Eindhoven University of Technology

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Efimov physics for different two-body interaction models

Rademaker, T.J.

Award date:2014

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Efimov physics for differenttwo-body interaction models

T.J. Rademaker

June 27, 2014 CQT2014-3

Bachelor Thesis

Supervisor:dr. ir. S.J.M.M.F. Kokkelmans

Applied Physics:Coherence and Quantum Technology

Eindhoven University of Technology

Abstract

We predict less universal behavior of the parameters determining the Efimov effect in three-body physics. We solve the Skorniakov-Ter-Martirosian equation and observe five differentmodels that describe two-body interactions. We see that the zero-range model accuratelydescribes the Efimov physics for diverging scattering length. The Efimov bound states arecompletely determined by the scattering length and the three-body parameter. In the modelwith the finite-range potential we find that the cutoff parameter takes over the role of thethree-body parameter and is related to the van der Waals length that describes long-rangebehavior of the potential. We also find unexpected universal physics in a narrow Feshbachresonance. In this model the effective range and the scattering length completely determinesthe Efimov trimers. The most extensive model takes into account the narrow resonance, thefinite-range and the coupling between the open and closed channel. We observe that for largebackground scattering less universal behavior applies; the cutoff parameter is not relatedlinear to the van der Waals length anymore. All models reproduce the universal scaling ratioof 515 between the energies of the bound states. Lastly, we make a start with considerations ofoff-shell two-body interactions. This can potentially lead to new surprising universal featuresin Efimov physics.

Contents

1 Introduction 11.1 Few-body physics and the Efimov effect . . . . . . . . . . . . . . . . . . . . . 11.2 Parameters in few-body physics . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Solving the Efimov bound states . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Outline report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Ultracold two-body interactions 72.1 Scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Off-shell scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Coupled channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Efimov physics 133.1 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Three-body recombination rate . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Skorniakov Ter-Martirosian equation . . . . . . . . . . . . . . . . . . . . . . . 16

4 Model setup 194.1 Three-body scattering matrix in momentum space . . . . . . . . . . . . . . . 194.2 Two-body scattering amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Separable scattering amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Results 255.1 Comparison between several two-body interactions . . . . . . . . . . . . . . . 255.2 Off-shell two-body scattering matrix . . . . . . . . . . . . . . . . . . . . . . . 295.3 Recombination rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Conclusion 31

A Angular averaging 35

B Additional figures for Efimov bound states 37

Chapter 1

Introduction

1.1 Few-body physics and the Efimov effect

The Efimov effect has theoretically been predicted by Vitaly Efimov in the early seventies[Efimov, 1970], [Efimov, 1971]. It describes how an infinite number of bound states arises in athree-body system for diverging scattering length, due to resonant ultracold two-body inter-actions. Arbitrarily close to the Feshbach resonance, where the scattering length a → ±∞,an accumulation of weakly-bound states is formed. We can regard these three-body boundstates as three Borromean rings. These rings are bound together as long as all rings areclosed. When one particle is removed from the bond, or equivalently when one ring is cut,the other two atoms, rings, are not bound together anymore. One may call this some sort ofentanglement.

The Efimov effect only appears when the interparticle interaction becomes very strong.The energy of the Efimov trimers falls off like −1/r2. This is compensated by the potentialof the centrifugal barrier which falls off like 1/r2. In the ultracold limit there is no centrifugalbarrier, which is the main reason that in the ultracold regime the most favorable conditionsto observe this effect can be reached. Also, the control over the parameters like the scatteringlength is the best, as the particles can be easily trapped and manipulated.

Long since Efimovs prediction, it was impossible to observe the Efimov effect, becauseatoms could not be cooled to a temperature of several nK. In 1995, E.A. Cornell and C.E.Wiemann observed a Bose-Einstein condensate (BEC) for which they received the Noble Prizein 2001. Using combined laser and magnetic cooling equipment and evaporative cooling, thetemperature of the atoms is reduced to such a degree that the atoms’ wave-nature dominatesover their particle-nature. All waves ‘sense’ each other, and they start to behave like anenormous, single superatom. In 2006, a group in Innsbruck used these cooling techniques onthree cesium atoms in the gas-state, placed in a vacuum chamber. On the very edge of con-densation the group observed the typical recombination rate, modified by the existence of thefragile Efimov bound states [Kraemer, 2006]. Their breakthrough brought renewed attentionto the Efimov effect these last years, manifesting itself in better predictions by theorists forthe observations. The experimentalists try to verify these new predictions, which goes on andon, until the process of few-body physics is completely understood.

1

CHAPTER 1. INTRODUCTION

Figure 1.1: Energy of the Efimov bound states for their wavenumber K in terms of 1/a. They arescaled by a power of one fourth so more binding energies can be shown in one graph. The AAA and ADindicate the domain where three free atoms and where an atom and a dimer are allowed respectively.The boundaries of the regions are indicated by the stripes. The two-body threshold is the boundarybetween the AD and T region and falls off like 1/a2. The free-atom threshold is for negative scattering

length on the x-axis. The three-body parameters a(−)0 and κ∗ indicate respectively the scattering length

at zero energy and the energy of the trimer at diverging scattering length from the deepest bound state.The figure is taken from [Braaten and Hammer, 2006]

.

2

1.2. PARAMETERS IN FEW-BODY PHYSICS

In Fig. 1.1 the characteristic for the trimer binding energies is shown. The axes arescaled by a power four, in order to show more trimer states in one figure. The scaling be-tween two consecutive energies at the point of diverging scattering length on the y-axis isE(n)

E(n+1) → e2πs0 ≈ 515 where s0 = 1.00624 is a constant. Also, the scaling between the start-

ing points of all trimers on the x-axis converges to a constant, namely k(n)

k(n+1) → eπs0 ≈ 22.7.

1.2 Parameters in few-body physics

In Fig. 1.1 there are seemingly an infinite number of bound states both in the accumula-tion point of diverging scattering length and for decreasing energy K to −∞. The latter isnonphysical, because it states that when the range of interaction decreases (a decreases), thebinding energy increases; there is no ground state. This can be done an infinite number oftimes, yielding tighter and tighter systems. The Englishman [Thomas, 1935] called this theThomas collapse, which is caused by the use of a zero-range model for a scattering process.This is artificially avoided by setting a lower limit on the bound-state energies. [Fedorov andJensen, 2001] introduced a short range length scale called the three-body parameter κ∗. When

[Kraemer, 2006] observed Efimov trimers in the laboratory, it became easier to measure a(−)0

than κ∗, and because κ∗ and a(−)0 are both parameters that belong to the same Efimov trimer,

a(−)0 is from that point on called the three-body parameter. a

(−)0 accounts for the short-range

three-body physics and secures the ground state of the three-body system. In recent years, ithas come to attention that the three-body parameter also seems to be a universal parameter,independent of the short-range details of the potential. It scales proportional with a value of∼ 9.8 to the van der Waals length, a length scale related to the long-range nature of the vander Waals potential.

Another parameter to describe the three-body system more accurately is the effectiverange Re, which is related to the resonance width near the Feshbach resonance. It has beenintroduced by [Petrov, 2004]. Petrov showed that the system is completely determined bytwo parameters, the effective range Re and the scattering length a. Last years it becameclear that the zero-range model, which only incorporates the contact-potential, introducesanother problem in the description of the binding and collision process in narrow Feshbachresonances. The contact-potential is rather nonphysical, because it states that particles onlyinteract when they establish themselves on exactly the same spot. Particles occupy space,therefore a collision takes place when particles are near each other, before they are on top ofeach other.

[Thøgersen et al., 2008] introduced the finite-range potential, as a correction on the contactpotential. The finite-range brings along a third parameter besides the scattering length andthe effective range. It is called the cutoff parameter Λ and it describes the range of attractionor repulsion in collisions. In the intermediate regime of a narrow Feshbach resonance, wherethe scattering length is only slightly larger than the effective range, less universal behaviourof the Efimov bound states occurs. With finite-range corrections, the shifts in universalityare trying to be explained, although it is an open issue whether the corrections may explainthe experimental findings.

