Post on 30-Mar-2023
Efimov Physics in Fermionic
Lithium atoms
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the DegreeDoctor of Philosophy in the Graduate School of The Ohio State
University
By
Daekyoung Kang, B.S.
Graduate Program in Physics
The Ohio State University
2011
Dissertation Committee:
Eric Braaten, Advisor
Richard Furnstahl
Mohit Randeria
Gregory Lafyatis
Abstract
Efimov physics refers to universal phenomena that are characterized by discrete scal-
ing behavior in three-body systems consisting of particles that interact with a large
scattering length. The most well-known example is Efimov trimers, a sequence of
universal bound states that in the case of infinite scattering length have a geometric
spectrum with an accumulation point at the three-particle threshold. Efimov physics
is also manifested in scattering processes through log-periodic dependence on the col-
lision energy or on the scattering length. In experiments with trapped ultracold gases,
the most dramatic features associated with Efimov physics are resonant enhancements
of loss rates from an Efimov trimer near a scattering threshold. This thesis presents
studies of Efimov physics in the three lowest hyperfine states of fermionic 6Li atoms.
We calculate the spectrum of the Efimov trimers as a function of the magnetic field.
We calculate the three-body recombination rate at threshold, which exhibits loss res-
onances and interference minima associated with Efimov physics. We also calculate
the relaxation rate of diatomic molecules due to inelastic collision with an atom,
which also exhibit loss resonances and local minima. We compare our results with
experimental measurements using trapped ultracold gases of 6Li atoms.
ii
Acknowledgments
From the bottom of my heart, I want to thank my advisor Eric Braaten for his infinite
support and patient guidance, without which my research would have been impossible.
His rigorous intuition and warmhearted spirit are virtues for a physicist that I should
strive to achieve throughout my lifetime. I also appreciate Mim Braaten’s generous
caring for Eric’s group and friends.
Jungil Lee provided me a chance to jump into research during undergraduate
school. He is not only an excellent physicist but also a thoughtful educator who keeps
his eyes on his students and tries to give them more opportunities. His advice and
concern make me feel warm inside. I especially thank him for his encouragement,
without which I would not have been able to start the PhD program in OSU.
Lucas Platter was more like a co-advisor to me than a collaborator. He taught
me not only the numerical techniques for solving the three-body problem but also
an attitude of looking out for future directions. I am deeply grateful for his caring,
especially during the period when he was a postdoc at OSU.
It was an honor to collaborate with Geoffrey Bodwin, from whom I learned a
scholar’s faithfulness. I was very glad to collaborate with Hans-Werner Hammer on
the thesis topic. I was always happy to talk to Pierre Artoisenet about anything. He
tried to answer even my stupid questions and he found the answer most of the time. I
would like to congratulate Christian Langmack on his wedding with Ghazal and wish
him good luck with both his new life and his research. I am also in debt to Chaehyun
iv
Yu, Jong-Wan Lee, and Hee Sok Chung for their collaboration and discussions. I
personally relied much on Heechang Na and I thank him for his time.
My last thanks go to my family. Without my wife Hee Joo Choi and her endless
love, my research could not have been completed. My parents always waited patiently
without any doubts and any demands. I especially thank my father-in-law Jaiyul
Choi for his attention to my study and for his constant encouragement. I dedicate
this thesis to my family.
v
Vita
December 14, 1979 . . . . . . . . . . . . . . . . . . . . . . . . . Born—Jinju, South Korea
February, 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S., Korea University, Seoul, SouthKorea
vi
Publications
E. Braaten, D. Kang, and L. Platter, Universal relations for identical bosons fromthree-body physics, Phys. Rev. Lett. 106, 153005 (2011).
H. W. Hammer, D. Kang, and L. Platter, Efimov Physics in Atom-Dimer Scatteringof 6Li Atoms, Phys. Rev. A 82, 022715 (2010).
P. Artoisenet, E. Braaten, and D. Kang, Using Line Shapes to Discriminate betweenBinding Mechanisms for the X(3872), Phys. Rev. D 82, 014013 (2010).
E. Braaten, D. Kang, and L. Platter, Short-Time Operator Product Expansion forrf Spectroscopy of a Strongly-interacting Fermi Gas, Phys. Rev. Lett. 104, 223004(2010).
E. Braaten, H. W. Hammer, D. Kang, and L. Platter, Efimov Physics in 6Li Atoms,Phys. Rev. A 81, 013605 (2010).
E. Braaten, D. Kang, J. Lee, and C. Yu, Optimal spin quantization axes for quarko-nium with large transverse momentum, Phys. Rev. D 79, 054013 (2009).
E. Braaten, H. W. Hammer, D. Kang, and L. Platter, Three-body Recombinationof Fermionic Atoms with Large Scattering Lengths, Phys. Rev. Lett. 103, 073202(2009).
E. Braaten, D. Kang, J. Lee, and C. Yu, Optimal spin quantization axes for thepolarization of dileptons with large transverse momentum, Phys. Rev. D 79, 014025(2009).
E. Braaten, D. Kang, and L. Platter, Exact Relations for a Strongly-interactingFermi Gas near a Feshbach Resonance, Phys. Rev. A 78, 053606 (2008).
H. S. Chung, J. Lee, and D. Kang, Cornell Potential Parameters for S-wave HeavyQuarkonia, J. Korean Phys. Soc. 52, 1151 (2008).
E. Braaten, H. W. Hammer, D. Kang, and L. Platter, Three-Body Recombination ofIdentical Bosons with a Large Positive Scattering Length at Nonzero Temperature,Phys. Rev. A 78, 043605 (2008).
vii
G. T. Bodwin, H. S. Chung, D. Kang, J. Lee, and C. Yu, Improved determinationof color-singlet NRQCD matrix elements for S-wave charmonium, Phys. Rev. D 77,094017 (2008).
D. Kang, T. Kim, J. Lee, and C. Yu, Inclusive Charm Production in Υ(nS) Decay,Phys. Rev. D 76, 114018 (2007).
G. T. Bodwin, E. Braaten, D. Kang, and J. Lee, Inclusive Charm Production in χb
Decays, Phys. Rev. D 76, 054001 (2007).
E. Braaten, D. Kang, and L. Platter, Universality Constraints on Three-Body Re-combination for Cold Atoms: from 4He to 133Cs, Phys. Rev. A 75, 052714 (2007).
D. Kang and E. Won, Precise Numerical Solutions of Potential Problems UsingCrank-Nicholson Method, J. Comput. Phys. 227, 2970 (2008).
G. T. Bodwin, D. Kang, and J. Lee, Potential-model calculation of an order-v2
NRQCD matrix element, Phys. Rev. D 74, 014014 (2006).
G. T. Bodwin, D. Kang, and J. Lee, Reconciling the light-cone and NRQCD ap-proaches to calculating e+e− → J/ψ + ηc, Phys. Rev. D 74, 114028 (2006).
D. Kang, J.-W. Lee, and J. Lee, Color-evaporation-model prediction for σ(e+e− →J/ψ +X) at B factories, J. Korean Phys. Soc. 47, 777 (2005).
D. Kang, J.-W. Lee, J. Lee, T. Kim, and P. Ko, Inclusive production of 4 charmhadrons in e+e− annihilation at B factories, Phys. Rev. D 71, 071501 (2005).
D. Kang, J.-W. Lee, J. Lee, T. Kim, and P. Ko, Color-evaporation-model calculationof e+e− → J/ψ + cc+X at
√s = 10.6 GeV, Phys. Rev. D 71, 094019 (2005).
