, tn)} = Sum of permutations of fn(rn, tn) times product of Θ(tn − tn′) As the sum of permutations will have N ! options of time-ordering included as terms, the Dyson series is factored by 1 N ! . S(∞,−∞) = 1 + ∞∑ N=1 1 N ! (−i) N N∏ n=1 T{ ∫ ∞ −∞ dt(n)HI(tn)} = 1 + ∞∑ N=1 1 N ! (−i) N ∫ ∞ −∞ N∏ n=1 dtn ∫ N∏ n=1 d3rnT{ N∏ n=1 VI(rn, tn)} In this theory the interaction potential VI(rn, tn) will be composed of a sum of products of field operators ψ(rn, tn) and ψ†(rn, tn). In order to act with the S operator on any initial and final states, it is possible to commute/anticommute the field operators constituting VI such that all creation operators will be acting on ⟨f| and all annihilation operators will be acting on |i⟩. In the bosonic case, each such commutation between creation/annihilation operators at (rn, tn) to annihilation/creation operators at (rn′ , tn′) will add δ3(rn−rn′). Because the final and initial states are themselves expressible as creation and annihilation operators acting on vacuum states |0⟩ and ⟨0|, with prefactors e±i(rf,i·kf,i−tf,iωf,i) those operators may also be commuted until all creation and annihilation operators are acting on vacuum. By Wick’s theorem, only the δ-functions that were used to per- form this normal-ordering procedure will remain, factored by the coupling constants and subject to the time-ordering Heaviside step-functions. These δ-functions are collected into the time-ordering of creation and annihilation operator pairs, which constitute the retarded propagators in position space: T{ψ†(rn, tn), ψ(rn′ , tn′)} = Θ(tn − tn′)(ψ(rn′ , tn′)ψ†(rn, tn) + δ3(rn − rn′)) + Θ(tn′ − tn)ψ(rn′ , tn′)ψ†(rn, tn) Θ(tn − tn′)δ3(rn − rn′) = DR(rn − rn′ , tn − tn′) The Sfi matrix for the theory determined by the Hamiltonian in equation (3.1) is then expressible in position space as Sfi = δfi + ⟨0| ∞∑ N=1 1 N ! ( −i 4 )NλNMN ∫ ∞ −∞ N∏ n=1 dtn ∫ N∏ n=1 d3rn∏ n,n′ DR(rn − rn′ , tn − tn′) × e−i(ri·ki−tiωi)ei(rf·kf−tfωf)|0⟩. The first δfi term does not contribute to interactions, and subtracting it produces the iTfi transition matrix. Only fully connected products of propagators contribute 15 3. Zero-Range Model: Effective Quantum Field Theory to the transition matrix, that is from starting n′ = i to final n = f all interrim n′, n must form causal paths from i to f. The term MN results as the possible distinct permutations of propagator products are added. Each vertex may be permuted with any other, leading to N ! sets of permutations of propagator products. For an interaction of two creation and two annihilation field operators at each vertex, it is possible to permute each field twice, leading to (2!)2 = 4 sets of propagator product terms for each vertex, 4N . However, such exchanges are not distinct if the internal propagators between vertices are identical, leading to the emergence of a symmetry factor S2 = 2 in the case of identical bosons: iTfi = ⟨0| ∞∑ N=1 (−iλ S )N ∫ ∞ −∞ N∏ n=1 dtn ∫ N∏ n=1 d3rn distinct∏ n,n′ DR(rn − rn′ , tn − tn′)e−i(ri·ki−tiωi)ei(rf ·kf −tfωf )|0⟩ In momentum space, Fourier transformation yields the following result for the scat- tering amplitude A of the interaction leading from initial to final coherent states: iTfi = ⟨0| ∞∑ N=1 (−iλ S )N distinct∏ n,n′ DR(kn − kn′ , ωn − ωn′)|0⟩δ3(kf − ki)δ(ωf − ωi) ≡ iA(kf,ki, ωf, ωi) × δ3(kf − ki)δ(ωf − ωi) 3.1.2 The Scattering Amplitude: Green’s Function and Feyn- man Diagrams In momentum space, the free atom propagator becomes with +iϵ provided by the Heaviside step-functions in time ensuring it has the properties of a retarded propa- gator, Dψ,ψ†(q, ωq) = i ωq − q2 2m + iϵ . (3.7) We can confirm that this is consistent with the retarded propagator in position space through Fourier transform: F [Dψ,ψ†(q, ωq)] = ∫ dq3 (2π)3 ∫ ∞ −∞ dω 2π e−iω(t2−t1)eiq(r2−r1) ωq − q2 2m + iϵ (3.8) which after a shift ω → ω + p2 2m gives a weighted Heaviside step-function from its integral representation: F [Dψ,ψ†(q, ωq)] = Θ(t2 − t1) ∫ dq3 (2π)3 e iq(r2−r1)e−i( q2 2m −iϵ)(t2−t1) where after the limit ϵ → 0 is taken the terms within the integral collect into plane waves of type eiqrn−i q2 2m tn (2π) 3 2 and succinctly become delta functions δ3(r2 − r1) upon integration for all q due to the orthonormality of these functions. 16 3. Zero-Range Model: Effective Quantum Field Theory Figure 3.1: The fully connected two-body Green’s function in momentum space iA2 is built up from a sum of contributions at each loop level. To form an additional loop, the previous diagram in the iteration must be connected to a new vertex with two additional propagators. Lower diagram shows the Green’s function as the Lippmann-Schwinger equation. In addition to the free propagator it is necessary to determine the tree-level 4-point vertex. It is found as the N = 1 term of the sum in the Tfi matrix. In momentum space the first order vertex of a 2-body interaction at 0 loop level becomes: F0 ψ†ψ†,ψψ = −ic1λ0, (3.9) where c1 = 1 for identical bosons. The Feynman rules for two-body scattering are shown in appendix A.1, figure A.1. The only fully-connected way to construct the contributing amplitudes to two-body scattering is shown diagrammatically in figure 3.1. To obtain the interaction at all loop levels, the contributions must be summed over to infinite number of loops to produce the fully connected and amputated two-body interaction diagram iA2: iA2 = −λ0c1 1 − λ0c12 ∫ dk3 (2π)3 ∫ ∞ −∞ dω 2π [ i ω − k2 2m + iϵ i (ωq − ω) − (q3+q4−k)2 2m + iϵ) ] −1 . (3.10) A symmetry term c12 = c1 2 = 1 2 , the result of exchangeable internal propagators in figure 3.1, is included for identical bosons. For distinguishable bosons, the symmetry factor is c1 = c12 = 1 4 due to the non-symmetry of the tree-level diagram and the internal propagators not being interchangeable. Let Iω = ∫∞ −∞ dω 2π [ i ω− k2 2m +iϵ i (ωq−ω)− (q3+q4−k)2 2m +iϵ) ]. The contour integration over fre- quency ω to simplify the propagator loop is demonstrated in figure 3.2. Start by changing the integration variable from real ω to complex z: Iω = [ ∮ dz 2π i z − ( k2 2m − iϵ) −i z − (ωq − (q3+q4−k)2 2m + iϵ) − ∫ C dz 2π i z − ( k2 2m − iϵ) −i z − (ωq − (q3+q4−k)2 2m + iϵ) ] 17 3. Zero-Range Model: Effective Quantum Field Theory Figure 3.2: Contour integral over the complex frequency plane in counterclockwise orientation for looped-propagator factor. The propagator product has poles at z1 = ωq − (q3+q4−k)2 2m + iϵ in the upper half plane and z2 = k2 2m − iϵ in the lower half plane. The contour encloses only pole z1 and can be found as its residual. The integral over the real axis equates the contour integral as the contribution from the semicircular path C vanishes for large radius of C. The integral along the far curve C vanishes due to the contribution of the curve length being diminished by the value of the denominator as |z| → ∞, meaning the contour integral equates the real line integral. Then the Residue theorem can be used to evaluate the contour integral over the upper plane pole. Iω = 2πi 2π lim z→(ωq− (q3+q4−k)2 2m +iϵ) (z − (ωq − (q3 + q4 − k)2 2m + iϵ)) × [ 1 z − ( k2 2m − iϵ) 1 z − (ωq − (q3+q4−k)2 2m + iϵ) ] = i ωq − (k− q3+q4 2 )2 m − (q3+q4)2 4m + iϵ Setting Iω back into iA2, the result for the scattering amplitude of identical bosons is: A2 = −λ0 1 − λ0 2 ∫ dk3 (2π)3 1 ωq − (k− q3+q4 2 )2 m − (q3+q4)2 4m + iϵ  −1 . (3.11) Denote the integral part as I2. Because this integral is divergent, an ultraviolet regularization scheme must be used to compute it. The ultraviolet cutoff of the momentum Λ is according to the scaling