3

CHAPTER 1. INTRODUCTION

1.3 Solving the Efimov bound states

So far, several methods have been introduced to solve the Efimov bound states. One of thesefocuses on reparameterizing the system in terms of hyperspherical coordinates in real spaceby using the hyperspherical adiabatic approximation, and afterwards solving the Faddeevequations. This has first been proposed by [Fedorov and Jensen, 1993]. Another method isto solve the system in momentum space, which is more natural, because in the characteristicfigure (Fig. 1.1, the energy of the bound states is expressed in terms of the wavenumber K.Integration of the effective range into the system goes naturally through a better approxima-tion of the scattering amplitude.

A few possible solution strategies that have been applied are these of [Petrov, 2004],[Massignan and Stoof, 2008] and [Pricoupenko and Lasinio, 2011]. Petrov, who revealed novelphysics of three-body collisions by introduction of the effective range, solves the Schrodingerequation in the center of mass reference frame of three bosons. He defines an operator LEwhich determines the system in Eq. 9 of this paper. By making a simplification to solve thisequation he obtains well-known results in the region of small detuning, and when insertingthis solution again into the equation, he obtains new corrections, valid in the region whereRe ∼ a. [Massignan and Stoof, 2008] also study narrow Feshbach resonances, but they startfrom the Lipmann-Schwinger equation. They define a three-body scattering matrix from atwo-channel model, which includes the two-body matrix T2. By separating the collisions of theT2 matrix into background and transition processes new information can be extracted fromthe scattering process. Due to the pole structure of the resulting energy dependent T2(E)matrix, the Efimov physics is reproduced across the whole resonance. This model is ideal todescribe three-body physics in narrow Feshbach resonance with a large background scatter-ing length. The last method described here is from Pricoupenko. By analyzing a separabletwo-channel model, and considering the scattering amplitude f(E) of the two-body system,a model for the three-body recombination rate is recovered, which includes the backgroundscattering length. Like [Massignan and Stoof, 2008], Pricoupenko has focused on narrow Fes-hbach resonances in the region where a shallow dimer exists.

The Pricoupenko model is explored further in this report. Levinsen [2013] has providednotes and a Mathematica notebook, with which several two-body interactions can be anal-ysed. Through numerics and input for the two-body scattering matrix the Efimov bound

states, the universal three-body parameter a(−)0 and the scaling factor between two bound

states are computed accurately.

4

1.4. OUTLINE REPORT

1.4 Outline report

The report is structured in the following way. First, we study scattering theory and Feshbachresonances (Chapter 2) and Efimov physics (Chapter 3) extensively to solve the Efimov boundstates in momentum space. We describe five different models for two-body interactions incor-porating 1 to 4 parameters, in Chapter 4. The models are valid in different regimes for Re, Λand abg. We predict and discuss how well these models are able to reproduce the three-bodyparameter and the universal scaling in Chapter 5. Lastly, in this chapter directions for furtherresearch are given to evaluate two-body off-shell interactions and to find the recombinationrate from this interaction potential.

5

CHAPTER 1. INTRODUCTION

6

Chapter 2

Ultracold two-body interactions

The Efimov bound states are three-body bound states. To understand what happens whenthree ultracold particles interact with each other, we first need to master ultracold two-bodyinteractions, because the interaction that determines three-body physics is two-body scatter-ing. The main interaction between two particles at low temperature has a short-range natureand will be determined by low-energy collisions. This chapter is restricted to describing theselow-energy collisions (Section 2.1), deriving an off-shell two-body scattering amplitude (Sec-tion 2.2), interpreting the two-channel concept (Section 2.3) and understanding the Feshbachresonance in Section 2.4. After this chapter, the interactions between two particles with ap-proximately zero kinetic energy should be clear, because the report is structurally progressingon these concepts.

2.1 Scattering theory

For the introduction and explanation of the scattering concepts, we follow [Braaten and Ham-mer, 2006]. For a more detailed description of scattering theory, we recommend Taylor [1987]or Sakurai [1993].

The quantum mechanical scattering can be understood by considering two particles withseparation distance r, opposite momenta p = ±~k and equal kinetic energy E = ~2k2/mwhere k and m denote the wavenumber and mass of the particles. The stationary wavefunction ψ(r) takes into account the elastic scattering process. For r→∞, the particle-wave,propagating in the z-direction is described by a wave function consisting of an unaffectedincoming plane wave eikz and an outgoing spherical wave eikz/r, given by

ψ(r) = eikz + fk(θ)eikz

r, (2.1)

where fk(θ) secures the scattering amplitude, dependent on the wavenumber k and theangle θ at which collision has taken place. The larger fk(θ), the larger the effect of thescattering process on the wave function. Then ψ(r) behaves more and more as the sphericalwave that came to existence after the collision. The scattering amplitude may be decomposedinto the contribution of an infinite number of discrete angular momenta L by expanding it in

7

CHAPTER 2. ULTRACOLD TWO-BODY INTERACTIONS

terms of the Legendre polynomials PL(cos θ). This decomposition is called the partial-waveexpansion and it reads

fk(θ) =1

k

∞∑L=0

(2L+ 1)cL(k)PL(cos θ). (2.2)

The coefficients cL(k) are constrained by the unitary condition, that is |cL(k)| ≤ 1. Thephase shifts δL(k) follow in this way from

cL(k) = eiδL(k) sin δL(k) =sin δL(k)

cos δL(k)− i sin δL(k)=

1

cot δ(k)− i. (2.3)

For ultracold interactions only S-wave interactions are taken into account, thus reducingthe decomposition of fk(θ) into only one term, namely the one with lowest angular momentumL = 0. By inserting the coefficient c0(k) in terms of δ0(k) from Eq. 2.3 into the scatteringamplitude and noting that P0(cos θ) = 1, the scattering amplitude in Eq. 2.2 simplifies to

fk(θ) =1

k cot δ0(k)− ik. (2.4)

In the remainder of the report, δ0(k) will be called δ(k), omitting the 0, because onlyS-wave interactions with angular momentum L = 0 are to be considered in the ultracoldlimit. If the atoms are identical fermions, for which S-wave interactions are forbidden, P-wave scattering dominates in the low-energy limit. Considering all the previous, the mostimportant parameter in Efimov physics, the scattering length a, can be defined in the low-energy limit of k → 0 as

limk→0

fk(θ) = −a. (2.5)

This is the first order term of the so called effective range expansion, where the phase shiftδ(k) is expanded in terms of k2

k cot δ(k) = −1

a+

1

2Rek

2 − 1

4P ∗k4 (2.6)

where the parameter Re is called the effective range and P ∗ is called the shape param-eter. The physical meaning of the scattering length can best be understood by noting thatlow-energy particles are not actually going to touch, when interacting, since their de Brogliewavelength is too large. They will attract each other in some potential, then scattering takesplace and the particles move away from each other. The precise scattering potential is unim-portant in this process. The scattering length is a measure which tells on what length scalethe details of the potential becomes important. The effective range describes on what lengthscale the range of the potential is and lastly the shape parameter P ∗ predicts the shape of thepotential. From the effective range expansion in Eq. 2.6 we see that in the lowest energy limitonly the scattering length determines the system, because the effective range runs quadratic

8

2.2. OFF-SHELL SCATTERING

in k. In this report, only Re and a are considered important parameters; P ∗ is neglected.

Another important concepts in scattering theory are the scattering matrices. The S-matrixcontains the information present in the scattering process. This unitary matrix is defined as

S(k) = e2iδ(k) (2.7)

where the factor 2 is for conventional purposes. It is linked to the scattering amplitude asf(k) = S(k)−1

2ik which might have become clear from Eqs. 2.3 and 2.4. The T-matrix is relatedto the S-matrix as

T (k) =S − 1

2πi(2.8)

and will be called the scattering matrix throughout the report, although strictly speakingit is a transition matrix. The T-matrix takes in a central place in this report, as will becomeclear in Chapter 4.The Efimov bound states will namely be solved by a numerical model thatis built on an implicit integral equation in the three-body scattering matrix T3(k).

2.2 Off-shell scattering

A brief derivation of the off-shell two-body scattering matrix of [van Dongen, 2014] is givenbelow. It makes use of known concepts in scattering theory like the phase shift δ and thescattering length a.