viii
Fields of Study
Major Field: Physics
Studies in:
Atomic physics: Efimov physics in ultracold atomsHigh energy physics: QCD phenomenology and heavy quarkonium
Advisor: Eric Braaten
ix
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivVita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapters
1 Introduction 1
1.1 6Li atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Alkali atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Hyperfine spin states . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 2-body system at low energy . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Natural scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Two-body scattering . . . . . . . . . . . . . . . . . . . . . . . 91.2.3 Universality with large scattering length . . . . . . . . . . . . 121.2.4 Scattering lengths of 6Li atoms . . . . . . . . . . . . . . . . . 16
1.3 Efimov physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.1 Efimov trimers . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.2 Loss of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.3 Observations of Efimov trimers . . . . . . . . . . . . . . . . . 27
1.4 Effective field theory approach . . . . . . . . . . . . . . . . . . . . . . 291.4.1 Identical bosons . . . . . . . . . . . . . . . . . . . . . . . . . . 291.4.2 Fermions with three spin states . . . . . . . . . . . . . . . . . 35
1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2 Three-body Recombination for Negative Scattering Lengths 39
2.1 Three-body recombination and optical theorem . . . . . . . . . . . . 392.2 STM equations at threshold . . . . . . . . . . . . . . . . . . . . . . . 402.3 Recombination into deep dimer . . . . . . . . . . . . . . . . . . . . . 412.4 Equal negative scattering lengths . . . . . . . . . . . . . . . . . . . . 422.5 Unequal negative scattering lengths . . . . . . . . . . . . . . . . . . . 44
x
2.6 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Efimov trimer spectrum and three-body recombination 50
3.1 Theoretical formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1.1 Three-body recombination rates . . . . . . . . . . . . . . . . . 513.1.2 STM equations . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1.3 Three equal scattering lengths . . . . . . . . . . . . . . . . . . 553.1.4 Homogeneous STM equations and Efimov trimers . . . . . . . 573.1.5 Dimer relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Low-field universal region . . . . . . . . . . . . . . . . . . . . . . . . 603.2.1 Three-body recombination revisited . . . . . . . . . . . . . . . 603.2.2 Efimov trimers . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 High-field universal region . . . . . . . . . . . . . . . . . . . . . . . . 663.3.1 Measurements of three-body recombination . . . . . . . . . . . 663.3.2 Efimov trimers . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.3 Predictions for three-body recombination . . . . . . . . . . . . 713.3.4 Atom-dimer resonance . . . . . . . . . . . . . . . . . . . . . . 733.3.5 Many-body physics . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 793.5 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Dimer relaxation 82
4.1 Dimer relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 STM equations for non-zero energy . . . . . . . . . . . . . . . . . . . 854.3 Zero temperature results . . . . . . . . . . . . . . . . . . . . . . . . . 874.4 Finite temperature results . . . . . . . . . . . . . . . . . . . . . . . . 944.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 964.6 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Outlook 99
xi
List of Figures
Figure Page
1.1 Hyperfine energy levels of 6Li atoms . . . . . . . . . . . . . . . . . . . 61.2 Illustration of Feshbach resonance . . . . . . . . . . . . . . . . . . . . 151.3 Scattering lengths of 6Li atoms in the low-field region . . . . . . . . . 181.4 Scattering lengths of 6Li atoms in the high-field region . . . . . . . . 191.5 Energy spectrum of Efimov trimers for identical bosons . . . . . . . . 211.6 Illustration of three-body recombination . . . . . . . . . . . . . . . . 251.7 Illustration of dimer relaxation . . . . . . . . . . . . . . . . . . . . . . 261.8 Diagrams for 2-atom amplitude . . . . . . . . . . . . . . . . . . . . . 311.9 Integral equation for atom-diatom amplitude . . . . . . . . . . . . . . 34
2.1 Coefficients C and D in the three-body recombination rate . . . . . . 432.2 Three-body recombination rate in the low-field region . . . . . . . . . 452.3 Three-body recombination rate in the high-field region . . . . . . . . 47
3.1 Three-body recombination rate in the low-field region . . . . . . . . . 613.2 Energy and width of the Efimov trimer in the low-field region . . . . 643.3 Three-body recombination rate in the high-field region . . . . . . . . 673.4 Energies of the Efimov trimers in the high-field region . . . . . . . . . 693.5 Widths of the Efimov trimers in the high-field region . . . . . . . . . 703.6 Three-body recombination rates in the high-field region . . . . . . . . 723.7 Dimer relaxation rate near the atom-dimer resonance . . . . . . . . . 75
4.1 Relaxation rate for the 23-dimer . . . . . . . . . . . . . . . . . . . . . 894.2 Relaxation rate for the 13-dimer . . . . . . . . . . . . . . . . . . . . . 914.3 Relaxation rate for the 12-dimer . . . . . . . . . . . . . . . . . . . . . 924.4 Relaxation rate for the 23-dimer at nonzero temperature . . . . . . . 95
xii
Chapter 1
Introduction
The year 2011 is the 100th anniversary of the discovery of superconductivity. A
century of study has enabled great achievements in understanding the mechanism
for this phenomenon and for the related phenomenon of superfluidity. However, our
understanding has not reached the level to achieve the ultimate goal: superconduc-
tivity at room temperature. Toward the advancement of this goal, ultracold atoms
can play a role as a laboratory to improve our understanding of this phenomenon
because the fundamental interactions in these systems are simple and experimentally
controllable. There have been extensive investigations of ultracold atoms consisting
of fermionic atoms with two spin states [1]. This system has a superfluid phase at low
temperature. As the interaction strength of the atoms is varied, the mechanism for
superfluidity exhibits a smooth crossover from the BCS mechanism (Cooper pairing of
atoms) to the BEC mechanism (Bose-Einstein condensation of diatomic molecules).
For fermionic atoms with three spin states, there is the possibility of new superfluid
phases and new mechanisms for superfluidity [2, 3, 4, 5, 6]. The first experiments
with such a system have been carried out using the three lowest hyperfine states of
6Li atoms [7, 8].
Fermionic atoms with three spin states also open up new possibilities in few-
body physics. If the pair scattering lengths are large, there are remarkable three-
1
body phenomena which do not occur in fermions with two spin states. If the three
scattering lengths are infinitely large, there is an infinite sequence of three-atom bound
states called Efimov trimers with a geometric spectrum and an accumulation point
at the three-atom threshold [9, 10]. The ratio of the binding energies of successive
Efimov trimers is approximately 1/515. This remarkable three-body phenomenon is
characterized by discrete scale invariance. Universal phenomena associated with the
discrete scale invariance are referred to as Efimov physics [11, 12]. In this thesis, we
present our studies of Efimov physics in the three lowest hyperfine states of 6Li atoms.
In Sec. 1.1, the basic properties of alkali atoms and their hyperfine states are
reviewed. In Sec. 1.2, the low energy physics in the two-body system and its universal
behavior are explained. In Sec. 1.3, Efimov physics and its observations are discussed.
In Sec. 1.4, our theoretical framework is described. We outline the following chapters
in Sec. 1.5.
1.1 6Li atoms
In this section, we review the basic properties of alkali atoms and their hyperfine
states.
1.1.1 Alkali atoms
The types of atoms that are most easily cooled to ultra-low temperatures are the
alkali atoms that lie below hydrogen in the periodic table. They are lithium (Li),
sodium (Na), potassium (K), rubidium (Rb), and cesium (Cs). In this section, we
review the basic properties of alkali atoms due to their constituents.
The constituents of an atom are protons, neutrons, and electrons. Their electric
charges in units of the proton charge are +1, 0, and −1, respectively. The proton
and neutron are much heavier than an electron; their masses are larger by a factor
2
of about 1840. The structure of an atom consists of a tiny massive core called the
atomic nucleus, surrounded by clouds of electrons that are arranged in shells. The
nucleus consists of protons and neutrons. The number of protons in the nucleus
determines the element of the atom. For example, lithium (Li) atoms have a nucleus
with 3 protons. In an electrically neutral atom, the number of electrons is equal to the
number of protons. The total number of protons and neutrons in the nucleus is called
the atomic mass number and it determines the isotope of the atom. The isotope is
commonly specified by giving the atomic mass number as a pre-superscript. The most
common isotopes of lithium are 6Li and 7Li. The nucleus of these isotopes contain
three and four neutrons, respectively. The electronic structure of an alkali atom such
as Li consists of closed shells of electrons plus a single electron in the outermost shell.
The mass m of a 6Li atom is approximately six times that of a proton. A convenient
conversion constant for 6Li atoms is
h
m= 1.0558× 10−4 cm2/s, (1.1)
where h is Planck’s constant.
All elementary particles can be classified into two categories: bosons and fermions.
Collections of identical particles have dramatically different behavior depending on
whether they are bosons or fermions. For identical bosons, the quantum state must
be symmetric under exchange of any two bosons. This implies that any number of
identical bosons can occupy the same quantum state. The ground state of a many-
body system of N noninteracting identical bosons is a Bose-Einstein condensate, in
which all N bosons occupy the lowest-energy quantum state. For identical fermions,
the quantum state must be antisymmetric under exchange of any two fermions. This
implies the Pauli exclusion principle, which states that two identical fermions cannot
occupy the same quantum state. The ground state of a many-body system of N
3
noninteracting identical fermions consists of a single fermion occupying each of the
N lowest-energy quantum states. Protons, neutrons, and electrons are all fermions.
Composite particles, such as atoms, can also be classified as either fermions or
bosons. A composite particle is a boson if its constituents include an even number
of fermions and it is a fermion if its constituents include an odd number of fermions.
Since a neutral atom contains an equal number of protons and electrons, it is a boson
if its nucleus includes an even number of neutrons and a fermion if its nucleus includes
an odd number of neutrons. For example, a 6Li atom is a fermion and a 7Li atom is
a boson. The majority of the experiments with ultracold fermionic atoms have been
carried out using 6Li atoms.