limit an inverse of the natural length scale, which in this case is the van der Waals length, so Λ ∼ 1 lvdW . Before introducing the cutoff, first factor the mass out of the momentum terms in the denominator, and 18 3. Zero-Range Model: Effective Quantum Field Theory collect the terms as such: I2 = ∫ dk3 (2π)3 −m (k − q3+q4 2 )2 + ( √ ( q3+q4 2 )2 − ωqm− iϵ)2 After a variable change y = k − q3+q4 2 , it is possible to change the evaluation into a spherical integral and impose an ultraviolet cutoff condition ∞ → Λ: I2 = − m (2π)3 4π ∫ Λ 0 dy y2 y2 + ( √ ( q3+q4 2 )2 − ωqm− iϵ)2 the integral is completed through addition of zero x2 x2+f(x) = x2+f(x) x2+f(x) − f(x) x2+f(x) : I2 = − m 2π2 Λ − ( √ ( q3+q4 2 )2 − ωqm− iϵ)2√ ( q3+q4 2 )2 − ωqm− iϵ × arctan ( (Λ − q3+q4 2 )√ ( q3+q4 2 )2 − ωqm− iϵ )  . It should be supposed that the ultraviolet cutoff Λ is far greater than any energy scale reached by two-body scattering in the ultra-cold regime. The arctan term can be simplified to arctan(∞) = π 2 . Abreviate ∆2 = ( q3+q4 2 )2 −ωqm− iϵ, then denoting this 2-loop integral as I2(q3 + q4, ωq) gives: I2(q3 + q4, ωq) = − m 2π2 [Λ − π 2 ∆]. (3.12) Therefore the off-shell scattering amplitude is Aid, boson(ωq, q3, q4) = − [ 1 λ0 + m 4π2 Λ − m 8π∆ ]−1 , (3.13) Ageneral boson(ωq, q3, q4) = − [ 4 λ0 + mr π2 Λ − mr 2π∆ ]−1 , (3.14) where mr is the reduced mass of the distinguishable bosons, m = 2mr if the masses are identical. The equations (3.13) and (3.14) simplify by performing a trans- formation of the energy and momenta into the center-of-mass frame which sets q3cm + q4cm = 0 so that ∆2 cm = −ωqcmm − iϵ. The on-shell amplitude that corre- sponds to a limitation of the energy to the case where the sum of the energy is equal to the total kinetic energy of the particles, that is the total energy sufficient for these two atoms to exist with given momenta q3, q4. Set ωq to be the threshold energy: ωq = q2 3 2m + q2 4 2m = q2 m . (3.15) Evaluating the amplitude for this energy will give the T (ωq, q1, q2, q3, q4) transition matrix element: Tzero range = A2δ( q2 m − (ωq1 + ωq2))δ3(q3 + q4 − q1 − q2). (3.16) 19 3. Zero-Range Model: Effective Quantum Field Theory 3.1.3 Renormalisation of the Two-Body Coupling Coeffi- cient The renormalisation condition can be formulated so that in the low energy limit of q2 → 0, the result should be exactly the same as the result of quantum mechanical scattering at low energy in the scaling limit lvdW → 0, such that all terms of the effective range expansion in section 2.3, equation (2.26) of k cot δ0(k) were eliminated aside from - 1 a . Tscatter = lim q2→0 [Tzero range] −8π m a = lim q2→0 −[ 1 λ0 + m 4π2 Λ − m 8π (−q2) 1 2 ]−1 a = m 8π [ 1 λ0 + m 4π2 Λ]−1 This gives an explicit definition of the bare coupling coefficient in terms of scattering length: λ0 = 8πa m [1 − 2a π Λ]−1 (3.17) Reinsert these expressions into the expression of the Tzero range matrix: lim ϵ→0 Tzero range = 8πa m(iaq − 1) , ( or 4πa m(iaq − 1) ) . (3.18) Scale-invariant fix-points of a can be found for the zero-range Hamiltonian with the definition of new constants: Ĉ1 = Λ 4 8πa m [1 − 2a π Λ]−1. (3.19) The meaning of the running constant is such that the differential equation Λ d dΛĈ1 = β(Ĉ1), (3.20) maps out the possible Ĉ1 as a trajectory of Λ, which can be viewed as the variety of Hamiltonian theories depending on the choice of cutoff, or scale. A fix-point in this trajectory where Ĉ1 is independent of Λ represents a scale-invariant Hamiltonian with the coupling defined at that fixed point. The fix-point of Ĉ1 in turn gives the values of scattering length a at which theory is scale-invariant. The two factors now determine the solutions for when β(Ĉ1) = 0, which are the fixpoints of the renor- malisation group flow and therefore correspond to a scale-invariant Hamiltonian: 2aΛ π − 2aΛ = 0 → a = 0 2aΛ π − 2aΛ = 1 π 2aΛ − 1 = 1 −1 → a = ±∞ The theory is therefore scale invariant in the non-interactive case a = 0 and the resonant limit |a| → ∞. Because the theory is dependent exclusively on the length scale of a, it must treat the interaction potential as contact through the s-wave scattering channel. This consequence of the scaling limit gives the model its zero- range property. 20 3. Zero-Range Model: Effective Quantum Field Theory 3.2 Three-Body Interactions In order to produce low-energy three-body interactions, it is userful to rewrite the Hamiltonian in equation (3.1) with quantum field operators Ψ signifying the presence of two-atom bound states, known as dimer fields, with the two-body dimer-atom interaction bare coupling constant γ0. In the case of two identical boson species this will be: H = ∫ dr3 [ 1 2m∇ψ† · ∇ψ − λ0 4 Ψ†Ψ + λ0 4 (Ψ†ψψ + ψ†ψ†Ψ) + γ0 36Ψ†ψ†Ψψ ] . (3.21) The dimer fields are not independent from the atomic fields, as they are subjected to the constraints: 0 = λ0 4 Ψ − λ0 4 ψψ − γ0 36ψ †Ψψ, Ψ = ψψ 1 − γ0 9λ0 ψ†ψ . Therefore if the dimer-atom coupling constant γ0 = 0, the dimer field is the product of two atomic fields. With this definition of the dimer field, up to a truncation to the three-body sector the Hamiltonians in equation (3.21) is the zero-range Hamiltonians in equation (3.1) with the added terms γ0 36(ψ†ψ)3 or γ0 36(ψ† 1↑ψ1↑)2(ψ† 2↓ψ2↓) respectively. This means it is an adequate effective field theory for three-body interactions under the condition that the two-body dimer exists and its threshold energy is within the low-energy limit. The free particle propagator for the atom is the same as in the two-body model, Dψ,ψ†(q, ωq). A bare, non-causal propagator is introduced for the dimer field by the self-interaction term λ0 4 (Ψ†Ψ): D0 Ψ,Ψ† = 4i λ0 , (3.22) which in fact is 4 times the inverse of the vertex of two-body interaction F0 2 . The recombination interactions are represented at the tree level by three-point vertices: F0 2ψ†,Ψ = F0 Ψ†,2ψ = −iλ0 2 , (3.23) and the dimer-atom scattering is represented by the four-point vertex: F0 Ψ†,ψ†,ψ,Ψ = −i γ0 36 . (3.24) The factor 1 2 = 2 4 in the three-point vertex occurs as the diagrams are symmetric under exchange of atom fields in the exchange of identical bosons. The four-point vertex has no symmetry and the factor 1 36 persists from the definition of the inter- action Hamiltonian. 21 3. Zero-Range Model: Effective Quantum Field Theory Figure 3.3: The exact boson dimer propagator is found diagrammatically. For each loop level the contributing diagram is multiplied by an additional pair of decay and formation 3-point vertices and two looped atom propagators which are integrated over all indeterminate momenta and energies. Each loop level has a symmetry factor 1 2 . The propagator loop integral is computed exactly as in equation (3.12) to be I2(p, ωp) = − m 2π2 [Λ − π 2 √ (p2 4 − ωpm− iϵ)]. The general case is analogously determined with the mass m replaced with double of the reduced mass 2mr. 3.2.1 The Exact Dimer Propagator The T -matrix receives ultraviolet-divergent contributions to the exact propagator for the diatomic field from the products of the recombination diagrams. These ultraviolet-divergent contributions are identical to the ultraviolet divergence of the two-body scattering amplitude iA2 developed in section 3.1, which was proven to be renormalisable using ultraviolet cutoff regularization scheme, because they are represented by equivalent diagrams. The exact propagator shown in figure 3.3 to form an infinite geometric series. In the general case of mass ratios, the mass is replaced according to m 2 → mr during the internal ω contour integral evaluation: iDΨ,Ψ†;m(p, ωp) = i 4 λ0 + 1 2(i 4 λ0 )2(−iλ0 2 )2iI2(p, ωp;m) + 1 4(i 4 λ0 )3(−iλ0 2 )4(iI2(p, ωp;m))2 + ... DΨ,Ψ†;m(p, ωp) = 4 λ0 ∞∑ n=0 [ λ0 2 I2(p, ωp;m) ]n = 4 λ0 1 + λ0 2 m 2π2 [Λ − π 2 √ (p 2 4 − ωpm− iϵ)] −1 The bosonic dimer propagator’s renormalisation utilizes the same renormalization condition as the two-body scattering amplitude because the combined diagram F0 2ψ†,ΨDΨFΨ†,2ψ ∼ A2. Therefore with λ0 = 8πa m [1 − 2a π Λ]−1: DΨ,Ψ†(p, ωp;m) =  8πa 4m(1 − 2a π Λ)2 − λ2 0m 32π √ (p 2 4 − ωpm− iϵ) −1 , (3.25) and factoring out λ0 gives DΨ,Ψ†(p, ωp;m) = 32π λ2 0 m a −m √ (p 2 4 − ωpm− iϵ) −1 . (3.26) The dimer exists only when the scattering length a > 0. Then the dimer’s self- energy E2Ψ is the pole of this propagator, which is at ωp = p2 4m − 1 ma2 + iϵ and lies in 22 3. Zero-Range Model: Effective Quantum Field Theory Figure 3.4: The integral equation of Green’s function A3 is formed by representing all looped terms as the tree terms connected to the Green’s function by propagators of undetermined energy and momenta. The recombination vertices diagram has two undetermined atomic propagators and one undetermined diatomic propagator as part of the loop, while the four-point vertex diagram has one of each undetermined propagator. the upper half complex plane. It’s possible to define a field renormalisation constant ZΨ: ZΨ = 64π λ2 0am 2 The field renormalisation constant is applied to create this definition of the exact propagator: DΨ,Ψ†(p, ωp;m) := ZΨ am2 2 1 m a −m √ (p2 4 − ωpm− iϵ) (3.27) Together with the vertices and atomic propagator, the exact dimer propagator serves as the building element for the Feynman diagrams of this effective field theory. The Feynman rules for this approach to three-body scattering are shown in appendix A.2, figure A.2. 3.2.2 The Atom-Dimer Scattering Amplitude The next stage is to derive the dimer-atom scattering amplitude iA3. The integral equation of this amplitude is shown diagrammatically in figure 3.4. It has been shown previously in section 3.1.2 that for identical bosons the result is independent of the choice of inertial reference frame, therefore the centre-of-mass frame is used where the total momentum of the incoming and outgoing particles is 0: q1 = −p1 and q2 = −p2. With total energy ωT , the equation for the dimer-atom scattering of 23 3. Zero-Range Model: Effective Quantum Field Theory identical bosons is formulated as: A3(q1, q2, ωT ) = −λ2 0 4 1 (ωT − ωq1 − ωq2) − (q1+q2)2 2m + iϵ − γ0 36 − 1 2 ∫ ∞ −∞ dω 2π ∫ dk3 (2π)3 [(λ 2 0 4 1 (ωT − ωq2 − ω) − (q2+k)2 2m + iϵ + γ0 36) × ( i ω − k2 2m + iϵ ZΨ m a −m √ (k2 4 − (ωT − ω)m− iϵ) )am2A3(q1, k, ωT )] In the treatment of identical bosons, A3(q1, q2, ωT ) = −λ2 0 4 1 (ωT − ωq1 − ωq2) − (q1+q2)2 2m + iϵ − γ0 36 − 1 2 ∫ ∞ −∞ dω 2π ∫ dk3 (2π)3 [(λ 2 0 4 1 (ωT − ωq2 − ω) − (q2+k)2 2m + iϵ + γ0 36) × ( i ω − k2 2m + iϵ ZΨ m a −m √ (k2 4 − (ωT − ω)m− iϵ) )am2A3(q1, k, ωT )] Given that the poles of the three-body amplitude are in the upper half complex plane, the only ω-pole of the above integral equation that is in the lower half complex plane is ω → k2 2m − iϵ. Proceeding in the same manner as for the two-body scattering amplitude in section 3.1.2, let Iω,3 = ∫ ∞ −∞ dω 2π [(λ 2 0 4 1 (ωT − ωq2 − ω) − (q2+k)2 2m + iϵ + γ0 36) × ( i ω − k2 2m + iϵ ZΨ m a −m √ (k2 4 − (ωT − ω)m− iϵ) )am2A3(q1, k, ωT )] after a change of the variable ω to z, we perform the contour integral of ω through the lower half plane. The real line integral equals the negative of the clockwise contour integral as the contribution of the large radius curve vanishes just as for C. This simplifies the integral as: Iω,3 = −(λ 2 0 4 1 (ωT − ωq2 − k2 2m) − (q2+k)2 2m + iϵ + γ0 36) × −ZΨ m a −m √ (k2 4 − ωTm+ k2 2 − iϵ) am2A3(q1, k, ωT )] We simplify further with the kinematic values ωq1 = q2 1 2m2 = q2 1 2m2 , ωq2 = q2 2 2m2 : A3(q1, q2, ωT ) = −λ2 0 4 1 ωT − q2 1 m − q2 2 m − q1·q2 m + iϵ − γ0 36 + 1 2(2π)3 ∫ dk3[(λ 2 0 4 1 ωT − q2 2 m − k2 m − q2·k m + iϵ + γ0 36) × −ZΨ m a −m √ (3k2 4 − ωTm− iϵ) am2A3(q1, k, ωT )] 24 3. Zero-Range Model: Effective Quantum Field Theory The total angular momentum l is set to l = 0 and the amplitude projected onto this sector. Let θ be the angle between q1 and q2 while φ is the angle between q1 and k. In the lower energy limit of the ultracold gas regime the scattering amplitude may be projected onto the states with angular momentum number l = 0 by averaging over cos θ in the tree level diagrams and cosφ in the loop level diagrams: A3,l=0(q1, q2, ωT ) = − ∫ 1 −1 d cos θλ 2 0 8 − m q1q2 −ωTm q1q2 + (q2 1+q2 2) q1q2 + cos θ − iϵ − γ0 36 + 1 2(2π)3 ∫ dk3[ ∫ 1 −1 d cosφ(λ 2 0 8 − m q2k −ωTm q2k + (q2 2+k2) q2k + cosφ− iϵ + γ0 36) × −ZΨ m a −m √ (3k2 4 − ωTm− iϵ) am2A3(q1, k, ωT )] = m q1q2 λ2 0 8 ln (−ωTm+ (q2 1 + q2 2) + q1q2 − iϵ −ωTm+ (q2 1 + q2 2) − q1q2 − iϵ ) − γ0 36 + 1 4π2 ∫ Λ 0 dk[k2(m 3aλ2 0 8q2k ln (−ωTm+ (q2 2 + k2) + q2k − iϵ −ωTm+ (q2 2 + k2) − q2k − iϵ ) − γ0am 2 36 ) × ZΨ m √ (3k2 4 − ωTm− iϵ) − m a A3(q1, k, ωT )] In order to remove the dependence on λ0, multiply by ZΨ to make the integral equation self-consistent as ZΨA3(q1, k, ωT ) → A3(q1, k, ωT ) and set the dimensionless running parameter G(Λ) ≡ −Λ2 m γ0 9λ2 0 : A3,l=0(q1, q2, ωT ) = 16π ma [ 1 2q1q2 ln (−ωTm+ (q2 1 + q2 2) + q1q2 − iϵ −ωTm+ (q2 1 + q2 2) − q1q2 − iϵ ) + G(Λ) Λ2 ] (3.28) + 16m π ∫ Λ 0 dk ( 1 8q2k ln (−ωTm+ (q2 2 + k2) + q2k − iϵ −ωTm+ (q2 2 + k2) − q2k − iϵ ) + G(Λ) Λ2 ) × k2 m √ (3k2 4 − ωTm− iϵ) − m a A3,l=0(q1, k, ωT ) The term G(Λ) must have a special dependence on Λ for the theory to become renormalisable. The parameter G(Λ) is not expected to depend on the two-body scattering length a, but as a dimensionless running constant it must instead be a function of a dimensionless ratio Λ Λ∗ where Λ∗ is the three-body parameter previously introduced as part of the Efimov effect in section 2.3, equation (2.31). 3.2.3 Renormalisation of Three-Body Term G(Λ) The integral in equation (3.28) is generally divergent, and so certain considerations must be made for the scattering amplitude A3,l=0 to perform renormalisation Λ → 1 lvdW . We must analyse the contributions that the scattering amplitude makes to the integral as the internal momentum k is increased, and see how it is possible to tune the scaling parameter G(Λ) to diminish the effect of these contributions on the modulus and phase of A3,l=0. 25 3. Zero-Range Model: Effective Quantum Field Theory First reviewing the region q2 → ∞ with G(Λ)/Λ2 → 0, the inhomogeneous term A3,l=0,inh(q1, q2, ωT ) = 2π ma [ 1 q1q2 ln (−ωTm+ (q2 1 + q2 2) + q1q2 − iϵ −ωTm+ (q2 1 + q2 2) − q1q2 − iϵ ) ] , (3.29) does not have a divergence for Λ → ∞ and can be neglected in the investigation of high momentum behavior. Then we set the ansatz A3,l=0(q1, q2, ωT ) = Aqs−1 1 , sup- posing that s is imaginary and the modulus of this scattering amplitude is dependent on q−1 1 , which will allow the integral to be convergent: qs−1 1 = 32m 8π ∫ Λ 0 dk ( 1 2q2k ln (−ωTm+ (q2 2 + k2) + q2k − iϵ −ωTm+ (q2 2 + k2) − q2k − iϵ ) ) k2ks−1 m √ (3k2 4 − ωTm− iϵ) − m a We can at this point eliminate −ωTm− iϵ and 1 a as they are much smaller than k2 and k respectively, and change the variable k = xq2 qs−1 2 = 4 π √ 3 1 q2 ∫ Λ 0 dxxs−1qs2 ln (x 2 + x+ 1 x2 − x+ 1) This produces the equation 1 = 4 π √ 3 ∫ Λ 0 dxxs−1 ln (x2+x+1 x2−x+1). It is evaluated as a Mellin transform, by first factorising the x polynomials within the logarithm with roots x1 = 1−i √ 3 2 , x2 = 1+i √ 3 2 and using the identity M [ln (x+y x−y )] = πys tan πs 2 : π √ 3 4 = ∫ Λ 