A scattering process may be represented by a square well with a certain depth V0 andwidth r0. One may see this process as one particle, standing still in the origin, and anotherparticle, approaching the origin from far away with wavenumber k and returning to r → ∞with wavenumber k’ 6= k. From Sakurai [1993], in the most general case where inelasticcollisions are taken into account, the wave function of the scattering process is given by

ψ(k, k’) =1

2ikr0(eikz + f(k’,k)eik’z (2.9)

and the scattering amplitude of the process is given by

f(k’,k) = −1

4(2π)3 2m

~2〈k’|V |ψ(+)〉. (2.10)

where ψ(+) is the wave function of the outgoing spherical wave. An schematic representa-tion of the process described by Eqs. 2.9 and 2.10 is given in Fig. 2.1. This implicit equationis the Lipmann-Schwinger equation, which is a solution of the time-independent Schrodingerequation. One solution strategy uses an iterative procedure to find the off-shell scatteringamplitude f(k’,k). This strategy is called the Born approximation. We are not going to usethis approach. Instead, we try to find the wave function of the outgoing wave ψ(+). Considerthe quantum well in Fig. ??. The wave function ψ(+) (during the) scattering off the well isgiven by

9

CHAPTER 2. ULTRACOLD TWO-BODY INTERACTIONS

Figure 2.1: The process of off-shell two-body scattering described by the Lipmann-Schwinger equation.In the region where scattering takes place a particle may scatter a great number of times, although theprobability that this happens decreases expontentially. Figure is taken from [Sakurai, 1993]

ψ(+)(r, k, k’) =

A eiδ

r sin(kr) r ≤ r0

1(2π)3/2

1kr sin(k’r + δ) r > r0

where δ is the phase shift after scattering and A is the amplitude of the wave inside thesquare well. Both parameters can be found by considering the continuity conditions at r = r0.Filling in ψ(+)(r ≤ r0, k, ’) in the equation for the off-shell scattering amplitude f(k’,k) (Eq.2.10) and calculating the Fourier transform over the width of the square well results in

f(k’,k) =ei(arctan(k(r0−a))−kr0) (k’(r0 − a) cos(k’r0)− sin(k’r0))

k’(

1 + k2+k’2

mV0 ~2

)√1 + k2(r0 − a)2

. (2.11)

2.3 Coupled channels

In Fig. 2.2 the potentials governing the collision process are schematically represented.At large distance two free atoms approach each other, due to an attractive, long-rangeV (r) ∝ 1/r6 van der Waals potential, with a total energy E larger than the dissociationthreshold or rather called collision threshold εthr = εα + εβ. The hyperfine and angular mo-mentum quantum numbers of the atoms determine the collision channels. In the remainderof the report, we will assume there is only one open channel. Due to the Zeeman effect, themagnetic field strength influences the total energy of a hyperfine state, e.g. εthr. Thus, theenergy of the bound states in both the open and closed channels may be tuned by applyingan external magnetic field.

At short distance, where the collision takes place, and when the closed channel happensto have a bound state at εQ near E, a transition between both channels is allowed, because

10

2.4. FESHBACH RESONANCES

Figure 2.2: Schematic representation the potentials of a coupled open and closed channel. Figure istaken from Kokkelmans [2014]

there is resonant coupling between the channels. In this region a so called Feshbach resonanceappears, in the point of diverging scattering length. After the transition, one of the atoms istrapped in the potential of the energetically closed channel. This is the case if an inelasticcollision takes place and energy is passed on from one atom to the other. This process isevaluated off the energy shell, abbreviated to off-shell.

The focus on this report will be on evaluating off-shell processes, extending the case whereonly on-shell collisions are taken into account for solving the Efimov bound states. For theultracold two-body interactions, only the Feshbach resonance needs to be discussed, whichwill be done in the next section.

2.4 Feshbach resonances

The Feshbach resonance is a multi-channel resonance, occurring when the background scatter-ing of the open channel and the resonant scattering of the closed channel interfere with eachother. The channels have different spin configurations and therefore an external magneticfield B may be used to tune the scattering length a(B) of a collision process. The scatteringlength is the diverging parameter in a Feshbach resonance with a characteristic shape givenby

a(B) = abg

(1− ∆B

B −B0

)(2.12)

where abg is the background scattering length, ∆B represents the width of the Feshbachresonance and B0 is the field of the resonance. In Fig. 2.3 the characteristic shape is plot.The scattering length can be tuned to arbitrary large values. The results in this report arevalid in the regime of large scattering lengths, where a >> Re (small detuning), and in theregime of intermediate detuning, where a ∼ Re and ∆B is small.

11

CHAPTER 2. ULTRACOLD TWO-BODY INTERACTIONS

Figure 2.3: Characteristic shape of the feshbach resonance. At the field of resonance B0, a → ±∞,far away from this field of resonance, a → abg. ∆B determines whether width of the resonance. Thesmaller ∆B, the narrower the Feshbach resonance.

12

Chapter 3

Efimov physics

In this chapter we will describe the surprising universal features of Efimov physics (Section3.1) and also the less universal features on which we focus in this report. We will derive thethree-body recombination rate in Section 3.2 (which is measured in the laboratory) and givethe reader the structure of the derivation of the STM-equation in Section 3.3 (used for solvingthe Efimov bound states in momentum space).

3.1 Universality

The bound states have universal features. Universal features should immediately grab the at-tention of every theoretical physicist, because the predicted effects are valid for many systemsand not only for one special case. For the Efimov bound states this means that the interactionpotential is described by a few parameters, such as the scattering length a and the three-bodyparameter κ∗. The details of the potential are irrelevant, which means that the results arevalid for a variety of systems. The universality of the Efimov physics manifests itself in thedifferent fields of physics it applies to. The theoretical prediction has been expounded byEfimov in the field of nuclear of physics. 40 decades later, the experimental observation havebeen done by Kraemer in the field of ultra-cold gases. So Efimov physics is applicable in boththe field of nuclear physics and the ultra-cold gases.

An example of universality concerns the universal binding energy of atomic particles. Con-sider e.g. a helium-4 atom and a neutron. The van der Waals force takes care of the bindingsenergy of the helium-4 atom with a characteristic scattering length a dozen nanometer. Theneutron is bound by the much stronger nuclear force between the quarks, for which the scat-tering length is several femtometer. The binding energy Eb → ~2

ma2for both the helium-4

atom and the neutron, though these systems vary some several orders of magnitude. This iswhat universality in physics comprises: a result valid for a variety of systems without knowingor being interested in the exact details of these systems.

A parameter which describes less universal behavior of Efimov physics is the effective rangeRe from the effective range expansion of Eq. 2.6. In a Feshbach resonance Re is linked to the

13

CHAPTER 3. EFIMOV PHYSICS

Figure 3.1: Universality occurs mostly at one side of the resonance, not across. All three figuresare experimental observations of the recombination rate L3 compared to theoretical predictions. In theupper part of the figure, we note that for small negative scattering length the behavior is not predictedcorrectly. In the middle figure there is universality at both sides of the figure resonance and in the lowerfigure show that for large scattering lengths the recombination rate is not predicted correctly comparedto the observations. Figures are results from [Zaccanti et al., 2009] (upper figure), [Gross et al., 2009](middle figure) and [Pollack et al., 2009] (lower figure)

resonance width R∗. They are defined as

R∗ = 2Re =~2

mabgµB∆B(3.1)

where the factor 2 is for conventional purposes in the effective range expansion. m isthe mass of the two-particle system, abg is the background scattering length, µB is the Bohrmagneton and ∆B is the width of the resonance. In a narrow Feshbach resonance, where ∆Bis small, Re is large, and less universal behavior applies. This regime is called the regime ofintermediate detuning. The effective range is necessary in two-body interactions to determinethe interaction potential. From Fig. 3.1 becomes clear that at the edges of the figure, the zero-range model does not describe the few-body system accurately anymore. A surprising featureof universality arose after the introduction of the finite-range correction by [Thøgersen et al.,2008]. In the zero-range model, the introduction of the three-body parameter was necessaryto find a lower bound state, a ground state for the Efimov trimers. The cutoff parameter Λ,which describes the finite-range potential, could be linked to the three-body parameter. Withthe introduction of the finite-range [Thøgersen et al., 2008] killed two birds with one stone,because the finite-range potential makes more physical sense than the zero-range model, andit also replaces the role of the three-body parameter κ∗.