1.1.2 Hyperfine spin states
An alkali atom in its electronic ground state has multiple spin states. There are two
contributions to its spin: the electronic spin S with quantum number s = 12
and the
nuclear spin I with quantum number i. The 2(2i + 1) spin states can be labeled
|ms,mi〉, where ms and mi specify the eigenvalues of Sz and Iz. The Hamiltonian for
a single atom includes a hyperfine term that can be expressed in the form
Hhyperfine =2Ehf
(2i+ 1)h2I · S. (1.2)
This term splits the ground state of the atom into two hyperfine multiplets with
energies differing by Ehf . The eigenstates can be labeled by the eigenvalues of the
hyperfine spin F = I + S. The associated quantum numbers f and mf specify the
eigenvalues of F 2 and Fz. The eigenvalues of Hhyperfine are
Ef,mf=f(f + 1)− i(i+ 1)− 3
4
2i+ 1Ehf . (1.3)
4
The two hyperfine multiplets of an alkali atom consist of 2i+ 2 states with f = i+ 12
and 2i states with f = i − 12. For example, a 6Li atoms has nuclear spin quantum
number i = 1. The two hyperfine multiplets consist of four states with f = 32
and
two states with f = 12. The f = 3
2multiplet is higher energy by Ehf . The frequency
associated with the hyperfine splitting is Ehf/h ≈ 225 MHz.
In the presence of a magnetic field B = Bz, the Hamiltonian for a single atom
has a magnetic term. The magnetic moment µ of the atom is dominated by the
term proportional to the spin of the electron: µ = µS/( 12h). The magnetic moment
µ of an alkali atom such as Li is approximately that of the single electron in the
outermost shell: µ ≈ −2µB, where µB is the Bohr magneton. The magnetic term in
the Hamiltonian can be expressed in the form
Hmagnetic = −2µ
hS ·B. (1.4)
If B 6= 0, this term splits the two hyperfine multiplets of an alkali atom into 2(2i+1)
hyperfine states. In a weak magnetic field satisfying µB Ehf , each hyperfine mul-
tiplet is split into 2f + 1 equally-spaced Zeeman levels |f,mf 〉. In a strong magnetic
field satisfying µB Ehf , the states are split into a set of 2i+1 states with ms = +12
whose energies increase linearly with B and a set of 2i + 1 states with ms = −12
whose energies decrease linearly with B. Each of those states is the continuation
in B of a specific hyperfine state |f,mf 〉 at small B. It is convenient to label the
states by the hyperfine quantum number f and mf for general B, in spite of the
fact that those states are not eigenstates of F 2 if B 6= 0. We denote the eigenstates
of Hhyperfine + Hmagnetic by |f,mf ;B〉 and their eigenvalues by Ef,mf(B). The two
eigenstates with the maximal value of |mf | are independent of B:
∣
∣f = i+ 12,mf = ±(i+ 1
2);B
⟩
=∣
∣ms = ±12,mi = ±i
⟩
. (1.5)
5
Figure 1.1: The hyperfine energy levels as a function of the magnetic field 6Li atoms.Figure from Ref. [7]
Their eigenvalues are exactly linear in B:
Ef,mf(B) =
i
2i+ 1Ehf ∓ µB. (1.6)
If B 6= 0, each of the other eigenstates |f,mf ;B〉 is a linear superposition of the two
states |f = i− 12,mf〉 and |f = i+ 1
2,mf 〉.
The dependence of the hyperfine energy levels of 6Li atoms on the magnetic field
is illustrated in Fig. 1.1. At B = 0, the hyperfine multiplets with f = 32
and f = 12
are
split by Ehf . The magnetic energy scale µB is comparable to the hyperfine splitting
Ehf when B is about 80 Gauss. At higher magnetic fields, the three ms = −12
states
decrease linearly with B, while the three ms = +12
states increase linearly.
Thus far all experiments with fermionic atoms with three spin states have been
carried out using 6Li atoms in the three lowest hyperfine spin states. For simplicity,
6
we label these hyperfine spin states by integers in order of increasing energy:
|1〉 = |12,+1
2〉, (1.7)
|2〉 = |12,−1
2〉, (1.8)
|3〉 = |32,−3
2〉. (1.9)
1.2 2-body system at low energy
In this section, we describe the interactions between two atoms with very low en-
ergy and we introduce the concept of the scattering length. Then, we explore the
universality of 2-body physics for large scattering length.
1.2.1 Natural scale
According to the wave-particle duality of quantum mechanics, a particle with momen-
tum p behaves like a wave with the de Broglie wavelength λ = 2πh/p. In a gas with
temperature T , the typical momentum of a particle with mass m is p ≈√mkBT . In
ultracold atom experiments, the typical temperatures are lower than a micro-Kelvin
(µK). Thus the de Broglie wavelength of a 6Li atom in the experiment is greater than
105 a0, where a0 is the Bohr radius: a0 ≈ 5.29×10−11m. Such an atom can not resolve
any structure that is smaller than the wavelength of the atom. If the wavelength of
the atom is larger than the size of an atom, which is typically a few Bohr radii, it
cannot resolve the atomic structure. Thus, the atoms can be accurately described by
pointlike particles. If the wavelength of the atom is larger than the range of the force
between atoms, it cannot resolve the details of the interactions. The force could just
as well be replaced by a short-range force with a suitably adjusted strength or even
by a zero-range force. This makes a detailed description of the forces between atoms
unnecessary.
7
The force between two atoms can be specified by a potential U(r) which gives
the potential energy as a function of the separation r of the atoms. The potential
between two neutral atoms is highly repulsive at short distances that are comparable
to the Bohr radius and it is attractive at longer distances. The repulsion between
the outermost electron shells of the two atoms can change the charge distributions
of the shells, making the atoms electrically polarized. This deformation causes an
attractive force between the polarized atoms. This attractive potential is called the
van der Waals potential:
UvdW (r) = −C6
r6, (1.10)
where C6 is a constant that is different for each element. The constant C6 defines a
length scale called the van der Waals length `vdW :
`vdW =4
√
mC6/h2. (1.11)
This is the distance at which the kinetic energy p2/m ∼ h2/m`2vdW of a pair of
atoms is comparable to their potential energy |UvdW (`vdW )| ∼ C6/`6vdW . For 6Li
atoms, `vdW ≈ 65 a0. The van der Waals length is the natural length scale for the
interaction between neutral atoms with sufficiently low energy. Atoms with de Brogile
wavelengths larger than `vdW are unable to resolve even the power-law tail of the
interatomic potential. Their interactions can therefore be described by a short-range
potential or even by a zero-range potential.
The constant C6 also determines the van der Waals energy scale given by
EvdW =h2
m`2vdW
. (1.12)
This is the typical size of the binding energy of the most weakly-bound diatomic
molecules. It also sets the temperature scale EvdW/kB below which we consider the
atoms to be ultracold. For 6Li atoms, this temperature is 6.8 mK. This is comparable
8
to the temperature Ehf/kB set by hyperfine splitting for 6Li atoms, which is about
11 mK. Since the typical wavelength of ultracold atoms is larger than `vdW , they
are unable to resolve details of the interaction potential. This makes it possible to
describe their interactions accurately by a few parameters.
1.2.2 Two-body scattering
In this section, we briefly review scattering of a 2-body system and define some of
the important scattering parameters at low energy.
Let us consider the scattering of a beam of atoms on a target. Part of the beam
is scattered by the target and the remainder of the beam passes through the target
unscattered. If a beam travels along the z axis, the wavefunction of the atom in
the absence of the target is a plane wave eikz, where k is the wavenumber of the
atom, which is determined by its energy: E = h2k2/(2m). The plane wave at a
large distance r from the target can be expressed as an infinite sum of incoming and
outgoing spherical waves [13]:
eikz −→ i
2k
∞∑
l=0
(2l + 1)il[
e−i(kr−lπ/2)
r− ei(kr−lπ/2)
r
]
Pl(cos θ), (r →∞)
(1.13)
where Pl(cos θ) is the Legendre polynomial. Assuming that the potential is rotation-
ally symmetric, the scattered waves are azimuthally symmetric. In the presence of
the target, the wavefunction ψ(r) of the atom at large distance r can still be de-
composed into incoming and outgoing spherical waves. Conservation of probability
requires that the outgoing spherical waves have the same amplitude as in the plane
wave but they can differ in phase:
ψ(r) −→ i
2k
∞∑
l=0
(2l+1)il[
e−i(kr−lπ/2)
r− e2iδl
ei(kr−lπ/2)
r
]
Pl(cos θ), (r →∞)
(1.14)
9
where δl is the phase shift due to the scattering, which depends on the wavenumber
k. The asymptotic wavefunction ψ(r) can be expressed as the sum of the incident
plane wave in Eq. (1.13) and an outgoing spherical wave:
ψ(r) −→ eikz + f(θ)eikr
r, (r →∞) (1.15)
where f(θ) is the scattering amplitude:
f(θ) =∞∑
l=0
2l + 1
k cot δl − ikPl(cos θ). (1.16)
A convenient observable associated with the scattering probability is the cross sec-
tion. The number of incident atoms per unit time and unit area is proportional
to their velocity times their probability density: (hk/m) × |eikz|2 = hk/m. Simi-
larly, the number of scattered atoms per unit time and unit area is proportional to
(hk/m)× |f(θ)eikr/r|2 = (hk/m)|f(θ)|2/r2. Taking the ratio of these two quantities
and integrating over the surface gives the cross section. The differential cross section
is therefore given by
dσ
dΩ= |f(θ)|2. (1.17)
The differential solid angle is dΩ = 2π sin θdθ. The cross section σ is obtained by
integrating over the scattering angle θ.