0 dxxs−1[ln (x+ x1 x− x1 ) + ln (x+ x2 x− x2 )] = π (e−iπs 6 + ei πs 6 )(eiπs 2 − ei π2 2 ) 2i cos πs 2 This gives the transcendental equation √ 3s 8 = sin πs 6 cos πs 2 . (3.30) and for two species of bosons with distinct masses m1, m2:√ 1 + 2 (1+ m1 m2 )s 2(1 + m1 m2 ) = sin πs 6 cos πs 2 . (3.31) The solution s = ±is0 must be fully imaginary, and s0 determines the Efimov spectrum scaling parameter λ described in section 2.5 and the general expression of A in the high momentum limit: A3,q2→∞ = A+q −1+is0 2 + A−q −1−is0 2 (3.32) We can now study the contribution to A3 in the momentum region 1 a , q1, √ mωT ≪ k ≪ Λ using this new ansatz from equation (3.32). We expect that these contri- butions affect the phase and amplitude of the log-periodic solution of A3 in the high-momentum limit, and that compensating for them with G(Λ) will lead to a 26 3. Zero-Range Model: Effective Quantum Field Theory scale-invariant result for all momenta, which will allow for theory renormalisation using low-energy observables such as the atom-dimer scattering length [21]. The scaling contributions to the amplitude denoted as A+ 3 become: A+ 3 = ∫ Λ k0≫ 1 a dk[ 1 k2 + G(Λ) Λ2 ] × (A+k is0 + A−k −is0) (3.33) The tuning for G(Λ) must be motivated by these contributions tending as 1 Λ2 : G(Λ) = A+ 1−is0 Λis0 + A− 1+is0 Λ−is0 A+ 1+is0 Λis0 + A− 1−is0 Λ−is0 (3.34) The coefficients A± can be chosen to depend on the three-body parameter Λ∗ so that A+ = 1 2 √ (1 + s2 0)Λ−is0 ∗ , A− = 1 2 √ (1 + s2 0)Λis0 ∗ . Inserting these into equation (3.34): G(Λ) = √ (1+is0) (1−is0)( Λ Λ∗ )is0 + √ (1−is0) (1+is0)( Λ Λ∗ )−is0√ (1−is0) (1+is0)( Λ Λ∗ )is0 + √ (1+is0) (1−is0)( Λ Λ∗ )−is0 = eis0 ln ( Λ Λ∗ )+i arctan (s0) + e−is0 ln ( Λ Λ∗ )−i arctan (s0) eis0 ln ( Λ Λ∗ )−i arctan (s0) + e−is0 ln ( Λ Λ∗ )+i arctan (s0) leading to the analytical function of what is called the three-body scaling term: G(Λ; Λ∗) = cos (s0 ln ( Λ Λ∗ ) + arctan (s0)) cos (s0 ln ( Λ Λ∗ ) − arctan (s0)) . (3.35) It is demonstrated using numerical testing in section 4.1.1 that this analytical re- sult for G(Λ) is an accurate renormalisation group limit cycle for the purposes of determining the atom-dimer scattering amplitude A3,l=0. 3.3 Observables of the Three-Body System The atom-dimer scattering amplitude provided by equation (3.28) inherits certain properties from the three-body scaling term G(Λ) that manifests themselves both as a dependence of the atom-dimer scattering length on the three-body parameter Λ∗ and as discrete scaling symmetries of the Efimov spectrum. These effects depend explicitly on the solution for the parameter s0, which is different for identical bosons and bosons with general mass ratios. Just as for two-body scattering, the T matrix element for atom-dimer scattering is found at minimum kinetic energy required for elastic scattering. For q2 = −q1, the T -matrix element becomes: Tq1,p1,q2,p2 = A3,l=0(q1, q1, 3q2 1 4m − 1 ma2 ). (3.36) 27 3. Zero-Range Model: Effective Quantum Field Theory The differential cross-section is in the case of bosons of identical masses: dσ dΩ = 2m 3k km 6π2 |A3,l=0(q1, q1, 3q2 1 4m − 1 ma2 )|2 = m2 9π2 |A3,l=0(q1, q1, 3q2 1 4m − 1 ma2 )|2, (3.37) and similarly to the discussion in section 2.4 for two-body scattering of non-identical particles, the effective range expansion applied to atom-dimer scattering states that for the atom-dimer scattering length aAD, f0 = −aAD and dσ dΩatom,dimer = |aAD|2, it follows that in the low-energy limit for bosons q1 → 0 the atom dimer scattering length is given by: aAD = −m 3πA3,l=0(0, 0,− 1 ma2 ). (3.38) It is useful to also define the function C(q1, q2, ωT ) = ma 8π A3,l=0(q1, q2, ωT )(q2 1 − q2 2) − 1 a + √ 3q2 1 4 −mωT , (3.39) which at q1 = q2 = 0 reduces to C(0, 0,− 1 ma2 ) = −aAD. Poles of the scattering amplitude A3,l=0 constitute the spectrum of the three-body bound states known as trimers. For energies ωT near the threshold energies −E(n) 3 of these trimers, the Green’s function is factorised into two bound-state functions: A3,l=0(q1, q2, ωT ) = B(n)(q1)B(n)(q2) (ωT + E (n) 3 ) (3.40) Setting this into equation (3.28) gives for the case of the identical bosons: B(n)(q2) = 4 π ∫ Λ 0 dk ( 1 2q2k ln (−ωTm+ (q2 2 + k2) + q2k − iϵ −ωTm+ (q2 2 + k2) − q2k − iϵ ) + G(Λ) Λ2 ) (3.41) × k2√ (3k2 4 − ωTm− iϵ) − 1 a B(n)(k) For the case of two distinct species of bosons with mass ratio m1 m2 = rm, the scattering amplitude is modified by other symmetry factors and the mass is replaced with the reduced mass. We only consider the case where the scattering length of the distinct species a12 is tuned by Feshbach resonance to be much greater than the scattering length of two identical atoms a11 and a22, as this precludes the involvement of corresponding dimer states and simplifies the bound state equation for two m1 and one m2 atoms to take the form: B(n) 112(q2) = 16 π ∫ Λ 0 dk 1 + rm 32q2k ln ( −2ωTmr + (q2 2 + k2) + 2 (1+rm)q2k − iϵ −2ωTmr + (q2 2 + k2) − 2 (1+rm)q2k − iϵ ) + G(Λ) Λ2  (3.42) × k2√ (k 2(1+ 4 1+rm ) 4 − 2ωTmr − iϵ) − 1 a12 B(n) 112(k) 28 4 Results This chapter presents the results acquired from numerical calculations of the zero- range model values discussed in sections 3.2 and 3.3. Gauss-Legendre quadrature was used to solve the bound state and three-body scattering amplitude equations formulated as Fredholm equations of the second kind. Confirmation of the renormalization utility of a three-body parameter dependent scaling term G(Λ; Λ∗) is done with results in section 4.1.1. Section 4.1.2 provides results for the dependence of the atom-dimer scattering length on the proportion Λ Λ∗ and on the diatomic scattering length. Section 4.2.1 demonstrates in detail the Efimov effect in systems of identical bosons. The discrete scaling symmetries of the Efimov effect have been satisfied for up to six trimer branches in both the binding energy spectrum at the resonant limit |a| → ∞ and for diatomic scattering lengths characterising the thresholds to the atom-dimer and triatom states with the trimer region. In addition to this, universal scaling curves are represented for the ratios of atom-dimer to diatom scattering lengths, and the ratio of the two shallowest dimers with the dimer binding energy. Section 4.2.2 provides some of the results for the emergence of the Efimov effect in search of a critical value of the mass ratio rm = m1 m2 at which the fermionic system produces a spectrum of trimers that satisfy a discrete scaling symmetry. 