14

3.2. THREE-BODY RECOMBINATION RATE

Figure 3.2: In the red box the recombination process is depicted schematically. The recombination ofEq. 3.2 is an inelastic collision between three identical particles where a dimer is formed and kineticenergy is released, which means the atom flies away fast. The figure is an adjusted version of a figureof the review of [Grimm and Ferlaino, 2010]

3.2 Three-body recombination rate

The three-body recombination rate αrec is an indirect measure for the number of Efimovstates formed. This is, because an Efimov state is a long-lived state, which will recombine ona certain moment, according to the process of identical particles A in the following inelasticcollision,

A+A+A→ A2 +A, (3.2)

which is schematically represented in the red box of Fig. 3.2. Here, A represents an atomand A2 a dimer. Before the collision, the three particles are free particles with no bindingenergy, while after the collision the dimer is formed at the two-body threshold (the blue line)and the released binding energy is transformed into kinetic energy of the dimer and the atom.Using the calculated recombination rate from [Esry et al., 1999]

K3 =k

3!µσrec (3.3)

where the factor 3! comes from recombining atoms in a BEC, theoretically predicted by[Kagam et al., 1985] and experimentally verified by [Burt, 1997]. k is the wave number of theincoming energy, µ = m/3 is the reduced mass of the three-body system (m is the mass of

15

CHAPTER 3. EFIMOV PHYSICS

one atom) and σrec is the recombination cross-section, given by

σrec =1152π2

k5|SA+A+A→A2+A|2. (3.4)

[Esry et al., 1999] derived this quantity from Fermi’s Golden Rule. The three-body scat-tering process A2 + A → A2 + A is the actual process which is being studied. The onlyinelastic channel of this process is A2 + A → A + A + A, which is exactly the time-reverseof the process we have studied in the aforementioned. Therefore, the imaginary part of thephase shift, which is related to inelastic scattering, determines the recombination rate. In thetime-reversal a multiplication factor 3/4 is added, which is explained by the conservation offlux at large distance from a two-body system to three-body system.

For theoretical considerations, it is more convenient to introduce the true recombinationrate αrec which is defined in terms of K3 as

αrec =3

3!K3. (3.5)

The factor 3 is explained by noting that both the dimer and the atom are ejected from thecondensate during recombination and the 3! is there to avoid double counting. Consideringall the above from Eqs. 3.3 - 3.5, αrec can be written as

αrec =3

6· 3

4· 1152π2

6 k4m/3(1− |SA2+A→A2+A|2) =

216π2

mk4(1− e−4δim) (3.6)

where δim is the imaginary part of the phase shift. The expression for αrec, resulting fromthis transparent derivation, is proportional to a4. Also, it shows log periodic behavior in a,because of the scattering length dependence in the phase shift. The resulting atom loss rateis given in three dimensions by

n = −αrecn3 (3.7)

where n denotes the density of the condensate.

3.3 Skorniakov Ter-Martirosian equation

Some words are spent on the derivation of the STM-equation. For completeness we recom-mend the reader to read the original proof [Skorniakov and Ter-Martirosian, 1957] or notesfrom Bruun [2013].

The derivation starts with a diagrammatical approach. In Fig. 3.3 the atom-dimer scat-tering only valid for identical bosons is schematically represented. Note that this process isessentially a ladder operator, which reminds us of the Lipmann Schwinger equation. Theresulting scattering matrix T3 indeed satisfies the Lipmann Schwinger equation. In analyticalform the equation for atom-dimer scattering reads

T3(k, k’, P ) = G(P − k− k’)− 2i

∫d3p

dΩ

2π4G(P − k, k’)G(p)T2(P − p)T3(p, k’, P ) (3.8)

16

3.3. SKORNIAKOV TER-MARTIROSIAN EQUATION

Figure 3.3: Schematic representation of the atom-dimer scattering process to find the three-bodyscattering matrix T3.

where k is the momentum of the incoming particles, k’ is the momentum of the outgoingparticles, P is the total momentum of the incoming particles and p is an integration variable.Also, G(k) = (E−k2/2m+i0)−1 is the bare Green’s function that is evaluated in the complexplane. The dependence of the frequency Ω can be integrated out by using complex analysisand closing the integration contour in the lower half-plane. Both T2(P − p) and T3(p, k’, P )are analytical functions of Ω in this region, therefore the integral is given by the sum of theintegrands residue in its poles. Only the pole in E = p2/2m is found in the lower half plane.When taking the above into account, the resulting integral equation reads

T3(k, k’, E) =1

E + i0− k2/m− k’2/m− kk’/m+

2

∫d3p

(2π)3

1

E + i0− k2/m− p2/m− kp/mT2(E − 3p2/4m)T3(p, k’, E) (3.9)

where E + i0 is the total energy in the center-of-mass reference frame, approaching thereal axis in the complex plane from above. This is the general result of [Skorniakov andTer-Martirosian, 1957]. The T3(k) scattering matrix consists of the first term that describesnon-scattering, and the integral term that considers (repeated) atom-dimer scattering. Fora total trimer energy E3 ≤ 0, the non-scattering term disappears. This is explained bynoting that the homogeneous Lipmann-Schwinger equation for bound states consists only ofa scattering state. We recall Eq. 2.1 where the wave function after the collision process isthe sum of the incoming plane wave and the outgoing spherical wave. For bound states theresimply is no such thing as an incoming plane wave, so it should neither be present in thescattering matrix. Taking on-shell input where k’ = k for Eq. 3.9 and neglecting the firstterm on the r.h.s. results in

T3(k,E) = 2

∫d3p

(2π)3

1

E − k2/m− p2/m− kp/mT2(E − 3p2/4m)T3(p,E). (3.10)

17

CHAPTER 3. EFIMOV PHYSICS

18

Chapter 4

Model setup

The Efimov bound states may be found from the three-body scattering matrix T3(k) by solvingthe STM equation. The STM equation gives an expression in momentum space for T3(k) interms of the two-body scattering matrix T2(k). The numerical model to solve this equationis explained in Section 4.1. Assuming the on-shell condition, several two-body scatteringmatrices can be used to solve the Efimov bound states. These are the following, listed withincreasing complexity, being characterized by an increasing number of parameters:

I Contact potential; scattering length a.

II Effective range expansion; narrow resonance characterized by a and Re

III Finite-range potential; broad resonance and Gaussian cutoff characterized by a and Λ

IV Narrow resonance and Gaussian cutoff; a, Re and Λ

V Narrow resonance, Gaussian cutoff and background scattering; a, Re, Λ and abg

In Section 4.2 the t-matrices are described for each of the models and in Section 4.3 theseparability of the t-matrices is described.

4.1 Three-body scattering matrix in momentum space

Following Levinsen [2013], define the off-shell two-body scattering matrix T2(E + i0) withinitial momenta ki and final momenta kf at a total energy E

〈kf |T2(E + i0)|ki〉 = ξ(kf )ξ(ki)t(E + i0) (4.1)

where ξ(k) = e−k2/Λ2

is the Gaussian cutoff and Λ is the cutoff parameter. The scatteringamplitude is obtained by considering the on-shell condition where k = |ki| = |kf | =

√mE

and m the mass of the system

f(k) = −m4πξ2(k)t(k2/m) =

1

k cot δ(k)− ik. (4.2)

19

CHAPTER 4. MODEL SETUP

We note that the off-shell T2(E+i0)-matrix consists of an energy dependent scattering am-plitude and a momentum dependent Gaussian cutoff. By splitting T2(E + i0) the SkorniakovTer-Martirosian equation from Eq. 3.10 is represented in the following way:

t−1(E − 3εk/2)T3(k) = 2

∫d3p

(2π)3

ξ(|k− p/2|)ξ(|p− k/2|)E − k2/m− p2/m− k · p/m

T3(p). (4.3)

Here T3(k) is the three-body scattering matrix and εk = k2/2m is the dispersion. p standsfor the momentum, regarded as the integration variable. Angular averaging of Eq. 4.3 resultsin the closed-form integral equation

t−1(E−3εk/2)T3(k) =m

2π2

∫dp ξ(|k−p/2|)ξ(|p−k/2|) p

klog

E − p2/m− k2/m+ k · p/mE − p2/m− k2/m− k · p/m

T3(p).

(4.4)

For details on this calculation, see the Appendix A. Note that T3(k) is on both sides ofthe equation: Eq. 4.3 is an implicit equation in T3(k). This kind of equation is a numericallysolvable homogeneous Fredholm equation of the second kind. In Chapter 19 of the bookNumerical Recipes by Press and Flannery [2007] an extensive explanation is given about howto solve this type of implicit integral equation.