If the potential is short-ranged with no power-law tail, it is known that the
k2l+1 cot δl for small scattering energy E = h2k2/(2m) can be expanded in powers
of k2 [14]:
k2l+1 cot δl =∞∑
n=0
cl,nk2n. (1.18)
The coefficients cl,n are called effective range parameters. For the S-wave phase shift,
the leading terms in the effective range expansion are
k cot δ0 = −1
a+
1
2rek
2 + · · · , (1.19)
10
where a is the scattering length and re is the effective range. The natural magnitudes
for the effective range coefficients is determined by the length scale ` set by the range
of interaction. By dimensional analysis, cl,n can be expressed as `2n−2l−1 multiplied
by a dimensionless coefficient. In the absence of an enhancement mechanism, we
expect the dimensionless coefficient to be order 1. An example of an enhancement
mechanism is a bound state that is very close to the two-atom threshold. If the bound
state is in the S-wave (l = 0) channel, the scattering length a is large compared to
`. Upon inserting k cot δl from Eq. (1.18) into the scattering amplitude in Eq. (1.16),
one can see that that the S-wave (l = 0) term dominates the amplitude at low energy.
The higher partial waves (l > 0) are suppressed by (k`)2l.
The low energy expansions in Eqs. (1.18) and (1.19) express the information about
the potential that is relevant at low energy in terms of a few parameters, such as a
and re. The coefficients of higher powers of k in the effective range expansion are less
important, because they are suppressed by powers of the energy.
The potential between atoms is not short-ranged, because it has the power-law
tail at large r given by the van der Waals potential in Eq. (1.10). Consequently,
the low energy expansions in Eq. (1.18) break down [15]. Since the van der Waals
potential decreases as a high power of r, some of the terms in the expansion are the
same as for a short-range potential. For the S-wave phase shift, the two leading terms
in the effective range expansion still have the form in Eq. (1.19). The expansion of
k cot δ1 still starts at order k−2, so the P-wave term in the cross section is suppressed
by k2 at low energy. For all higher partial waves (l ≥ 2), the expansion of k cot δl
start at order k−4, so the corresponding terms in the cross section are suppressed by
k4. Therefore, the S-wave term still dominates at sufficiently low energy [15, 11]. At
extremely low energy, the only relevant interaction parameter is the scattering length
a.
11
1.2.3 Universality with large scattering length
As discussed in the previous section, in the generic case when the effective range
parameters have natural values set by the range `, low energy scattering can be
treated systematically by expanding in powers of the energy. In this subsection, we
discuss the case of an unnaturally large scattering length |a| ` and we introduce
the concept of universality.
We consider two particles with a large scattering length |a| ` and with energy
small compared to the scale h2/m`2 set by the range. For convenience, we will refer
to the particles as atoms. We will see that this system has nontrivial properties that
are completely determined by the scattering length. We will refer to these properties
as universal. This adjective is appropriate because different systems with a large
scattering length will have identical low-energy behavior up to one overall length
scale that is set by a. If we insert the expansion in Eq. (1.19) into the S-wave term
in the amplitude in Eq. (1.16), there are two terms that are not suppressed. These
terms define the universal scattering amplitude:
f(k) =1
−1/a− ik . (1.20)
By inserting Eq. (1.20) into Eq. (1.17) and integrating over the solid angle, we obtain
the total cross section:
σ(k) =4π
1/a2 + k2. (1.21)
Thus the cross section for low-energy scattering has a nontrivial form that is com-
pletely determined by the scattering length.
If the scattering length a is large and positive, there is a diatomic molecule with
universal properties. We will refer to this bound state as the shallow dimer. Quan-
tum mechanics implies that bound states are associated with poles in the scattering
12
amplitude f(k) for complex values of the momentum k. If f(k) has a pole on the
positive imaginary axis at k = iκ, then there is a bound state with binding energy
h2κ2/m. The scattering amplitude in Eq. (1.20), has a pole at k = i/a. If a > 0, this
pole is in the upper half-plane of the complex variable k, so there is a corresponding
bound state. The universal expression for the binding energy of the shallow dimer is
ED =h2
ma2(a > 0). (1.22)
The typical separation of its constituents is a. In addition to the shallow dimer,
there may also be diatomic molecules whose binding energies are of order h2/(m`2),
or larger. We will refer to them as deep dimers, because they are much more deeply
bound than the shallow dimer. A deep dimer has no universal properties. Its binding
energy is much larger than that of the shallow dimer. The typical separation of its
constituents is order ` or smaller, so it is much smaller than the shallow dimer.
The limit of large scattering length |a| ` is closely related to the zero-range
limit `→ 0. The zero-range limit can be achieved by taking the range of the interac-
tion potential to zero while simultaneously increasing its depth so that the scattering
length remains fixed. The limit is independent of the shape of the potential. In
the zero-range limit, the universal scattering amplitude in Eq. (1.20) becomes ex-
act up to arbitrarily high energies. The universal expression for the binding energy
in Eq. (1.22) also becomes exact. If there are any deep dimers, their binding ener-
gies become infinitely large. Since the universal results are the same in both limits,
we will sometimes use the phrases large scattering length, zero-range, and universal
interchangeably.
In the limit a → ∞, the universal cross section approaches 4π/k2, which is the
maximum value allowed by unitarity. The limit a → ±∞ is therefore called the
unitary limit. In this limit, there is no length scale associated with the interactions.
13
Thus the system has a symmetry under scaling the spacial coordinates by an arbitrary
positive factor λ and the time by a factor λ2. This symmetry is called scale invariance.
The scale invariance of the unitary limit manifests itself at finite scattering length
by simple scaling behavior under simultaneous scaling of a and kinematic variables.
For example, when a and the momentum variable k are scaled by the factors λ and
λ−1, the cross section in Eq. (1.21) is changed by a factor λ2: σ(λ−1k;λa) = λ2σ(k; a).
The binding energy in Eq. (1.22) also shows the scaling behavior. When a is scaled
by λ, ED is changed by a factor λ−2: ED(λa) = λ−2ED(a). This scaling behavior is
a general feature of the system with large scattering length. It follows from the fact
that the scattering length a is the only interaction parameter that sets a length scale
in the zero-range limit.
Universality is important, because it relates phenomena in various fields of physics.
There are examples of systems with large scattering lengths in nuclear physics and
high energy physics as well as in atomic physics. In nuclear physics, the best example
is the neutron, whose two spin states interact with a large negative scattering length.
In high energy physics, a good example is the charm mesons D∗0 and D0, which form
a very weakly bound state called the X(3872) and therefore must have a large positive
scattering length. A classic example in atomic physics is 4He atoms, whose scattering
length is about +200 a0, which is much larger than the van der Waals length scale
`vdW ≈ 10 a0. Another example in atomic physics is the three lowest hyperfine states
of 6Li atoms at a large magnetic field. Each pair of spin states interacts with a
scattering length −2160 a0, which is much larger than the van der Waals length scale
`vdW ≈ 65 a0.
Atomic physics is unique in that it is also possible to tune the scattering lengths of
atoms experimentally. This can be accomplished by adjusting the magnetic field near
a Feshbach resonance. A Feshbach resonance arises when a diatomic molecule is near
14
B
aa bg
B0+∆ B0
Figure 1.2: Dependence of the scattering length on the magnetic field near a Feshbachresonance.
the threshold for a pair of atoms. If the diatomic molecule has a magnetic moment
that is different from twice that of the atoms, its energy relative to the threshold
can be changed by a magnetic field. This also changes the scattering length a of the
atoms. Near the Feshbach resonance, the scattering length can be approximated by
a(B) = abg
(
1− ∆
B − B0
)
. (1.23)
The scattering length diverges at B = B0, which is the position of the Feshbach
resonance. It also vanishes at B = B0 + ∆. Far above or below the resonance, the
15
scattering length approaches abg. The dependence of the scattering length on the
magnetic field is illustrated in Fig. 1.2 for the case ∆ < 0. By adjusting the magnetic
field, the scattering length can be tuned to any desired value. In particular, it can
be made infinitely large by tuning B to B0. For a detailed discussion of Feshbach
resonances, the readers is referred to a review article [16].
1.2.4 Scattering lengths of 6Li atoms
In this subsection, we introduce our conventions for the scattering lengths of a
fermionic atom with three spin states. We also show how the scattering lengths
for the three lowest hyperfine states of 6Li atoms depend on the magnetic field.