4.1 Results for Scattering Amplitudes of Identical Bosons in the Zero-Range Model The results procured in this section used Gauss-Legendre quadrature procedure out- lined in the appendix B.3 to solve the integral equations for A3,l=0 and C numerically. The quadrature procedure for solving A3,l=0 used 15 quadrature points and interpo- lation, while the procedure for C used 100 quadrature points and no interpolation. The functions and variables are scaled by mass m and scattering length a. 4.1.1 Renormalisation of the Three-Body Scattering Am- plitude Figure 4.1 shows a comparison of the analytical function for G(Λ) in equation (3.35) from section 3.2.3 to most effective numerical tests ofG, as well as the A3,l=0 identical bosons scattering amplitude graphs evaluated at zero incoming atom momentum q2 = 0 and at total energy E = − 1 ma2 for increasing ultraviolet cutoffs. The results are not fully conclusive, as the best fit results do coincide with the analytical G(Λ), 29 4. Results 10 -4 10 -2 1 10 2 10 4 Outgoing Atomic Wavenumber q 1 [a -1 ] -25 -20 -15 -10 -5 0 5 A to m -D im er S ca tt er in g A m p li tu d e A 3 ,L = 0 (q 1 , 0 , -1 /( m a 2 )) [a ] Scattering Amplitude: Scaling Term G(Λ), Dependance on Cutoff Λ Λ = Λ 0 Λ = 2.5Λ 0 Λ = 5Λ 0 Λ = 10Λ 0 Λ = 15Λ 0 Λ = 20Λ 0 Λ = 25Λ 0 Λ = 50Λ 0 Λ = 100Λ 0 10 0 10 1 10 2 10 3 10 4 Multiples of Ultraviolet Cutoff Λ 0 [n] -20 -15 -10 -5 0 5 10 15 20 T h re e- B o d y S ca li n g T er m G Scattering Amplitude: Numerical Renormalisation Tests of Three-Body Scaling Term G Compared to G(Λ) Numerical Tests G(nΛ 0 ) Figure 4.1: Left: Comparison of numerical tests of G to the analytic ansatz G(Λ). The result is consistent and shows promise for the validity of the renormalisation provided by inclusion of the three-body parameter Λ∗ = 0.9522a−1 in the theory. Λ0 = 2.165Λ∗ : G(Λ0) ≈ 0. Right: scattering amplitudes A3,L=0(q1, 0,− 1 ma2 ) for increasing ultraviolet cutoffs indicate a significant changes in the value of the am- plitude at q1 < 1 a , thereby affecting the atom-dimer scattering length result despite renormalisation of G(Λ). but the numerical procedure determining the scattering amplitude seems to be non- convergent for low values of q1. As an alternative a modified scattering amplitude C(q1, 0,− 1 ma2 ) given by equa- tion (3.39) is evaluated using a non-interpolative method with significantly higher number of quadrature points described in appendix B.3.1, and uses a higher ultra- violet cutoff Λ0 = 1152Λ∗ : G(Λ0) ≈ 0. For modified scattering amplitude function C(q1, 0,− 1 ma2 ) with G = 0, the expected log-periodic behavior for high outgoing momenta q1 are shown in figure 4.2. This confirms the assessment made in sec- tion 3.2.3 that the high-momentum form of the scattering amplitude A3,l=0 would asymptotically tend to the form of log-periodic q1−is0 . Using the arguments from section 3.2.3, the effect of including a non-zero G term in C on the amplitude and phase of the function can be seen in figure 4.3. Using the analytical function of G(Λ) given in equation (3.35), the renormalisation of the three-body is demonstrated for large values of the ultraviolet cutoff in the results from figures 4.4 and 4.5. The inclusion of the analytical parameter G(Λ) forces the function C(q1, 0,− 1 ma2 ) evaluated with any ultraviolet cutoff Λ back into the same phase and amplitude as the one evaluated for Λ0. As the three-body parameter G is the same for the atom-dimer scattering amplitude A3,l=0 as it is for the modified function C, the renormalisation group limit cycle is satisfied, and the low-momentum value of A3,l=0 becomes scale-invariant, confirming that the physical observable atom-dimer scattering length of the theory is renormalised. 30 4. Results 10 1 10 2 10 3 Outgoing Atomic Wavenumber q 1 [a -1 ] -15 -10 -5 0 5 10 15 M o d if ie d A to m -D im er S ca tt er in g A m p li tu d e F u n ct io n C (q 1 , 0 , -1 /( m a 2 )) [a ] High-Momentum Behavior of Atom-Dimer Scattering Amplitude Comparison to Log-Periodic Fitted Functions Λ = Λ 0 F = 0.47, f = s0, φ = π-0.03 Λ = 2.5×Λ 0 F = 0.85, f = s0, φ = π/2+0.63 Λ = 5×Λ 0 F = 3.7, f = s0, φ = 3π/2-0.05 Λ = 10×Λ 0 F = 0.62, f = s0, φ = 5π/4+0.05 Λ =15 ×Λ 0 F = 0.48, f = s0, φ = 9π/8+0.03 Figure 4.2: Comparison of the C(q1, 0,− 1 ma2 ) function displayed as solid lines with fitted log-periodic functions F cos (f ln (q1) + ϕ). It is demonstrated that for all cutoffs, the frequency f is s0 ≈ 1.00624 and that the phase and amplitude of the function are dependent on the cutoff Λ. 4.1.2 Results for the Atom-Dimer Scattering Length The modified scattering amplitude function C(q1, 0,− 1 ma2 ) is determined indepen- dently by the parameters of diatomic scattering length a and the three-body pa- rameter Λ∗ when the ultraviolet cutoff Λ is set at a specific value in terms of the scattering length. This is shown by the results of scattering amplitude graphs in fig- ure 4.6 as a and Λ∗ are varied. Additionally it is demonstrated that the atom-dimer scattering length aAD = −C(0, 0,− 1 ma2 ) depends on the three-body parameter Λ∗. 4.2 Results for Three-Body Bound State Energy in the Zero-Range Model The results in this section implemented Gauss-Legendre quadrature using 20 quadra- ture points through the procedure outlined in appendix B.2. 4.2.1 Results for Bound State Energy of Identical Bosons The Efimov effect and its universal discrete scaling symmetries have been reproduced in 6 trimer branches using the bound state equation (3.41) in section 3.3 for identical bosons as seen in figure 4.7. The values of the binding energies increase in depth by a factor of approximately λ2 = 22.72 in the resonant limit while the diatom scattering lengths associated with trimer-three atom threshold a = a+ and the trimer-dimer threshold a = a− decrease by the factor λ as shown in figure 4.8. Because the solution parameter for identical bosons s0 ≈ 1.00624, this is consistent with the 31 4. Results 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 Outgoing Atomic Wavenumber q 1 [a -1 ] -4 -3 -2 -1 0 1 2 3 M o d if ie d A to m -D im er S ca tt er in g A m p li tu d e F u n ct io n C (q 1 , 0 , -1 /( m a 2 )) [a ] Effect of Three-Body Scaling Term G on Atom-Dimer Scattering Amplitude Λ = Λ 0 , G = 0 Λ = 2.5×Λ 0 , G = 0 Λ = 2.5 ×Λ 0 , G = -18 Λ = 2.5 ×Λ 0 , G = -3 Λ = 2.5 ×Λ 0 , G = -0.5 Λ = 2.5 ×Λ 0 , G = 3 Λ = 2.5 ×Λ 0 , G = 18 Figure 4.3: Comparison of modified scattering amplitude function C(q1, 0,− 1 ma2 ) visualises the effect of the three-body scaling term. Solid lines are with G = 0 evaluated at ultraviolet cutoffs Λ = Λ0 and Λ = 2.5Λ0, dashed lines are with non-zero values of G evaluated at ultraviolet cutoff Λ = 2.5Λ0. The phase and amplitude of the dashed lines are shifted, while the frequency remains unchanged. This confirms that there may exist G(Λ) such that leaves the scattering amplitude invariant for all momenta q1 < Λ. relation for the discrete scaling parameter λ = e π s0 described in section 2.5, and the Efimov trimer spectrum of equation (2.30) is developed with n∗ := 0. Additionally, comparison to atom-dimer scattering length as well as the ratios of first and second trimer binding energies to the dimer binding energy ED shown in figure 4.9 are consistent with the universal scaling curves found in literature [18]. For quantum degenerate gases, we view these universal results of the zero-range Model in the three-body sector as predictions for the reactions of optically trapped Bose-Einstein gases and condensates in the ultracold regime to the tuning of the diatomic scattering length using Feshbach resonance. The universal model predicts, besides the scaling symmetries viewed in figures 4.7 and 4.8, the existance of in- finitely many trimer branches at ever shallower energies [18]. These trimer branches indicate directly a strongly interacting quantum degenerate gas is formed that al- lows for three-atom bound states at ranges of large scattering lengths [−∞, a(−)] and [a(+),∞] in the ultracold regime. For our results, trimer branches (1) and (2) remain within the ultra-cold regime and the threshold points a(−), a(+) where these trimer branches intersect with the x-axis and the dimer line ED signify values of a at which the bound trimer state is in resonance with a three-atom state and an atom-dimer state respectively. When such resonances occur in trapped Bose-Einstein gases, the binding energy of the trimer can be released into the kinetic energy of the three atoms at the x-axis threshold, which occurs for certain negative scattering lengths a, or into the kinetic energy of the dimer-atom pair at the ED threshold for specific positive scattering lengths a [11]. These energies may exceed the confinement properties of the optical trap, leading to escaping gas atoms at these resonance points, called loss effects. 