Because the l.h.s. of Eq. 4.3, t−1(E−3εk/2), has both a term dependent on the scatteringlength 1/a and a momentum dependent term it can be split up. We will give an examplefor the easiest case, where the closed channel potential corresponds to a contact potential,thus being the most universal, because this model is described by one parameter. Thenk cot δ(k) = −1/a, and we may write t−1(k2/m) as

t−1(k2/m) =m

4πξ2(k)(1/a+ ik). (4.5)

Now separate the 1/a term and ik term in t−1(E− 3εk/2) and insert the energy E = −3q2

4to find

ξ2(k)1

aT3(k) = ξ2(k)

√3

2

√k2 + q2 T3(k)+

2

π

∫dp ξ(p)ξ(k)

p

klog

3q2/4 + k2 + p2 − k · p3q2/4 + k2 + p2 + k · p

T3(p)

(4.6)

where the simplification ξ(|k − p/2|)ξ(|p − k/2| = ξ(p)ξ(k) and ξ(k) = e−k2/Λ2

is used.Due to universality, the choice of the cutoff function should not influence the results.

The Fredholm equation can be solved by considering the eigenvalues of the kernel. Theycorrespond to the binding energies of the Efimov bound states. The kernel consists of thelog term and the split t-matrix on the r.h.s. consists of the momentum dependent term

ξ2(k)√

32

√k2 + q2. Choosing an appropriate grid for the momenta k (namely the often used

and most convenient Gaussian Quadrature Weights) with nk grid points and looping through

20

4.2. TWO-BODY SCATTERING AMPLITUDES

the energy-grid q will result in the well known Efimov bound states. For every energy q, wefind nk eigenvalues. The eigenvalues correspond to the reciprocal of the scattering length ofthe energy of a certain Efimov binding. These energies on the y-axis are mapped onto thex-axis and in this way the Efimov bindings are built up. Usually when solving equations, apoint on the x-axis is mapped onto the y-axis. Now it is done exactly the other way round,because the term with 1/a on the l.h.s. represents the quantity on the x-axis.

In the paragraph above subtle physical and mathematical reasoning is applied, concerningthe Gaussian cutoff ξ(k). In physical terms, the zero-range model is artificially transformedinto a finite-range potential by the existence of the Gaussian cutoff. One would desire tostay with the zero-range model, but the integral equation to find T3(k) is only defined fora certain cutoff function ξ(k). Mathematically, the cutoff function determines the momen-tum grid. One energy layer at a time is projected onto the momentum grid 1/a scaled by ξ2(k).

Another way to approach this problem is by considering the interaction potential in mo-mentum space and real space. The zero-range model, contact potential, or simply the deltafunction, has a constant Fourier transform

1√2π

∫ ∞−∞

δ(x)eikxdx =1√2π

(4.7)

through all of momentum space. To calculate the T3(k) matrix there has to be a cutoff atsome point in momentum space. Because of its implicit integral of form, this is the only waythe T3(k) matrix is defined. A rectangle function in momentum space corresponds to a sincfunction in real space. For a large enough cutoff parameter Λ, the sinc function is approxi-mately a Gaussian cutoff, which in its turn becomes a delta function in the limit of Λ→∞.A fine coincidence is that the Fourier transform of a Gaussian function is a differently-scaledGaussian function. For the scattering process this means that the interaction potential inreal space follows a Gaussian function and that the way the particles are scattered is beingfiltered by a Gaussian function on the magnitude of their momentum.

4.2 Two-body scattering amplitudes

In this section the scattering amplitudes are found for five closed channels potentials. Westart with the most extensive description of the two-body interaction, where four parametersare taken into account, namely the scattering length a, the effective range Re, the finite-rangepotential with Gaussian cutoff Λ and the background scattering length abg, which secures thecoupling between an open and closed channel). Model V may also be interpreted with anenergy dependent scattering length a, which takes into account the momentum dependenceof the closed channel:

a(k) = abg

(1− ∆B

B −B0 − k2/mµrel

). (4.8)

This is indeed a momentum dependence correction term on the scattering length from the

21

CHAPTER 4. MODEL SETUP

usual behavior of the scattering length, close to the Feshbach resonance from Eq. 2.12. Theinverse of the scattering length may be written as

a−1(k) = a−1

(1−

(1− a

abg

)k2

k2 +Q2

). (4.9)

Here Q2 = 2abgRe(1−abg/a) is some parameter-dependent constant. Taking into account the

finite-range potential with Gaussian cutoff ξ(k) = ek2/Λ2

and Eq. 4.9, the two-body scatteringamplitude is written as

4π/m

tV (E + i0)= a−1 −

(a−1 − a−1

bg

) k2

k2 +Q2+ ik erfc(−

√2ik/Λ) e−2k2/Λ2

. (4.10)

The t-matrix of Model IV follows from Eq. 4.10 in the limit of abg → 0 in model V, soneglecting the open-closed channel coupling:

4π/m

tIV (E + i0)= a−1 +

Re2k2 + ik erfc(−

√2ik/Λ) e−2k2/Λ2

(4.11)

The t-matrices of Model I - III can be obtained in limits of Re → 0 and Λ→∞, neglectingthe effective range and reducing the finite-range potential to an contact potential.

In the limit of Re → 0 model IV becomes a contact potential with a finite-range potential,which is exactly model III

4π/m

tIII(E + i0)= lim

Re→0

4π/m

tIV (E + i0)= a−1 − ik erfc(−

√2ik/Λ) e−2k2/Λ2

. (4.12)

When the cutoff parameter vanishes in the limit of Λ→∞, model IV reduces to model IIas

4π/m

tII(E + i0)= lim

Λ→∞

4π/m

tIV (E + i0)= a−1 +

Re2k2 + ik. (4.13)

For both limits of Re → 0 and Λ → ∞ the simplest case is found, namely model I witht-matrix

4π/m

tI(E + i0)= a−1 + ik (4.14)

where only the contact potential determines the scattering amplitude.

22

4.3. SEPARABLE SCATTERING AMPLITUDES

4.3 Separable scattering amplitudes

The t-matrices need to satisfy one condition explicitly, namely that they are separable in a0th order term 1/a and a 1st and 2nd momentum dependent term. All five models satisfythis condition more or less, except that the momentum dependent term of Model V is alsodependent on the scattering length a. There is no input of the first hand for the scatter-ing length, in contradiction to e.g. the effective range or the background scattering length.Therefore, this input has to be generated and this is done iteratively. As starting point, usethe t-matrix of Model IV to find the values of 1/a at which the energies are projected. Usethese values for 1/a in the term with k2 to find the first iteration. Many more iterations maybe found, although this is unnecessary, because convergence is fast. Even when abg is chosenlarge, so there will be some deviations in the values of 1/a, in three iterations, the three-body

parameter a(−)0 is calculated within an accuracy of a tenthousandth.

23

CHAPTER 4. MODEL SETUP

24

Chapter 5

Results

In the first section of Chapter 5 the Efimov bound states are found for all five two-bodyinteractions. We comment on the results for the three-body parameter and the scaling andshow two figures for the characteristic energy bound states. The most interesting is how in

different models, the value of the parameter a(−)0 is determined by the finite-range Λ or the

width of the resonance R∗. In Section 5.2 some comments are given on the off-shell two-bodyscattering amplitude that has roughly been derived in Section 2.2. We check the result in theon-shell limit and give an outlook for further research. In Section 5.3 an example how onemight calculate the recombination rate is given. Here we also indicate where further researchmight lead to and how the problem could be approached.

5.1 Comparison between several two-body interactions

In Fig. 5.1 the Efimov bound states are solved for the zero-range interaction of Model I.The universal scaling is well reproduced: the scaling for the first five trimers converges to theexpected value of 22.69. The three-body parameter is not predicted accurately, as expected.The three-body parameter had to be fixed, to get a state with the lowest energy and pre-vent the Thomas collapse. Precisely for this reason [Thøgersen et al., 2008] introduced thefinite-range correction to solve this problem, so we cannot expect the zero-range model to find

the universal a(−)0 . The found value is a

(−)0 = 17.8Λ, which is a peculiar outcome, because

this shows the dependence of a(−)0 on the cutoff parameter Λ, while this model is the contact

interaction without a finite-range correction! We commented on this at the end of Section 4.1and concluded that this is an inconsistency from our model when using the zero-range modelas two-body interaction.