We label the three spin states of the atom by the integers 1, 2, and 3. We
denote the scattering length of the pair ij by either aij = aji or ak, where (ijk) is
a permutation of (123). The two-body physics for fermions with two distinct spin
states that have a large pair scattering length aij is very simple in the zero-range
limit. The scattering amplitude for the pair ij with relative wavenumber k is given
by
fij(k) =1
−1/aij − ik, (1.24)
which is Eq. (1.20) with a replaced by aij. For positive aij , there is a weakly-bound
diatomic molecule with constituents i and j that we will refer to as either the (ij)
dimer or the ij-dimer. Its binding energy is h2/(ma2ij). We refer to the (12), (23), and
(13) dimers collectively as shallow dimers. As mentioned in previous section, there
are also deep dimers whose binding energies are comparable to or larger than the van
der Waals energy scale and are insensitive to changes in the large scattering lengths.
In the case of 6Li atoms, the three spin states are the hyperfine states given
in Eqs. (1.7), (1.8), and (1.9). The pair scattering lengths a12, a23, and a13 have
Feshbach resonances near 834 G, 811 G, and 690 G, respectively [17]. Beyond these
16
Feshbach resonances, all three scattering lengths approach the triplet scattering length
−2140 a0, which is large and negative. The zero-range approximation should be
accurate if |a12|, |a23|, and |a13| are all much larger than `vdW ≈ 62.5 a0 and it should
be at least qualitatively useful if |aij | > 2`vdW. There are two regions of the magnetic
field in which all three scattering lengths are larger than 2`vdW ≈ 125 a0: a low-field
region 122 G < B < 485 G and a high-field region B > 608 G. These two universal
regions are separated by a non-universal region in which all three scattering lengths
go through zeros. Efimov physics in the universal regions will be characterized by
values of κ∗ and η∗ that may not be the same in the two regions. In general, these
parameters may be expected to vary slowly with the magnetic field, just like the
scattering length away from a Feshbach resonance. In a sufficiently narrow region of
magnetic field, they can be treated as constants. While their values could in principle
be calculated from microscopic atomic physics, in practice they have to be determined
by measurements of 3-body observables.
In Fig. 1.3, the three scattering lengths a12, a23, and a13 are shown as functions
of the magnetic field in the low-field region from 0 to 600 G [18]. Throughout most
of this region, the smallest scattering length is a12. It satisfies |a12| > 2 `vdW in the
interval 122 G < B < 485 G and achieves its largest value −290 a0 = −4.6 `vdW near
320 G. This interval therefore contains a universal region in which all three scatter-
ing lengths are negative and relatively large. The zero-range approximation should
be quantitatively useful in the middle of this interval, but it becomes increasingly
questionable as one approaches the edges.
In Fig. 1.4, the three scattering lengths a12, a23, and a13 are shown as functions
of the magnetic field [18] in the high-field region from 600 G to 1200 G. This region
includes the Feshbach resonances in a12, a23, and a13 near 834 G, 811 G, and 690 G,
respectively. Beyond these Feshbach resonances, all three scattering lengths approach
17
0 200 400 600B [G]
-1.0
-0.5
0.0
0.5
1.0
a [u
nits
of 1
03a 0]
122313
Figure 1.3: The scattering lengths in units of 103a0 for the three lowest hyperfinestates of 6Li atoms as functions of the magnetic field B in the low-field region from0 to 600 G [18]. The two vertical dotted lines mark the boundaries of the universalregion in which the absolute values of all three scattering lengths are greater than2 `vdW.
the triplet scattering length −2140 a0. For B > 637 G, the absolute values of all
three scattering lengths are larger than 2140 a0 ≈ 34 `vdW. An estimate of the
lower boundary of this universal region is 608 G, where the smallest scattering length
is a13 = 125 a0 ≈ 2`vdW. The zero-range approximation should be very accurate
throughout most of this high-field region. The physics in this universal region is rich,
with the Feshbach resonances marking the boundaries between regions in which 3, 2,
1, or 0 of the three scattering lengths are positive.
1.3 Efimov physics
In this section, we describe the energy spectrum of Efimov trimers for identical bosons
and for fermions with three spin states. We also describe how Efimov physics can be
observed through the loss of trapped atoms and we summarize important experimen-
18
600 800 1000 1200B [G]
-5
0
5
10
a [u
nits
of 1
03a 0]
122313
Figure 1.4: The scattering lengths in units of 103a0 for the three lowest hyperfinestates of 6Li atoms as functions of the magnetic field B in the high-field region from600 G to 1200 G [18]. The three vertical lines mark the positions of the Feshbachresonances.
tal observations as of July 2011.
1.3.1 Efimov trimers
Three particles with large scattering length also have universal properties, but they
are much more intricate than those for two particles. These universal properties were
discovered by Vitaly Efimov [9, 10] and developed in a series of subsequent papers.
The most dramatic consequence is the existence of a sequence of universal 3-body
bound states that are now called Efimov trimers. The spectrum of Efimov trimers is
particularly simple and remarkable in the unitary limit in which the scattering length
is taken to infinity. There are an infinite number of geometrically spaced low-energy
bound states with an accumulation point at zero energy. In the case of identical
bosons, the binding energies of two successive trimers differ by a multiplicative fac-
tor of λ20 ≈ 515, where λ0 = eπ/s0 ≈ 22.7. The constant s0 is the solution to a
19
transcendental equation:
s0 cosh(πs0/2) =8√3
sinh(πs0/6). (1.25)
The numerical value of s0 is approximately 1.00624. The spectrum of Efimov trimers
in the unitary limit can be expressed as
E(n)T = λ
2(n∗−n)0
h2κ2∗
m(a = ±∞) , (1.26)
where κ∗ is the binding wavenumber of the trimer labeled n∗ and n is an integer.
In the 2-body sector, the unitary limit is characterized by scale invariance. How-
ever scale invariance requires the binding energies of discrete bound states to be either
0 or∞. Thus the existence of Efimov trimers indicates that scale invariance is violated
in the 3-body sector. However, there is a remnant of that symmetry. The Efimov
spectrum in Eq. (1.26) is compatible with discrete scale invariance with the discrete
scaling factor λ0. We will refer to universal phenomena associated with discrete scale
invariance in the three-body sector as Efimov physics [11].
In the 2-body sector, the scattering length is the only length scale provided by
interactions in the zero-range limit. In the 3-body sector, the discrete spectrum of
Efimov states in the unitary limit requires a 3-body parameter that provides another
length scale in addition to the scattering length. If there were no such parameter,
the system would have continuous scale invariance in the unitary limit. One simple
choice for the 3-body parameter is the wavenumber κ∗ defined by the spectrum of
Efimov trimers in the unitary limit in Eq. (1.26). Discrete scale invariance requires
that any dependence of physical quantities on the parameter κ∗ much be log-period
with discrete scaling factor λ0. The universal properties in the 3-body sector for
identical bosons are determined by the scattering length a and the 3-body parameter
κ∗.
20
1/a
K
x 22.7
x 22.7
Figure 1.5: Binding energies of three successive Efimov trimers (red curves) for iden-tical bosons. Arrows indicate the discrete scaling invariance along the directions ofthe arrows. The variable K is defined in Eq. (1.27).
The binding energies of three successive Efimov trimers are illustrated in Fig. 1.5
using variables that are particularly well suited to exhibiting the discrete scale invari-
ance. The horizontal axis is the inverse scattering length a−1. The vertical axis is an
energy variable K that also has a dimension (length)−1:
K = sign(E)√
m|E|/h. (1.27)
According to the Efimov effect, there are infinitely many trimer states along the
vertical axis (a = ±∞) as one approaches the origin from below. Only three of those
infinitely many trimers are shown in the figure. The spectrum of Efimov trimers is
K = −λn∗−n0 κ∗, so the ratio of the positions of two adjacent trimers on the axis is
λ0 ≈ 22.7. In the figure, the straight line approaching the origin at a 45 angle
with respect to the horizontal axis represents the threshold for atom-dimer scattering
21
states, which is K = −1/a. For a > 0, the trimers disappear through the atom-
dimer threshold at values of the scattering length a = λn∗−n0 a∗ that differ by the
discrete scaling factor. There is a threshold resonance in atom-dimer scattering at
each of these scattering lengths because of the trimer state near the threshold. The
horizontal axis corresponds to the threshold for three-atom scattering states. For
a < 0, the trimer states cross the three-atom threshold at values of the scattering
length a = λn∗−n0 a′∗ that differ by the discrete scaling factor. There is a resonance in
three-atom scattering at each of those values of the scattering lengths because of the
Efimov trimer near the 3-atom threshold. The scattering lengths a∗ and a′∗ at which
the Efimov trimers cross the thresholds differ from 1/κ∗ by universal multiplicative
constants: a∗ = 0.071/κ∗ and a′∗ = −1.5/κ∗.