32 4. Results 10 -1 1 10 10 2 10 3 10 4 -2 -1 0 1 2 Λ = 2.5Λ 0 , G = -1.29488 10 -1 1 10 10 2 10 3 10 4 -2 -1 0 1 2 Λ = 5Λ 0 , G = 26.6141 10 -1 1 10 10 2 10 3 10 4 -2 -1 0 1 2 Λ =10Λ 0 , G =1.09996 10 -1 1 10 10 2 10 3 10 4 -2 -1 0 1 2 Λ = 15Λ 0 , G = 0.449715 10 -1 1 10 10 2 10 3 10 4 -2 -1 0 1 2 M o d if ie d A to m -D im er S ca tt er in g A m p li tu d e C (q 1 , 0 , -1 /( m a 2 )) [a ] Λ =20Λ 0 , G = 0.132975 10 -1 1 10 1 10 2 10 3 10 4 -2 -1 0 1 2 Λ =25Λ 0 , G =-0.092658 10 -1 1 10 10 2 10 3 10 4 10 5 Outgoing Atomic Wavenumber q 1 [a -1 ] -2 -1 0 1 2 Λ =50Λ 0 , G = -1.00280 10 -1 1 10 10 2 10 3 10 4 10 5 -2 -1 0 1 2 Λ =100Λ 0 , G = -11.1335 Figure 4.4: Renormalisation demonstrated with modified scattering ampli- tude C(q1, 0,− 1 ma2 ). The solid black line in each graph indicates the function C(q1, 0,− 1 ma2 ) evaluated with ultraviolet cutoff Λ0 and G(Λ0) = 0. Solid red line is for a variable cutoff Λ and G = 0, dashed green line is for analytical solution G(Λ), blue and yellow dash-dotted lines are 2×G(Λ) and 2 5 ×G(Λ) respectively. Evidently the numerical solution provides the renormalisation required to shift the phase and amplitude of C back to the Λ0 line. 33 4. Results 10 -1 1 10 10 2 10 3 10 4 10 5 -2 -1 0 1 2 Λ = 250Λ 0 , G = 0.903954 10 -1 1 10 10 2 10 3 10 4 10 5 -2 -1 0 1 2 Λ = 500Λ 0 , G = 0.034749 10 -1 1 10 10 2 10 3 10 4 10 5 -2 -1 0 1 2 Λ =1000Λ 0 , G = -0.776755 10 -1 1 10 10 2 10 3 10 4 10 5 10 6 -2 -1 0 1 2 Λ = 1500Λ 0 , G = -1.80967 10 -1 1 10 10 2 10 3 10 4 10 5 10 6 -2 -1 0 1 2 M o d if ie d A to m -D im er S ca tt er in g A m p li tu d e C (q 1 , 0 , -1 /( m a 2 )) [a ] Λ =2000Λ 0 , G = -4.54618 10 -1 1 10 1 10 2 10 3 10 4 10 5 10 6 -2 -1 0 1 2 Λ =2500Λ 0 , G = 129.267 10 -1 1 10 10 2 10 3 10 4 10 5 10 6 Outgoing Atomic Wavenumber q 1 [a -1 ] -2 -1 0 1 2 Λ =5000Λ 0 , G = 1.16853 10 -1 1 10 10 2 10 3 10 4 10 5 10 6 -2 -1 0 1 2 Λ =10000Λ 0 , G = 0.163500 Figure 4.5: Continuation of figure 4.4 for even greater ultraviolet cutoffs. Although the renormalisation still provides scale-invariance, it should be noted that the zero- range model is not expected to provide accurate predictions for the high-momentum region and it is meaningless to study ultraviolet cutoffs that exceed the natural energy scale of the system by several orders of magnitude. 34 4. Results 10 -3 10 -2 10 -1 1 10 1 10 2 -8 -4 0 4 8 M o d if ie d A to m -D im er S ca tt er in g A m p li tu d e F u n ct io n C (q 1 , 0 , -1 /( m a 2 )) [a 2 ] Scattering Length |a| = 10 4 × a 2 Λ * = 1/900 a 2 -1 , a < 0 Λ * = 1/900 a 2 -1 , a > 0 Λ * = 1/30 a 2 -1 , a < 0 Λ * = 1/30 a 2 -1 , a > 0 Λ * = a 2 -1 , a < 0 Λ * = a 2 -1 , a > 0 Λ * = 30 a 2 -1 , a < 0 Λ * = 30 a 2 -1 , a > 0 Λ * = 900 a 2 -1 , a < 0 Λ * = 900 a 2 -1 , a > 0 10 -3 10 -2 10 -1 1 10 1 10 2 -8 -4 0 4 8 Scattering Length |a| = 10 2 × a 2 10 -3 10 -2 10 -1 1 10 1 10 2 -0.1 0 0.1 0.2 Scattering Length a = 10 -2 a 2 10 -3 10 -2 10 -1 1 10 1 10 2 -4 0 4 Scattering Length |a| = a 2 10 -3 10 -2 10 -1 1 10 1 10 2 Atomic Outgoing Wavenumber q 1 [a 2 -1 ] -0.010 -0.005 0 0.005 0.010 Scattering Length a = -10 -2 a 2 10 -4 10 -2 1 10 2 10 4 Three-Body Parameter Λ * [a 2 -1 ] 0 1 2 3 Atom-Dimer Scattering Length Dependence on Three-Body Parameter Λ * , a = a 2 a AD Figure 4.6: Graphs 1-5 show modified atom-dimer scattering amplitudes for a wide set of diatomic scattering lengths a and a variety of three-body parameters Λ∗. Blue, purple, red and orange lines are for negative scattering lengths a < 0 while teal, green, brown and yellow lines are for positive scattering lengths a > 0. Graph 6 shows the locally linear dependence of the atom-dimer scattering length on the order of magnitude of the three-body parameter. Ultraviolet cutoff Λ = 200a−1 2 for all measurements. 35 4. Results -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 Inverse Scattering Length 1/a [m -1/4 ] -5000 -4000 -3000 -2000 -1000 0 B in d in g W av en u m b er K [ m -1 /4 ] The Efimov Effect: Binding Energy Wavenumber Dependence on Scattering Length Ultraviolet Cutoff Λ = 1.33×10 15 m -1 , Three-Body Parameter Λ * = 9×10 7 m -1 E T (1) E T (2) E T (3) E T (4) E T (5) E T (6) E D Figure 4.7: The six branches of trimers constituent of three identical bosons. The binding wavenumber and inverse scattering length are scaled by a transformation xi → xi ( √ x2 1+x2 2) 3 4 in order to fit all 6 trimer branches. The straight line in the fourth quadrant indicates the threshold to atom-dimer energy favorable systems, and the x axis for negative scattering lengths a < 0 is the threshold for three-atom energy favorable systems. Between these thresholds lies the domain of the bosonic trimer. The energy ET of a trimer branch can be computed as ℏ2K2 m where m is the mass of a given atomic species. 36 4. Results 10 -4 1 10 4 10 8 T ri m er B in d in g E n er g y E T(n ) [ K ] Efimov Effect: Trimer Binding Energies at the Resonant Limit E T (n) 1 10 2 10 4 10 6 10 8 10 10 10 12 10 14 T ri m er B in d in g E n er g y E T(n ) [ E T(1 ) ] Efimov Effect Discrete Scaling Symmetry: Trimer Binding Energies at Resonant Limit E T (n) /E T (1) (λ 2 ) (n-1) 1 2 3 4 5 6 Trimer Branch Number (n) 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 T w o -B o d y S ca tt er in g L en g th a (n ) [ m ] Efimov Effect: Absolute Two-Body Scattering Length at Trimer-Dimer and Trimer-Atom Thresholds a + (n) -a - (n) 1 2 3 4 5 6 Trimer Branch Number (n) 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 1 T w o -B o d y S ca tt er in g L en g th a (n ) [ a (1 ) ] Efimov Effect Discrete Scaling Symmetry: Two-Body Scattering Length at Trimer-Dimer and Trimer-Atom Thresholds a + (n) /a + (1) λ (1-n) a - (n) /a - (1) Figure 4.8: Discrete scaling symmetries and absolute values of trimer binding energies at the resonant limit and threshold scattering lengths. The absolute energy was computed using the mass of He4, and the scale of the temperatures of the deeper trimers far exceeds the energy scales of the ultracold regime necessary for the s-wave approximation to be valid. The threshold scattering lengths also decrease into the infinitesimal, with the lowest reaching a length lower than 10−15 m - the scale of the charge radius of a proton. It’s likely that these deeper trimer branches are not naturally realisable. 1.5 2 2.5 0 50 100 150 200 250 S ec o n d T ri m er -D im er B in d in g E n er g y R at io E T(2 ) /E D Universal Scaling Curve: Trimer-Dimer Binding Energy Ratio, Scattering Length a = a 2 , Comparison of First Two Trimers E T (2) /E D 1.5 2 2.5 First Trimer-Dimer Binding Energy Ratio E T (1) /E D 0 0.5 1 1.5 2 2.5 S ca tt er in g L en g th R at io a A D /a Universal Scaling Curve: Atom-Dimer Scattering Length Ratio, Scattering Length a = a 2 , Compared to Trimer-Dimer Binding Energy Ratio a AD /a Figure 4.9: Universal scaling curves of the ratios aAD a2 , E (1) T ED and E (2) T ED for various Λ∗. Scaling violations are apparent for these values as the three-body parameter Λ∗ is altered. 