Recall that the width of the resonance is linked to the effective range as R∗ = 2Re, becausein the following section, only the resonance width and not the effective range will be discussed.In Model II in a narrow Feshbach resonance the width of the resonance is taken into account,because R∗ may reach the same large values as a. The role of the three-body parameter isunexpectedly taken over by the resonance width R∗ when R∗ > 10. For several values ofR∗ the three-body parameter and the scaling has been calculated. The resonance width islinked to the van der Waals length as R∗ = 9.8/10.9Rvdw = 0.90Rvdw. The results are givenin Table 5.1. We notice that the first scaling deviates more from the universal value of 22.7,and that the second scaling already converges better to this value. This is explained by the

25

CHAPTER 5. RESULTS

-1.0 -0.5 0.5 1.0 1.5

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

Figure 5.1: Efimov bound states with a contact potential as two-body interaction. The universalscaling factor 22.7 between two trimers is reproduced quite exact, but the three-body parameter is not.In this case it is found to be −17.8/Λ. It should be added as an extra parameter to fix the lowest Efimovbound state

Table 5.1: The results of a(−)0 and the first and second scaling as a function of R∗. When the value

of R∗ transcends a dozen Bohr radiuses a0, R∗ shows universal behavior as it is linked to the van derWaals length Rvdw, which determines the range of the interaction

R∗[a0] a(−)0 [R∗] First scaling Second scaling

1 -26.5 24.8 22.85 -13.0 26.0 23.010 -11.6 26.3 23.050 -10.9 26.5 23.0100 -10.9 26.5 23.01000 -10.9 26.5 23.0

26

5.1. COMPARISON BETWEEN SEVERAL TWO-BODY INTERACTIONS

Table 5.2: The results of a(−)0 and the first and second scaling as a function of Λ. When the value of

Λ becomes of the order of 1 a−10 and lower, universal behavior shows up, because the cutoff parameter

Λ is directly linked to the van der Waals length Rvdw, which determines the range of the interaction

Λ[a−10 ] a

(−)0 [Λ−1] First scaling Second scaling

0.001 -9.24 18.5 22.00.01 -9.24 18.5 22.00.1 -9.24 18.5 22.01 -9.24 18.5 22.010 -9.49 18.8 22.0100 -3.77 12.1 20.9

Table 5.3: The results of a(−)0 and the first and second scaling as a function of Λ and R∗. When the

value of the dimensionless parameter R∗Λ << 1 the results of Model III are reproduced. For R∗Λ = 1the results for Model I are found and when R∗Λ >> 1 Model II is recovered.

R∗Λ a(−)0 First scaling Second scaling

0.01 -9.31 [Λ−1] 18.6 22.00.1 -9.97 [Λ−1] 19.4 22.21 -16.6 [R∗] or [Λ−1] 23.0 22.610 -10.1 [R∗] 26.3 23.0100 -10.7 [R∗] 26.5 23.0

fact that in the second scaling a is already more than 500 times larger, and then universalitythrough 1/a→ 0 dominates the less universal behavior described by R∗. In the limit R∗ → 0the zero-range model is approached and there is no dependence of the three-body parameteron the resonance width anymore.

Model III incorporates the finite-range correction and also in this case, role of the three-body parameter is taken over by the introduction of Λ. For Λ ∈ [0.001, 1] a−1

0 a value of

a(−)0 = −9.24 Λ−1 is found. In this case, Λ = −9.24/9.8Rvdw = 0.94Rvdw becomes a universal

parameter, which is linked to a known property of the system. In the numerical model achoice for Λ < 0.001 gives errors, but in the physical world there is no reason why Λ cannotbecome arbitrary close to 0. This would give a very, very wide range at which the interactionpotential works. The first scaling is lower than 22.7 and for higher orders the factor 22.7 isapproached. In Table 5.2 the results are shown. For Λ > 1 [a−1

0 ] the results begin to deviatefrom universality. This is as can be expected because for larger Λ the finite-range does notshow Gaussian dispersion in the range of interaction, but becomes some sort of distortedzero-range model. There is no dependence of three-body parameter on the cutoff parameteranymore.

Model IV can be considered for three cases, namely in the case where

• R∗Λ > 1 and Model IV → Model II. Here a(−)0 is completely determined by R∗.

• R∗Λ < 1 and Model IV → Model III. Here a(−)0 is completely determined by Λ.

27

CHAPTER 5. RESULTS

-1.0 -0.5 0.5 1.0 1.5

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

Figure 5.2: Efimov bound states taking into account the coupling between the open and closed channel.Here the effect of the narrow Feshbach resonance dominates over the effect of the background scattering.Numerical errors start to play a role for large energies, due to the iterative model. Like in Model IIand IV the universal scaling is approached and the three-body parameter is predicted as a function ofR∗.

Table 5.4: Results for the universal scaling and the three-body parameter for model V

Model a(−)0 First scaling Second scaling R∗ abg Λ

V a) -10.7 [R∗] 26.5 23.0 100 -0.1 1V c) -10.9 [Λ−1 19.9 22.1 0.1 -100 10V c) -8.5 [Λ−1 17.1 21.8 0.1 -100 1V c) -7.4 [Λ−1 1.3 18.6 0.1 -100 0.1

• R∗Λ = 1 and Model IV → Model I. Here a(−)0 is overdetermined, because one may

choose to consider a(−)0 as a function of R∗ or as a function of Λ.

The third bullet point presents some interesting physics. When the effect of the narrowFeshbach resonance and the effect of the finite-range potential exactly cancel, the simplestcase, namely the zero-range model, is recovered, and both Λ and R∗ determine the three-body

parameter but they cannot determine a(−)0 together. It is one or the other. The results are

presented in Table 5.3.

There are two parameter choices for Model V. Therefore, Model V is considered in threelimits, namely in the limit a) where R∗ >> abg, b) where R∗ ∼ abg and c) where R∗ << abg.In 5.2 the Efimov bound states from Model V a) are depicted. The two-body threshold showswide parabolic behavior, because of the large value of R∗. The missing Figs. for Model II,III, IV and V c) are shown in the Appendix.

In Table 5.4 the results from two-body interaction Model V a) and V c) of the universalscaling and the three-body parameter are presented. For Model V a) the same results apply

28

5.2. OFF-SHELL TWO-BODY SCATTERING MATRIX

as to Model II and IV: the correct scaling is approached and the three-body parameter isuniquely determined in terms of R∗. In Model V c) the value of abg is chosen to be minusa thousand times larger than R∗ in Model V c) such that there is a very broad Feshbachresonance. We note that the scaling in first instance is too low and converges to 22.7 forweaker bound states. The universal three-body parameter is related to the range of thepotential in terms of Λ, like in Model III. We note that the coupling between the open andclosed channel through the introduction of the background scattering length is the reasonthat the finite-range does not determine long-range behavior in the form of the van der Waals

length. For a smaller finite range, so a wider range of interaction, the value of a(−)0 in terms of

Λ starts to deviate more and more from the universal value of −9.8. Such a large backgroundscattering length allows for a smaller finite-range to approximate the zero-range model, thanin Model III.

There was not enough time to understand the behavior that follows from Model V b).The behavior of Model V in three limits may raise some issues. First of all, Model V b)needs to be implemented with the values R∗ = 55a0, abg = −20a0 from [Levinsen, 2013] and[Dyke et al., 2013]. This model should predict the experimental observations Dyke et al.have done with this parameter choice. Secondly, how does the background scattering lengthinfluences the finite-range potential and why does it do this? Another discussion point is thatthe resonance width R∗ and abg are not independent parameters. R∗ = ~2

2mabgµB∆B is called

the resonance width, but is actually a function of the actual resonance width ∆B and of abg.So when considering resonance widths, remember that the parameter that tunes R∗ is ∆B.Nevertheless, the influence of abg on R∗ may not be underestimated and as we have not takenthis relation into account in this research, results may become clear when doing so. Only inModel V this problem arises because for Model II and Model IV there is a fixed backgroundscattering length and the tuning of ∆ determines R∗.