Efimov trimers are sharp states with the spectrum in Eq. (1.26) only if there are
no deep dimers in the 2-body spectrum. If there are deep dimers, Efimov trimer can
decay into an atom and a deep dimer and this gives the trimer a width. The inclusive
effects of all the deep dimers can be taken into account by analytically continuing
the Efimov parameter κ∗ to a complex value that is conveniently expressed in the
form κ∗ exp(iη∗/s0), where κ∗ and η∗ are positive real parameters [19]. Making the
substitution κ∗ → κ∗ exp(iη∗/s0) on the right side of Eq. (1.26), we find that binding
energies of the Efimov trimers acquire imaginary parts. The imaginary part can be
interpreted as half of the decay width Γ(n)T of the Efimov resonance. The binding
energies and widths of the Efimov trimers are
E(n)T = λ
2(n∗−n)0
h2κ2∗ cos(2η∗/s0)
m(a = ±∞) , (1.28)
Γ(n)T = λ
2(n∗−n)0
2h2κ2∗ sin(2η∗/s0)
m. (1.29)
Efimov trimers can also exist in fermionic system. The simplest fermionic system
is identical fermions. Because of the Pauli exclusion principle, S-wave interactions are
22
prohibited so their scattering length is zero. Thus, identical fermions with extremely
low energy are essentially noninteracting. In the case of fermionic atoms with two spin
states, the two spin states can interact through a large scattering length. However in
the 3-atom sector, the interaction is not strong enough to produce the Efimov effect.
The simplest case in which the Efimov effect arises is fermionic atoms with three
spin states. The low-energy interaction of each pair ij of spin states is given by a
scattering length aij . Thus there are three independent scattering lengths: a12, a23,
and a13. The Efimov effect arises only if all three scattering lengths are large. The
discrete scaling factor for fermions with 3 spin states has the same value λ0 ≈ 22.7
as for identical bosons. The universal properties are completely determined by the 3
scattering lengths a12, a23, and a13 and by the 3-body parameters κ∗ and η∗.
We take the zero of energy to be the scattering threshold for three atoms in the
three different spin states. If ajk is positive, the scattering threshold for an atom of
type i and a (jk) dimer is −h2/(ma2jk). We will refer to this scattering threshold as
the i + (jk) atom-dimer threshold. A trimer whose constituents are atoms of types
1, 2, and 3 must have energy below the 3-atom threshold and below the i + (jk)
atom-dimer threshold if ajk > 0. In general the spectrum of Efimov trimers depends
in a complicated way on the three scattering lengths. However if the three scattering
lengths are all equal, the spectrum reduces to that for identical bosons, which is
illustrated in Fig. 1.5. In the case, we can write a12 = a23 = a31 = a. In the unitary
limit a = ±∞, the trimer energies are given by Eq. (1.26) or by Eq. (1.29) if there are
deep dimers. The Efimov trimers disappear through the 3-atom threshold at negative
values of a given by a = λn−n∗
0 a′∗. They disappear through the atom-dimer threshold
at positive values of a given by a = λn−n∗
0 a∗.
The first experimental studies of fermionic atoms with three spin states have been
carried out using the three lowest hyperfine states of 6Li atoms [7, 8]. Efimov physics
23
in the system is complicated because the three pairwise scattering lengths a12, a23,
and a31 all change with the magnetic field as illustrated in Figs. 1.3 and 1.4.
One of the most important theoretical developments in few-body physics in re-
cent years was the discovery that the universality of particles with large scattering
lengths extends to the 4-body sector. In papers published in 2004 and 2007, Platter,
Hammer, and Meissner showed that there are two universal four-body bound states
associated with each Efimov trimer [20, 21]. They calculated the binding energies of
these universal tetramers in a limited range of 1/a. In 2009, von Stecher, D’Incao,
and Green mapped out the spectrum of the universal tetramers over the entire range
of 1/a [22]. In particular, they calculated the scattering lengths at which the univer-
sal tetramers appear at the 4-atom threshold and they pointed out that they could
be observed through 4-atom loss resonances. Universality in the 4-body sector and
beyond remains an important frontier of few-body physics.
1.3.2 Loss of atoms
The easiest way to observe Efimov trimers in ultracold atomic gases is through reso-
nant enhancement of loss rates. The atoms are trapped in a potential created by a
magnetic field or by laser beams. An atom can escape from the trapping potential if
it acquires a kinetic energy that is larger than the depth of the potential through an
inelastic scattering process. The escaping atoms can usually not be observed directly,
but they can be observed indirectly through the decrease in the number of trapped
atoms. By appropriate choice of hyperfine states, one can arrange that two-body
inelastic processes are forbidden by conservation of energy. At sufficiently low densi-
ties, the dominant loss mechanisms will then be through inelastic scattering processes
involving three atoms. There are two such loss mechanisms: three-body recombination
and dimer relaxation [23, 24, 25].
24
Figure 1.6: Three-body recombination process: three atoms collide with small mo-menta, two of the three bind to form a dimer (which can be either a deep dimer or ashallow dimer), and the dimer and remaining atom recoil with large momenta.
Three-body recombination, which is illustrated in Fig. 1.6, is the inelastic scatter-
ing of three incoming atoms into a dimer and an atom. By conservation of energy, the
increase in the kinetic energies of the outgoing dimer and atom must be equal to the
binding energy of the dimer. The outgoing dimer can be either a deep dimer whose
binding energy is of order h2/(m`2vdW ), or a shallow dimer, whose binding energy is
h2/(ma2). If the kinetic energy of the outgoing atom or dimer is larger than the depth
of the trapping potential, it can escape from the trap. This results in a decrease in
the number of trapped atoms.
The three-body recombination rate can be resonantly enhanced by an Efimov
trimer near the three-atom threshold. Therefore, the observation of a three-atom loss
resonance is indirect evidence for an Efimov trimer. In the case of identical bosons,
Efimov trimers approach the three-atom threshold at negative values of a that differ
by the discrete scaling factor 22.7, as illustrated in Fig. 1.5. There will be a three-
atom loss resonance at each of these values of a. There are also interference minima
in the three-body recombination rate at positive values of a that differ by 22.7. These
interference features are not directly related to Efimov trimers, but they are another
manifestation of Efimov physics. In the case of fermions with three spin states, there
can be three-atom loss resonances if all three scattering lengths are negative. There
can be interference minima if at least one scattering length is positive. For the three
25
Figure 1.7: Dimer relaxation process: an atom and a shallow dimer collide withsmall momenta, two of three atoms form a different dimer (which can be either adeep dimer and a shallow dimer), and the outgoing dimer and atom recoil with largemomenta.
lowest hyperfine states of 6Li atoms, the three scattering lengths change dramatically
with the magnetic field because of the Feshbach resonances, as shown in Figs. 1.3 and
1.4. This complication leads to a rich structure of resonance peaks and interference
minima in the three-body recombination rate, as we will see in Chapters 2 and 3.
Dimer relaxation, which is illustrated in Fig. 1.7, is the inelastic scattering of an
incoming atom plus a shallow dimer into an outgoing atom plus a dimer. Sometimes
in this thesis, the dimer relaxation process is also called atom-dimer relaxation to
emphasize the incoming state. In the case of identical bosons, the outgoing dimer must
be a deep dimer. In the case of fermions with three spin states, the outgoing dimer
can be either a deep dimer or a shallow dimer with a different pair of constituents
that is more tightly bound. By conservation of energy, the increase in the kinetic
energies of the atom and dimer must be equal to the difference between the binding
energies of the incoming and outgoing dimers. If the kinetic energy of the outgoing
atom or dimer is larger than the depth of the trapping potential, it can escape from
the trap. This results in a decrease in the number of trapped atoms.
The dimer relaxation rate can be resonantly enhanced if there is an Efimov trimer
near the atom-dimer threshold. In the case of identical bosons, Efimov trimers ap-
proach the atom-dimer threshold at positive values of a that differ by the discrete
26
scaling factor 22.7, as illustrated in Fig. 1.5. There will be an atom-dimer loss reso-
nance at each of these values of a. In the case of fermions with three spin states, there
can be atom-dimer loss resonances if at least one of the three scattering lengths is
positive. If at least two of the three scattering lengths are positive, there can also be
interference minima in the dimer relaxation rates. The minima are a manifestation
of Efimov physics that has no analog in identical bosons [26]. For the three lowest
hyperfine states of 6Li atoms, the dramatic changes in the three scattering lengths
as a function of the magnetic field leads to a rich structure in the dimer relaxation
rates, as we will see in Chapter 4.
1.3.3 Observations of Efimov trimers
The first discovery of an Efimov trimer in atomic physics was made in August 2007
by a group at the University of Innsbruck led by Rudi Grimm [27]. They observed
a resonant enhancement in the three-body recombination rate in a gas of ultracold
bosonic 133Cs atoms with large negative scattering length. They also observed an
interference minimum in the recombination rate at a positive scattering length. In a
subsequent experiment with a mixture of 133Cs atoms and dimers, the Innsbruck group
observed an atom-dimer loss resonance [28]. Efimov trimers have also been observed
using other types of bosonic atoms. In January 2009, a group at the University of
Florence observed three-body recombination loss resonances in a mixture of 41K and
87Rb atoms [29]. They can be attributed to heteronuclear Efimov trimers that are
composed of both K and Rb atoms. In April 2009, the Florence group observed
the three-body recombination loss resonances associated with two successive Efimov
trimers in a gas of ultracold 39K atoms [30]. The ratio of the scattering lengths at
these resonances was consistent with the predicted discrete scaling factor of 22.7. In
June 2009, a group at Bar-Ilan University observed an Efimov loss resonance and an
27
interference minimum on opposite sides of a Feshbach resonance in gas of ultracold
7Li atoms [31]. In November 2009, a group at Rice University observed two Efimov
loss resonances and two interference minima on opposite sides of the same Feshbach
resonance [32].