37 4. Results These scattering lengths have a discrete scaling symmetry determined by the Efimov effect, and of express importance is that deeper trimers require the tuning of ever lower scattering lengths to observe these resonances and their loss effects - if the scattering length is tuned to be shorter than the natural length scale of the studied atomic species, its van der Waals length, the results and predictions of the zero- range model lose their validity. Although the universality of the zero-range model does also predict resonances for very high values of |a|, the binding energy levels for shallower trimers become ever lower and therefore harder to detect in the laboratory. Yet the research of the ultracold regime Bose-Einstein condensate continues, and perhaps in the future more of these bound states will be found, allowing for a better understanding of the strong interactions in bosonic quantum degenerate gases. 4.2.2 Results for Bound State Energy of Distinct Bosons The Efimov effect has been reproduced for mixtures of bosons with distinguishable masses. The mass ratios tested were m1 m2 = {0.9, 2, 5, 10} using equation (3.42), the bound state equation for two atoms of mass m1 and one atom of mass m2, and the scattering length of mixed atoms tuned greater than all other diatomic scattering lengths using Feshbach resonance. The results of the emergent Efimov spectrum trimer branches are shown in figure 4.10. It is demonstrated that increasing the mass ratio for m1 > m2 will produce more tightly spaced trimer branches. For masses m2 > m1, however the discrete scaling factor λ grows and the deeper trimer branches quickly fall outside the scope of the ultracold regime in order of magnitude. Figure 4.11 shows the plot of the solutions s0 of the transcendental equation (3.31) as a function of the mass ratio rm = m1 m2 and the results for the discrete scaling symmetry factors λ of these mixed boson gas systems, confirming that the Efimov effect is satisfied. 38 4. Results -10000 0 10000 Inverse Scattering Length 1/a 12 [m -1/4 ] -8000 -6000 -4000 -2000 0 B in d in g W av en u m b er K [m -1 /4 ] Efimov Effect Mixed Bosons r m = 0.9 -1000-500 0 500 1000 1500 -1500 -1000 -500 0 B in d in g W av en u m b er K [m -1 /4 ] Efimov Effect Mixed Bosons r m = 2 E T (1) E T (2) E T (3) E D -500 0 500 1000 -800 -600 -400 -200 0 Efimov Effect Mixed Bosons r m = 5 -600 -400 -200 0 200 400 600 Inverse Scattering Length 1/a 12 [m -1/4 ] -400 -200 0 Efimov Effect Mixed Bosons r m = 10 Figure 4.10: The binding energy branches of trimers constitutent of mixed bosons, two with mass m1 and one with mass m2, mass ratio rm = m1 m2 and mixed diatomic scattering length a12 tuned greater than self-pair scattering lengths a11, a22. In order to fit the branches, the inverse scattering length and the binding wavenumber are scaled xi → xi ( √ x2 1+x2 2) 3 4 . The dashed straight line indicates the dimer binding energy ED = − 1 mra2 12 . The three-body parameter is set for all graphs as Λ∗ = 9 × 107 m−1. 39 4. Results 0 5 10 15 20 25 30 35 40 45 50 Mass Ratio r m = m 1 /m 2 0 0.5 1 1.5 2 2.5 3 3.5 T ra n sc en d en ta l E q u at io n S o lu ti o n P ar am et er s 0 Transcendental Equation Solution Parameter s 0 Dependence on Mass Ratio r m s 0 (r m ) r m = 0.9, s 0 ≈ 0.327 r m = 2, s 0 ≈ 0.885 r m = 5, s 0 ≈ 1.508 r m = 10, s 0 ≈ 2.011 1 2 3 Trimer Branch Number (n) 1 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 T ri m er B in d in g E n er g y E T(n ) [ E T(1 ) ] Efimov Effect: Trimer Binding Energies at the Resonant Limit E T (n) /E T (1) , r m = 0.9 (λ 2 ) (n-1) , λ = 1499 E T (n) /E T (1) , r m = 2 (λ 2 ) (n-1) , λ = 34.87 E T (n) /E T (1) , r m = 5 (λ 2 ) (n-1) , λ = 8.031 E T (n) /E T (1) , r m = 10 (λ 2 ) (n-1) , λ = 4.770 Figure 4.11: Left: plot of the transcendental equation solution parameter s0 de- pendency on the mass ratio rm. The values of s0 cease to be real at rm ≈ 0.74 and the Efimov effect does not occur for mass ratios at or below this point. Right: the discrete scaling symmetries for the four trimer sets. The ratios of trimer binding energies at the resonant limit correspond well to the scaling factors λ2 = e 2π s0 pre- dicted by the Efimov effect. 40 5 Conclusion From the work outlined in this thesis, we have applied the zero-range model de- veloped in the context of nuclear physics to procure a set of predictions for the three-body physics of Bose-Einstein gases in the ultracold temperature regime. It has been successfully demonstrated for bosons that the model is applicable to the three-body sector using a parameter three-body Λ∗ with which a renormalisa- tion group transformation is performed so that the atom-dimer scattering ampli- tude and the atom-dimer scattering length become scale-invariant. It has also been demonstrated that the result for the atom-dimer scattering length will depend on this three-body parameter. The Efimov effect has been reproduced for bosons and the discrete scaling sym- metries inherent to the Efimov effect have been verified for the trimer branches’ threshold scattering lengths toward the three-atom state and the atom-dimer state, as well as for the binding energy at the resonant limit. A universal scaling curve has been reproduced that demonstrates the scaling violations of the atom-dimer scatter- ing length and the trimer binding energies at the resonant limit. The Efimov effect has also been confirmed to occur for gases composed of mixtures of bosons, and that the mass ratio directly impacts the energy level spacing of the bound three-body state spectrum. In conclusion, this thesis has demonstrated the applicability of the effective field theory known as the zero-range model, with the results parametrised by the spin- statistic, masses, and scattering lengths of the gas atoms with inclusion of an exper- imentally discernible three-body parameter Λ∗ for the renormalisation of the three- body problem, to describe the three-body physics of quantum degenerate gases in the ultracold regime. 41 5. Conclusion 42 Bibliography [1] V. Efimov, “Energy levels of three resonantly interacting particles,” Nuclear Physics A, vol. 210, no. 1, (1973). [2] H.-W. Hammer and L. Platter, “Efimov States in Nuclear and Particle Physics,” Annual Review of Nuclear and Particle Science, vol. 60, no. 1, (2010). [3] K. G. Wilson, “Renormalization Group and Strong Interactions,” Phys. Rev. D, vol. 3, 8 (1971). [4] J. Thomas and M. Gehm, “Optically Trapped Fermi Gases,” American Scien- tist - AMER SCI, vol. 92, (2004). [5] M. Köhl, H. Moritz, T. Stöferle, K. 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Waldburger, et al., “Evidence for Efimov quantum states in an ultracold gas of caesium atoms,” Nature, vol. 440, no. 7082, (2006). [12] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cor- nell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science, vol. 269, (1995). [13] D. B. Spalding, “The Molecular Theory of Gases and Liquids,” The Journal of the Royal Aeronautical Society, vol. 59, (1955). [14] E. Braaten, M. Kusunoki, and D. Zhang, “Scattering models for ultracold atoms,” Annals of Physics, vol. 323, (2008). [15] J. Dalibard, “Collisional dynamics of ultracold atomic gases,” (2002). [16] J. J. Sakurai, Modern quantum mechanics; rev. ed. Reading, MA: Addison- Wesley, (1994). [17] L. D. Landau and L. M. Lifshitz, Quantum Mechanics Non-Relativistic The- ory, Third Edition: Volume 3. Butterworth-Heinemann, (1981), isbn: 0750635398. 43 9904034 Bibliography [18] E. Braaten and H.-W. Hammer, “Universality in few-body systems with large scattering length,” Physics Reports, vol. 428, (2006). [19] P. F. Bedaque, H.-W. Hammer, and U. van Kolck, “Renormalization of the Three-Body System with Short-Range Interactions,” Physical Review Letters, vol. 82, no. 3, (1999). [20] S. Weinberg, The Quantum Theory of Fields. Cambridge University Press, (1995), vol. 1. [21] P. Bedaque, H.