5.2 Off-shell two-body scattering matrix

From [van Dongen, 2014] a two-body scattering matrix is computed off-shell, meaning thatboth elastic and inelastic collisions may take place. This two-body scattering matrix gives amore realistic view on the collision processes. Our eventual goal was to investigate the Efimovbound states which were produced with inelastic two-body interactions. Due to the delayedfinding of the off-shell scattering matrix, no more research could be done in this region. Still,it remains of particular interest, whether the scaling and three-body parameters are repro-duced with this model. Further research may turn out if this approach gives new insights.The derivation for the off-shell scattering matrix has briefly described in Section 2.2. We willcheck the result in the on-shell limit and give directions on how to compute the Efimov boundstates from the off-shell two-body scattering matrix.

The result for the off-shell two-body matrix is given in Eq. 2.11. We check this result bytaking a Taylor expansion of 1/f(k’,k) in the on-shell limit where k = k’ = k, to allow onlyelastic collisions, and we find

1/f(k) = −1

a− ik + r0

(1− r0

a+

r20

3a2

)k2 +

r30

3

(1− 2r0

a+

7r20

5a2− r3

0

3a3

)k4. (5.1)

29

CHAPTER 5. RESULTS

This may be compared to the known effective range expansion k cot δ(k) = − 1a + 1

2Rek2 +

P ∗k4 from Eq. 2.6 and the on-shell scattering amplitude 1/f(k) = k cot δ(k) − ik from Eq.4.2. An expression for the effective range and the shape parameter follow directly from this:

Re = 2r0

(1− r0

a+

r20

3a2

). (5.2)

The result from Eq. 5.2 coincides exactly with the result of [Kokkelmans, 2014] and [Flam-baum et al., 1999]. This indicates that the found off-shell two-body scattering amplitude maycorrectly describe the Efimov physics.

One of the challenges in this scattering amplitude is to relate several parameters in thedescription of the scattering amplitude like the width of the square well r0 to parameters fromEfimov physics like the effective range Re and the cutoff parameter Λ. Also, it is not directlyclear how the two wavenumbers k and k’ fit into this description. Further research may leadto new insights in the less universal behavior of Efimov physics.

5.3 Recombination rate

The numerical model that is presented in Chapter 4 finds the eigenvalues of the kernel of theSTM integral equation from which we construct the Efimov bound states. These eigenvaluescorrespond to the poles of the scattering matrix T3(k), as in

T =S − 1

2πi=e2ikR∗

2πi

κ− ikκ+ ik

− 1

2πi(5.3)

where k = iκ, κ > 0 corresponds to the simple pole in momentum space, located on theimaginary axis. Indeed, the S-matrix is unitary and the non-resonant scattering contributionis there in the form of an exponential e2ikr0 , equivalent to hard-sphere scattering. Here r0

is the non-resonant contribution to the scattering length, on the order of the range of theinteraction potential, which corresponds to Re. To reconstruct the S-matrix from the model,we may calculate the product of all poles in momentum space, which are situated on theimaginary axis, in such a way that unitary is guaranteed:

S = e2ikr0

N∏n=1

κn − ikκn + ik

(5.4)

where N is the number of poles and all N poles k = iκn, κn > 0 are located on theimaginary axis. Notice that the poles in the T-matrix correspond to poles in the S-matrix,therefore this reconstruction is allowed. This reconstructed off-shell S-matrix can be used tofind the recombination rate αrec, which follows a quadratic equation in S, following e.g. [Esryet al., 1999] and Newby, as described in Section 3.2. Further research may turn out how thepredicted recombination rate agrees with the experimental observations.

30

Chapter 6

Conclusion

The goal of the Bachelor Thesis was to predict non-universal effects in Efimov physics. Wewanted to solve the Efimov bound states in momentum space for several two-body interac-tions. In the report we have discussed five models with increasing complexity that describethe scattering process with increasingly more detail. The next part and the initial goal was toincorporate inelastic collisions into Efimov physics. The provided off-shell two-body scatter-ing matrix should serve as input for the three-body scattering matrix from which the Efimovbound states would follow. We had good hopes this would lead to new insights into thenon-universal aspects of Efimov physics.

The model we have used solves the Skorniakov-Ter-Martirosian (STM) equation, an im-plicit integral equation in T3(k). As input it needs the scattering amplitude from two-bodyinteractions in the ultracold limit. The scattering amplitude shows typical behavior as itcontains a constant 1/a term, a complex ik term and more momentum dependence in k2.The STM equation is a homogeneous Fredholm equation of the second kind and it is solvedin a numerical model with standard Gaussian Quadrature Weights. The eigenvalues of thekernel of the integral minus the k-dependent term in the two-body interaction make up theenergies of the Efimov bound states. These energies are mapped layer by layer onto the 1/a-momentum grid. In this way the Efimov bound states have been built up.

For diverging scattering length, the universal aspects of Efimov physics emerge. The zero-range model (Model I in the report) states that particles collide only when they are at thesame point space. Close to the field of resonance of a Feshbach resonance Model I describesthe universal Efimov effect very well. The universal scaling kn/kn+1 between two trimersconverges to 22.7 and the universal scaling between the energy of two consecutive trimersEn/En+1 approaches the factor 22.72 ≈ 515. With only the scattering length a and the

surprising universal three-body parameter a(−)0 = 9.8Rvdw that secures the bound state with

the lowest bound state, the system is completely determined. A peculiar aspect of our modelis that we make use of a cutoff function in momentum space to determine the integral inT3(k) and map the energies, the discovered eigenvalues on the y-axis, on the weighted inversescattering lengths on the x-axis. So to find the properties of the system of the zero-rangemodel, we effectively make use of a finite-range potential.

A less universal aspect of Efimov physics occurs in a narrow Feshbach resonance (Model

31

CHAPTER 6. CONCLUSION

II). An extra parameter Re is necessary to give accurate predictions for the system. We have

shown that for Re > 10, the effective range is directly linked to a(−)0 and to the van der Waals

length as Re = 12a

(−)0 /10.9 = 0.45Rvdw. Also, we have seen that the finite-range potential

in Model III shows this behavior. For the cutoff parameter Λ ≤ 1 in the Gaussian cutofffunction, the three-body parameter is determined in terms of Λ. Model IV incorporates botha Gaussian cutoff and a narrow Feshbach resonance. We determined three limits for the di-mensionless parameter R∗Λ and we recovered the previous models for R∗Λ < 1 (Model III),R∗Λ > 1 (Model II) and surprisingly also Model I when R∗Λ = 1. In the last case the three-body parameter is overdetermined, because with both Λ and R∗ we found the three-bodyparameter corresponding to the zero-range model, but we could only use one parameter todetermine the system. Here the effects of the narrow Feshbach resonance and the Gaussiancutoff exactly cancel and we recovered a contact interaction.

Model V is the most detailed model. It takes into account the effective range, the finiterange and also the resonant coupling between the open and closed channel. The parameterthat describes the last process is the background scattering length abg. We have distinguishedthree cases where a) Re >> abg, b) Re ∼ abg and c) Re << abg. In the regime of large back-ground scattering length (Model V c)), we note that the three-body parameter cannot bereplaced by the cutoff parameter of the Gaussian cutoff, because for larger Λ the deviationfrom the universal value of 9.8 becomes increasingly larger. In the regime where the effectiverange dominates background scattering (Model V a), Model II is recovered. We saw thatthe three-body parameter is determined completely by R∗. For Model V b), where the ef-fect of background scattering and the narrow Feshbach resonance are of the same order ofmagnitude, like in [Dyke et al., 2013], there was not enough time to find results. Concerningthe iterative procedure in Model V, its performance has to be improved to give the correctcharacteristic for the Efimov bound state.

We have shown the outline of the derivation for the off-shell two-body scattering ampli-tude and we showed that in the on-shell limit the correct coefficients in the effective rangeexpansion were recovered. The model described in this report should be able to find the Efi-mov trimers from this off-shell scattering matrix. Further research should be done to link theproperties of the square well (its width, its depth, the incoming and outgoing wavenumbers)to parameters in Efimov physics like the effective range and cutoff parameter and even thebackground scattering length. In the report we have derived the recombination rate in a con-ceptual insightful way. In this derivation, the recombination rate depends on the scatteringmatrix. With our model the implicit integral equation of the three-body scattering matrix issolved, but T3(k) is never known explicitly. By considering some of the poles on the imaginaryaxis in complex momentum space, the T-matrix may be recovered and the recombination ratecan be calculated. We have only indicated how to do this and have not tried to find results.The calculation of the recombination rate is a necessity to link theoretical predictions to ex-perimental observations and therefore this should definitely be examined in follow-up research.