The universal tetramers associated with Efimov trimer have been observed through
four-atom loss processes [20, 21, 22]. In March 2009, the Innsbruck group observed loss
features from two universal tetramers associated with an Efimov trimer in 133Cs atoms
[33]. In November 2009, the Rice group measured two sets of universal tetramers that
are associated with two successive Efimov trimers in 7Li atoms [32].
Efimov physics has also been observed in fermionic systems with three or more
spin states [9, 10]. There are three groups that have carried out experiments with
many-body systems consisting of 6Li atoms in the three lowest hyperfine states:
• a group at Pennsylvania State University led by Ken O’Hara, which we will
refer to as the Penn State group,
• a group at the University of Heidelberg led by Selim Jochim, which we will refer
to as the Heidelberg group,
• a group at University of Tokyo led by Masahito Ueda, which we will refer to as
the Tokyo group.
Several observations of Efimov features in 6Li atoms have been reported by these three
groups. The discovery of a three-body recombination loss resonance in the low-field
region was announced by the Heidelberg group in June 2008 [7] and by the Penn State
group in October 2008 [8]. The first theoretical analysis explaining these loss features
in terms of an Efimov trimer close to the three-atom threshold appeared in November
2008 [34]; it makes up Chapter 2 of this thesis. Another recombination loss resonance
in the high-field region was discovered by the Penn State group in August 2010 [35].
28
A more comprehensive analysis of Efimov physics in 6Li atoms was completed in
August 2009 [36]; it makes up Chapter 3 of this thesis. This work predicted two
resonances in atom-dimer relaxation due to an Efimov trimer near the atom-dimer
threshold. The resonances were identified in March 2010 by the Heidelberg group [37]
and by the Tokyo group [38]. A thorough analysis of Efimov physics in atom-dimer
relaxation was completed in June 2010 [39]; it makes up Chapter 4 of this thesis. The
most exciting recent experimental development in this field is the radio-frequency
association of atoms and dimers into Efimov trimers announced by the Heidelberg
group in June 2010 [40] and by the Tokyo group in October 2010 [41].
1.4 Effective field theory approach
Effective field theory (EFT) is a general method for describing the low-energy de-
grees of freedom of a system using the formalism of quantum field theory [42, 43].
The simplest EFT that can describe particles with a large scattering length is the
zero-range model, which describes point particles that interact only through contact
interactions. The EFT is particularly useful in exploring universality at large scat-
tering length, because all nonuniversal terms suppressed by `/a are set exactly to
zero. In this section, the EFT approach to the 2-atom and 3-atom problem using the
zero-range model will be discussed. In this section, we set h = m = 1 for simplicity.
1.4.1 Identical bosons
In this subsection, we discuss the EFT approach for identical bosons.
In the quantum field theory framework, an atom is annihilated by a quantum
field ψ(r, t) and is created by a quantum field ψ†(r, t) at space-time point r and t.
Symmetry under the exchange of identical bosons is implemented through equal-time
29
commutation relations:
[ψ(r, t), ψ(r′, t)] = 0, (1.30a)
[
ψ(r, t), ψ†(r′, t)]
= δ3(r − r′). (1.30b)
The Lagrangian density for a nonrelativistic free field is given by
Lfree = ψ†(
i∂
∂t+∇2
2
)
ψ. (1.31)
This is the kinetic term in the Lagrangian density for interacting atoms. It implies
that the Feynman propagator for an atom of energy E and momentum k is i/(E −
k2 + iε).
The interaction terms in the Lagrangian must respect the symmetries of the funda-
mental interactions. Power counting rules can be developed that indicate the relative
importance of all the possible interaction terms [44, 45]. The power-counting rules for
nonrelativistic particles with short-range interactions reveal that the most important
interaction is a contact interaction between the particles. We will not develop these
power-counting rules. Instead, we will simply argue that a contact interaction is a
natural choice to describe low-energy atoms because their long wavelengths prevent
them from resolving the structure of their interaction potential. The interaction terms
of the zero-range model are given by
Lint = −g2
4
(
ψ†ψ)2 − g3
36
(
ψ†ψ)3, (1.32)
where g2 and g3 are the coupling constants for the 2-atom and 3-atom contact inter-
actions, respectively. The factors 4 and 36 in the denominators are chosen to cancel
symmetry factors associated with permutations of identical bosons. The interaction
term in Eq. (1.32) implies that the Feynman rules for the two-atom and three-atom
vertices are −ig2 and −ig3, respectively.
30
= + + + · · ·
= +
Figure 1.8: Diagrams for 2-atom amplitude: (upper diagram) summation over allorder diagrams in g2. (lower diagram) Lippmann-Schwinger integral equation.
All effects of interactions in the two-atom sector can be encoded concisely in a
function of a single variable: the transition amplitude A(E) for the scattering of a
pair of atoms with total energy E in their center-of-momentum frame. For example,
the scattering amplitude f(E) in Eq. (1.20) is given by
A(E = k2) = 8πf(k). (1.33)
Fig. 1.8 shows two diagrammatic equations for the 2-atom transition amplitude. The
upper diagrammatic equation shows the perturbative expansion of the amplitude or-
der by order in the coupling constant g2. Because of the large scattering length,
these diagrams must be summed to all orders in g2. The lower diagrammatic equa-
tion in Fig. 1.8 is an alternative way to calculate the amplitude that exploits the
recursive nature of the perturbative expansion. The corresponding equation is called
Lippmann-Schwinger integral equation:
iA(E) = −ig2 + (−ig2)1
2
∫
d3k
(2π)3
i
E − k2 + iε
(
iA(E))
. (1.34)
The integral over the momentum k in Eq. (1.34) is ultraviolet divergent. It can be
calculated analytically by imposing a ultraviolet momentum cutoff |k| < Λ. The
31
Eq. (1.34) then reduces to an algebraic equation for A(E) whose solution is
A(E) = −[
1
g2
+Λ
4π2− 1
8π
√−E − iε
]−1
. (1.35)
This amplitude depends explicitly on the momentum cutoff Λ. It can be independent
of the cutoff only if the coupling constant g2 depends implicitly on Λ in such a way
that the cutoff dependences in Eq. (1.35) cancel out. The dependence on g2 and Λ
can be eliminated in favor of a physical quantity, such as the scattering length a. The
scattering amplitude f(k) defined by Eq. (1.33) is
f(k) =
[
8π
g2
+2Λ
π+ ik
]−1
. (1.36)
This has the same dependence on k as the universal scattering amplitude in Eq. (1.20)
if a is identified with the following function of g2 and Λ:
a =1
8π
[
1
g2
+Λ
4π2
]−1
. (1.37)
After using this equation to eliminate g2 from the transition amplitude in Eq. (1.35),
we find that it is independent of the ultraviolet cutoff:
A(E) = −8π
[
1
a−√−E − iε
]−1
. (1.38)
This procedure of removing the cutoff dependence by eliminating g2 in favor of a is
called renormalization. The parameter g2 is often referred to as a bare coupling con-
stant while a is the renormalized coupling constant. Since the transition amplitude
A(E) in Eq. (1.38) encodes all physical observables in the 2-body sector, the renor-
malization of g2 is sufficient to remove all dependence on the ultraviolet cutoff in the
2-atom sector. The fact that the renormalization of the 2-atom transition amplitude
is simple and analytic is very useful in the calculation of 3-atom amplitudes.
The transition amplitude for three atoms is much more complicated than that for
32
two atoms. The general amplitude in the center-of-momentum frame is a function
of 9 independent variables: three energies and two momentum vectors. However all
Efimov physics in the three-atom sector can be encoded in a much simpler function
AS(p, k;E) called the STM amplitude which is a function of only 3 variables: the total
energy E and two relative momenta. The STM amplitude satisfies an integral equa-
tion called the STM equation that was first derived by Skorniakov–Ter-Martirosian
[46].
The three-body problem for particles with large scattering length was not under-
stood within the EFT framework until 1999, when important progress was made by
by Bedaque, Hammer, and van Kolck [47, 48]. They introduced a diatom field d by
changing the interaction Lagrangian in Eq. (1.32) to
LBHvK =g2
4d†d− g2
4
(
d†ψ2 + ψ†2d)
− g3
36d†dψ†ψ. (1.39)
There is no kinetic term for d in the Lagrangian, so its equation of motion is
d− ψ2 − (g3/9g2) dψ†ψ = 0. (1.40)
The 3-atom contact interaction (ψ†ψ)3 in Eq. (1.32) has been replaced by an atom-
diatom contact interaction d†dψ†ψ and by an interaction that allows a transition
from a diatom to a pair of atoms and vice versa. The Lagrangian in Eq. (1.39) is
equivalent to that in Eq. (1.32). This can be seen in the 2-atom or 3-atom sector
simply by eliminating d using the equation of motion in Eq. (1.40). It is more difficult
to show that it is also equivalent in the N -atom sector with N ≥ 4.