-W. Hammer, and U. van Kolck, “The Three-Boson System with Short-Range Interactions,” Nuclear Physics A, vol. 646, no. 4, (1999). 44 A Appendix: Feynman Rules A.1 Zero-range model, two-body sector Figure A.1: The Feynman rules for the two-body sector zero range model includes the retarded causal propagator, the external legs for creation and annihilation op- erators, and the four point vertex. The energy and momenta are conserved on each vertex. For identical bosons a symmetry factor 1 2 is implemented for every loop. Undetermined momenta and energies are integrated over for each loop. I A. Appendix: Feynman Rules A.2 Zero-range model, three-body sector Figure A.2: The Feynman rules for the three-body sector zero range model in- cludes the retarded causal atom propagator, the bare and exact dimer propagators, the external legs for creation and annihilation operators, the three point dimer-atom vertices and the four point dimer-atom vertex. The energy and momenta are con- served on each vertex. For the computation of the exact dimer propagator with identical bosons a symmetry factor 1 2 is implemented for every loop. Undetermined momenta and energies are integrated over for each loop. II B Appendix: Numerical Methods B.1 Gauss-Legendre Quadrature The bound state equation for three-body interactions is a homogeneous Fredholm integral equation: B(n)(q2, ωT ) = 4 π ∫ Λ 0 dk ( 1 2q2k ln (−ℏωTm+ ℏ2(q2 2 + k2) + ℏ2q2k − iϵ −ℏωTm+ ℏ2(q2 2 + k2) − ℏ2q2k − iϵ ) + G(Λ) Λ2 ) × k2 1 ℏ √ (3ℏ2k2 4 − ℏωTm− iϵ) − 1 a B(n)(k, ωT ) = ∫ Λ 0 dkK(k, q2, ωT )B(n)(k, ωT ) Where the scale-dependent three body running constant G(Λ) is for a choice Λ∗: G(Λ) = cos (s0 ln ( Λ Λ∗ ) + arctan (s0)) cos (s0 ln ( Λ Λ∗ ) − arctan (s0)) (B.1) It is expected that there exist energies −E(n) 3 = ℏω(n) such that B(n)(q, ω(n)) are solutions of the bound state equation. These energies are also poles of the three- body scattering amplitude function Al=0. To find these energies it is necessary to set a condition for when the bound state equation is satisfied by the energy. Numerically, it is possible to implement Gauss-Legendre quadrature to approximate the integral as a sum of the function values K(k, q2)B(n)(k) evaluated at specific points, multiplied by corresponding weights: B(n)(q2i, ωT ) = N∑ j=1 wjK(kj, q2i, ωT )B(n)(kj, ωT ) (B.2) The points q2i and kj are abcissae - the roots of the Legendre polynomials P (N)(xj) = 0 while the weights are computed as: wj = 2 (1 − xj)2|∂xP (N)(x)|2xj (B.3) However, these choices of weights and abcissae are only valid for integrals taken on the interval [−1, 1]. The strength of the Gauss-Legendre quadrature method is that III B. Appendix: Numerical Methods for any Legendre polynomial of degree n+1, Pn+1(x) is orthogonal to all polynomials Q(x) of degree n or lower: ∫ 1 −1 Pn+1(x)Q(x)dx = 0. (B.4) If I let the integrated function be factorised as f(x) = Q(x)Pn+1(x) + r(x), f(x) can be a polynomial up to degree 2n+ 1 while the property in equation (B.4) holds for Q(x) and Pn+1(x). Therefore the integral for such an f(x) polynomial up to degree 2n+ 1 on [−1, 1] is:∫ 1 −1 f(x)dx = ∫ 1 −1 Q(x)Pn+1(x)dx+ ∫ 1 −1 r(x)dx (B.5) The first term is zero, meaning to integrate f(x) exactly it is sufficient to integrate the remainder r(x). The remainder has a degree of at most n, else it would have leading terms divisible by Pn+1(x) and therefore part of Q(x). Using n quadrature points the exact integral of r(x) is always found:∫ 1 −1 r(xi)dx = n∑ i=1 wir(xi), (B.6) and using the abcissae xi such that Pn+1(xi) = 0, r(xi) + Q(xi)Pn+1(xi) = r(xi) = f(xi). Therefore for any f(x) that behaves like an 2n + 1 polynomial, the use of Gauss-Legendre quadrature returns an exact integral ∫ 1 −1 f(x)dx = ∑n i=1 wif(xi). To evaluate the solution numerically, it is first necessary to transform the variables into a dimensionless form by casting them in units of two-body scattering length |a|: k̂ = |a|k q̂2 = |a|q2 ω̂T = |a|2ωT Λ̂ = |a|Λ Then the calculation becomes: B(n)(q2, ωT ) = 4 π ∫ Λ 0 dk ( 1 2q2k ln (−ℏωTm+ ℏ2(q2 2 + k2) + ℏ2q2k − iϵ −ℏωTm+ ℏ2(q2 2 + k2) − ℏ2q2k − iϵ ) + G(Λ) Λ2 ) × k2 1 ℏ √ (3ℏ2k2 4 − ℏωTm− iϵ) − 1 a B(n)(k, ωT ) B(n)(q̂2, ω̂T ) = 4 π ∫ Λ̂ 0 dk̂ |a|  |a|2 2q̂2k̂ ln (−ℏω̂Tm+ ℏ2(q̂2 2 + k̂2) + ℏ2q̂2k̂ − iϵ −ℏω̂Tm+ ℏ2(q̂2 2 + k̂2) − ℏ2q̂2k̂ − iϵ ) + |a|2G(Λ̂/|a|) Λ̂2  × k̂2 1 |a|2 1 ℏ|a| √ (3ℏ2k̂2 4 − ℏω̂Tm− iϵ) − 1 a B(n)(k̂, ω̂T ) B(n)(q̂2, ω̂T ) = 4 π ∫ Λ̂ 0 dk̂  1 2q̂2k̂ ln (−ℏω̂Tm+ ℏ2(q̂2 2 + k̂2) + ℏ2q̂2k̂ − iϵ −ℏω̂Tm+ ℏ2(q̂2 2 + k̂2) − ℏ2q̂2k̂ − iϵ ) + G(Λ̂/|a|) Λ̂2  × k̂2 1 ℏ √ (3ℏ2k̂2 4 − ℏω̂Tm− iϵ) − sign(a) B(n)(k̂, ω̂T ) IV B. Appendix: Numerical Methods An important feature of this transformation is that the explicit dependence on the scattering length of the solution has been codified into the sign of the scattering length: sign(a) and on the adjustment of the interval - given any cutoff Λ, the interval of integration now also depends on |a|. Once the solutions corresponding to ω̂T are found the measure of energy measured in Kelvin can be recovered by E(n) = ℏ kB |a|2 ω̂ (n) T . The goal is to discover how these energies depend on the scattering length. It is understood the frequency and momenta have distances measured in |a|. Because of the log-periodic behaviour of the kernel K, the quadrature points should be evaluated on a logarithmic grid. For this reason it is first best to perform the change of variables k′ = log (k + 1), dk dk′dk ′ = ek ′ dk′ and shift the interval from [0,Λ] to [0, ln (Λ + 1)], before finding the Gauss Legendre weights and abcissae for [0, ln (Λ + 1)]. The same transformation should be applied to q2 in order to evaluate the left hand side and right hand side on the same quadrature points. k = ek ′ − 1∫ Λ 0 dkK(k, q2, ωT )B(n)(k, ωT ) = ∫ ln(Λ+1) 0 ek ′ dk′K(ek′ − 1, eq′ 2 − 1, ωT )B(n)(ek′ − 1, ωT ) B(n)(eq′ 2 − 1, ωT ) = 4 π ∫ ln(Λ+1) 0  ln (−ωT m ℏ +e2k′ +e2q′ 2 +ek′+q′ 2 −3ek′ −3eq′ 2 +3−iϵ −ωT m ℏ +e2k′ +e2q′ 2 −ek′+q′ 2 −ek′ −eq′ 2 +1−iϵ ) 2ek′+q′ 2 − ek′ − eq ′ 2 + 1 + G(Λ/|a|) Λ2  × (e2k′ − 2ek′ + 1)√ ek′(3(e2k′ −2ek′ +1)dk 4 − ωT m ℏ − iϵ) − sign(a) B(n)(ek′ − 1, ωT ) Finally we replace the parameter ωT mℏ with the square of the wavenumber KE = sign(ωT ) √ |ωT |mℏ . All variables of the equation are now dimensionless. With the kernel thus newly defined, the solution for B′′(n)(q′ 2, KE) = B(n)(eq′ 2 − 1, KE) can be expressed as the integral on [−1, 1]: k′ = ln(Λ + 1) 2 k′′ + ln(Λ + 1) 2 B′′(n)(q′ 2, KE) = ∫ 1 −1 ln(Λ + 1) 2 dk′′K( ln(Λ + 1) 2 k′′ + ln(Λ + 1) 2 , q′ 2, KE) × B′′(n)( ln(Λ + 1) 2 k′′ + ln(Λ + 1) 2 , KE) The weights are therefore multiplied by the Jacobian factor dk′ dk′′ = ln(Λ+1) 2 and the abcissae are scaled and shifted by ln(Λ+1) 2 . Using these as points q2i, kj = ln(Λ+1) 2 k′′ j + ln(Λ+1) 2 in the discretization of the integral makes makes a linear sys- tem of N equations (B.2). B′′(n)(ki, KE) − N∑ j=1 wjK(kj, q2i, KE)B′′(n)(kj, KE) = 0 (δi,j − N∑ j=1 wjK(kj, ki, KE))B′′(n)(kj, KE) = 0 (idN −M(KE))B′′(n)(kj, KE) = 0 V B. Appendix: Numerical Methods Therefore for those KE where det(idN − M(KE)) = 0, Fredholm equation is sat- isfied. The associated energy indicates the trimer energy which in units of Kelvin depends on a2 as well as the solutions being distinctly different depending on the sign of a. An anolagous procedure is performed for the bound state equation associated with fermions of distinct masses. Mathematica uses the GaussianQuadratureWeights method to find the abcissae and weights for the interval [0, ln (Λ|a| + 1)] where Λ is the van-der-Waals length of helium, and the