Summarizing the above: we have researched five different models of two-body interactionswith increasing complexity and found out how in less universal two-body models the universalrelation of the three-body parameter to the potential range emerge. This research may first ofall be continued by producing results for Model V b) and comparing them to the experimental

32

observations from e.g. [Dyke et al., 2013]. The relevance of the resonant coupling of the open-closed channel will demonstrate itself. The other continuation point focuses on implementinganother model for the two-body interactions, namely the off-shell two-body scattering matrix.Also in this case the behavior of the universal parameters and scaling should be examinedand in the end one might decide to calculate the recombination rate to compare this toexperimental observations.

33

CHAPTER 6. CONCLUSION

34

Appendix A

Angular averaging

To find Eq. 4.4 from Section 4.1, we need to perform angular averaging in the following way

t−1(E − 3εk/2)T3(k) = 2

∫d3p

(2π)3

ξ(|k− p/2|)ξ(|p− k/2|)E − k2/m− p2/m− k · p/m

T3(p) (A.1)

=

∫p

∫φ

∫θ

p2 sin θ dp dφ dθ

4π3

ξ(|k− p/2|)ξ(|p− k/2|)E − k2/m− p2/m− k p cos θ/m

T3(p)

=

∫pp2dp

∫ π

θ=0

sin θ dθ

2π2

ξ(|k− p/2|)ξ(|p− k/2|)E − k2/m− p2/m− k p cos θ/m

T3(p).

Substituting u = cos θ and du = − sin θ dθ and considering the boundaries, Eq. A.1becomes

t−1(E − 3εk/2)T3(k) =

∫pp2dp

∫ 1

u=−1

du

2π2

ξ(|k− p/2|)ξ(|p− k/2|)E − k2/m− p2/m− k p u/m

T3(p) (A.2)

=m

2π2

∫p2dp

ξ(|k− p/2|)ξ(|p− k/2|)k p

logE − k2/m− p2/m− k p/mE − k2/m− p2/m+ k p/m

T3(p)

=m

2π2

∫dp ξ(|k− p/2|)ξ(|p− k/2|) p

klog

E − k2/m− p2/m− k p/mE − k2/m− p2/m+ k p/m

T3(p).

This result coincides with the result of [?].

35

APPENDIX A. ANGULAR AVERAGING

36

Appendix B

Additional figures for Efimov boundstates

Besides the figures for the Efimov bound states from two-body interactions of Model I andModel V a), the bound states for other models are also explicitly depicted. In Figs. B.1 -B.4 the bound states for model II, III, IV and V b) and c) are given. The numerical errors inFigs. ?? and ?? are caused by the several iterations to find the correct 1/a in the momentumdependent term of the separable two-body potential.

37

APPENDIX B. ADDITIONAL FIGURES FOR EFIMOV BOUND STATES

-1.0 -0.5 0.5 1.0 1.5

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

Figure B.1: Efimov bound states for a narrow Feshbach resonance.

-1.0 -0.5 0.5 1.0 1.5

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

Figure B.2: Efimov bound states with the finite-range potential as two-body interaction.

38

-1.0 -0.5 0.5 1.0 1.5

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

Figure B.3: Efimov bound states with the finite-range potential as two-body interaction in a narrowFeshbach resonance.

39

APPENDIX B. ADDITIONAL FIGURES FOR EFIMOV BOUND STATES

-1.0 -0.5 0.5 1.0 1.5

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

Figure B.4: Efimov bound states taking into account the coupling between the open and closed channel.Here background scattering dominates over the effect of the narrow Feshbach resonance. The numericalerrors come to existence through the iterative procedure.

40

BIBLIOGRAPHY

Bibliography

E. Braaten and H.W. Hammer. Universality in few-body systems with large scattering length.Physics Reports, 428:259 – 390, 2006.

Georg Bruun. Skorniakov-Ter-Martirosian equation for bosons and fermions, December 2013.

Ghrist E.W. Myatt C.J. Holland M.J. Cornell E.A. Wieman C.E. Burt, E.A. Coherence,correlations and collisions: what one lears about Bose-Einstein condensates from theirdecay. Phys. Rev. Lett., 79:337, 1997.

P. Dyke, S.E. Pollack, and R.G. Hulet. Finite-range corrections near a feshbach resonanceand their role in the efimov effect. Phys. Rev. A., 88:023625, 2013.

V. Efimov. Energy levels arising from resonant two-body forces in a three-body system.Physics Letters B, 33:563 – 564, 1970.

V. Efimov. Weakly-bound states of three resonantly-interacting particles. Sov. J. Nucl. Phys.,12:589 – 595, 1971.

B. D. Esry, Chris H. Greene, and James P. Burke. Recombination of three atoms in theultracold limit. Phys. Rev. Lett., 83:1751 – 1754, August 1999.

D.V. Fedorov and A.S. Jensen. Efimov effect in coordinate space Faddeev equations. Phys.Rev. Lett., 71:4103 – 4106, 1993.

D.V. Fedorov and A.S. Jensen. Regularization of a three-body problem with zero-rangepotentials. Journal of Physics A: Mathematical and General, 34:6003, 2001.

V.V. Flambaum, G.F. Gribakin, and C. Harabati. Analytical calculation of cold-atom scat-tering. Phys. Rev. A., 59:1999 – 2005, 1999.

R. Grimm and F. Ferlaino. Forty years of efimov physics: How a bizarre prediction turnedinto a hot topic. American Physical Society, 3:9, 2010.

N. Gross, Z. Shotan, S.J.M.M.F. Kokkelmans, and L. Khaykovich. Observation of universalityin ultracold li7 three-body recombination. Phys. Rev. Lett., 103:163202, 2009.

Y. Kagam, E.L. Surkov, and G.V. Shlyapnikov. Bose-condensation effect on inelastic processesin a gas. JETP Lett., 42:209, 1985.

S.J.J.M.F. Kokkelmans. World Scientific Review Volume, chapter 1. World Scientific, January2014.

41

BIBLIOGRAPHY

Mark M. Waldburger P. Danzl J.G. Chin C. Engeser B. Lange A.D. Pilch K. Jaakkola A.Nagerl H.-C. Grimm R. Kraemer, T. Evidence for Efimov quantum states in an ultracoldgas of cesium atoms. Nature, 440:315 – 318, March 2006.

Jesper Levinsen. Notes on Efimov physics with different two-body interactions, December2013.

P. Massignan and H.T.C. Stoof. Efimov states near a feshbach resonance. Phys. Rev. A, 78:030701, 2008.

Christopher A. Newby. Modeling three-body recombination rates in ultracold 85Rb aroundthe 155 G Feshbach resonance.

D.S. Petrov. Three-boson problem near a feshbach resonance. Phys. Rev. Lett., 93:143201,September 2004.

S.E. Pollack, Dries. D., and R.G. Hulet. Science, 326:1683, 2009.

Teukolsky S.A. Vetterling W.T. Press, W.H. and B.P. Flannery. Numerical Recipes - The Artof Scientific Computing. Cambridge University Press, 3rd edition, 2007.

L. Pricoupenko and M.J. Lasinio. Ultracold bosons in the vicinity of a narrow resonance:shallow dimer and recombination. Phys. Rev. A, 84:062712, 2011.

J.J. Sakurai. Modern Quantum Mechanics (Revised Edition). Addison Wesley, 1st edition,1993.

G.V. Skorniakov and K.A. Ter-Martirosian. Sov. Phys. JETP, 4:648, 1957.

J.R. Taylor. Scattering Theory. Robert E. Krieger Publishing Company, 3rd edition, 1987.

M. Thøgersen, D. V. Fedorov, and A. S. Jensen. Universal properties of efimov physics beyondthe scattering length approximation. Phys. Rev. A., 78:020501, Aug 2008.

L.H. Thomas. The interaction between a neutron and a proton and the structure of h3. Phys.Rev., 47:903 – 909, June 1935.

R. van Dongen. Off-shell scattering due to a finite square well potential, June 2014.

M. Zaccanti, B. Deissler, C. DErrice, M. Fattori, M. Jona-Lasinio, S. Muller, G. Roati,M. Inguscio, and G. Modugno. Observation of an efimov spectrum in an atomic system.Nature Physics, 5:586 – 591, 2009.

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