The diatom field trick of Ref. [47] allows the general 3-atom transition amplitude
to be reduced to the much simpler transition amplitude for an atom and a diatom.
The atom-diatom transition amplitude satisfies an integral equation that is equivalent
to the STM equation. Fig. 1.9 shows a diagrammatic integral equation for the atom-
33
Figure 1.9: (upper diagram) The integral equation for the atom-diatom amplitude.(lower diagram) The integral equation for the complete diatom propagator.
diatom transition amplitude. The single and the double lines represent atom and
diatom fields, respectively. The STM amplitude is the projection of the atom-diatom
transition amplitude onto the S-wave term. The STM integral equation is
AS(p, k;E) =16π
a
[
1
2pkln
(
p2 + pk + k2 − E − iεp2 − pk + k2 − E − iε
)
+H(Λ)
Λ2
]
+4
π
∫ Λ
0
dq q2
[
1
2pqln
(
p2 + pq + q2 − E − iεp2 − pq + q2 − E − iε
)
+H(Λ)
Λ2
]
× AS(q, k;E)
−1/a+√
3q2/4− E − iε. (1.41)
where p (k) is the relative momentum of the incoming (outgoing) atom and diatom in
the center-of-momentum frame. We refer the reader to Ref. [11] for the details of the
derivation of the STM equation. The dimensionless 3-atom coupling constant H(Λ)
34
is defined by
g3 = −9g22
Λ2H(Λ), (1.42)
where H(Λ) is a dimensionless log-periodic function of Λ that can be approximated
by
H(Λ) ≈ h0cos[s0 ln(Λ/Λ∗) + arctan s0]
cos[s0 ln(Λ/Λ∗)− arctan s0]. (1.43)
This renormalization condition defines a renormalized three-body parameter Λ∗. The
analytic approximation derived in Ref. [47] was Eq. (1.43) with h0 = 1. In Ref. [49],
it was found that the analytic approximation is accurate only to about 10%. It
was found however that to a numerical accuracy of about 10−3, H is given by the
expression in Eq. (1.43) with the multiplicative numerical constant h0 = 0.879 [49].
For the practical solution of the STM equation in Eq. (1.41), it is convenient to fix
the numerical value of H(Λ) and then tune Λ to reproduce a three-body observable,
such as the binding energy of an Efimov trimer. Given the numerical value of H(Λ),
one can use Eq. (1.43) to determine Λ∗ although it is not necessary. The use of the
approximate expression in Eq. (1.43) with a generic cutoff Λ introduces an uncertainty
of about 10−3 associated with renormalization of the 3-atom contact interaction. This
uncertainty can be avoided by the very simple choice H = 0. In this case, there is no
atom-diatom contact interaction and it is the ultraviolet cutoff Λ that plays the role
of the 3-body parameter.
1.4.2 Fermions with three spin states
In this subsection, we discuss the EFT approach for fermions with three spin states.
Much of the formalism is similar to that for identical bosons except for symmetry
factors. We will focus our discussion on aspects associated with the difference in
symmetry factors.
We denote the fermionic quantum fields for the three spin states by ψi(r, t) and
35
ψ†i (r, t), where i = 1, 2, and 3. Antisymmetry under the exchange of identical
fermions is implemented through equal-time anticommutation relations:
ψi(r, t), ψj(r′, t) = δij, (1.44a)
ψi(r, t), ψ†j (r
′, t)
= δij δ3(r − r′). (1.44b)
The Lagrangian density has a kinetic term for each spin state:
Lfree =∑
i=1,2,3
ψ†i
(
i∂
∂t+∇2
2
)
ψi. (1.45)
The interactions between the atoms consist of two-atom contact interaction between
pairs of atoms in different spin states and a three-atom contact interaction between
three atoms in different spin states. The interaction terms in the Lagrangian density
are given by
Lint = −∑
i>j
gij ψ†iψ
†jψiψj − g123 ψ
†1ψ
†2ψ
†3ψ3ψ2ψ1. (1.46)
Note that there are no symmetry factors multiplying the coupling constants, unlike
in the interaction term for identical bosons in Eq. (1.32).
All effects of interactions in the two-atom sector can be encoded in three function
of a single variable: the transition amplitude Aij(E) = Aji(E) for a pair of atoms in
different spin states i and j. For example, the scattering amplitude fij(k) for that
pair of atoms is given by
Aij(E = k2) = 4πfij(k). (1.47)
Note that there is a factor of 2 difference from the corresponding equation for identical
bosons in Eq. (1.33). The diagrammatic equations for the transition amplitudeAij(E)
are similar to those in Fig. 1.8, with the two boson lines replaced by a line for atom
36
i and a line for atom j. The Lippmann-Schwinger integral equation is
iAij(E) = −igij + (−igij)
∫
d3k
(2π)3
i
E − k2 + iε
(
iAij(E))
. (1.48)
Note that there is no symmetry factor of 12
in front of the integral, unlike Eq. (1.48) for
identical bosons. The amplitude Aij differs from that for identical bosons Eq. (1.35)
in the symmetry factor:
Aij(E) = −[
1
gij
+Λ
2π2− 1
4π
√−E − iε
]−1
. (1.49)
The renormalization of the coupling constant gij is given by
aij =1
4π
[
1
gij+
Λ
2π2
]−1
, (1.50)
where aij is the scattering length for the pair of spin state i and j.
The diatom field trick of Bedaque, Hammer, and van Kolck can be generalized to
the field theory for fermion with three spin states. Three diatom fields d12, d23, and
d13 are introduced through the interaction Lagrangian
LBHvK =∑
i<j
gij
(
d†ijdij − d†ijψiψj − ψ†iψ
†jdij
)
−g123
3
(
d†12d12ψ†3ψ3 + d†23d23ψ
†1ψ1 + d†13d13ψ
†2ψ2
)
. (1.51)
All Efimov physics in the 3-atom sector can be encoded in a set of 9 STM amplitudes
Aii′(p, k;E). They are the transition amplitudes from atom i and diatom jk to atom
i′ and diatom j ′k′, where (i, j, k) and (i′, j′, k′) are permutations of (1, 2, 3). The
Lagrangian with diatom fields can be used to derive the STM equations for these
amplitudes, which are a set of 9 coupled integral equations. The explicit forms for
these equations are given in Chapter 2, 3, and 4, where they are used to calculate
various aspect of Efimov physics for the three lowest hyperfine states of 6Li atoms.
37
1.5 Outline
In the following three chapters, we present theoretical studies on Efimov physics in 6Li
atoms that were published in Refs. [34, 36, 39]. In Chapter 2, we calculate the 3-body
recombination rate for fermionic atoms with three spin states and large negative pair
scattering lengths. We provide semi-analytic expressions for the cases of 2 or 3 equal
scattering lengths. We obtain numerical results for the case of the three lowest hyper-
fine states of 6Li atoms and compare with experimental measurements of three-body
recombination rate in the low-field region. In Chapter 3, we calculate the universal
predictions for the spectrum of Efimov states and for the 3-body recombination rate
with any combination of signs for the three pair scattering lengths. Using the position
and width of an observed Efimov loss resonance as input, we calculate the binding
energies and widths of Efimov trimers and the three-body recombination rate for 6Li
atoms in the high-field region. We predict two atom-dimer loss resonances associ-
ated with Efimov trimers disappearing through the atom-dimer threshold. We also
predict an interference minimum in the three-body recombination rate at a magnetic
field where the 3-spin mixture may be sufficiently stable to allow experimental study
of the many-body system. In Chapter 4, we calculate the atom-dimer relaxation rate
for atoms and dimers composed of fermionic atoms with three spin states. We give
detailed prediction for Efimov features in 6Li atoms in the high-field region, including
the two atom-dimer loss resonance and two local minima in the dimer relaxation rate.
We conclude in Chapter 5 with an outlook on Efimov physics in 6Li atoms.
38
Chapter 2
Three-body Recombination for
Negative Scattering Lengths
In this chapter, we present calculations of the 3-body recombination rate at threshold
in the zero-range limit for fermionic atoms with three spin states and large negative
pair scattering lengths. We provide semi-analytic expressions for the cases of 2 or
3 equal scattering lengths and we obtain numerical results for the general case of 3
different scattering lengths. We apply our general results to the three lowest hyperfine
states of 6Li atoms and compare with the first 3-body recombination rate measure-
ments for these atoms. The analysis presented in this chapter was carried out in
collaboration with Eric Braaten, Hans-Werner Hammer, and Lucas Platter and was
published in Physical Review Letters [34] in August 2009.
2.1 Three-body recombination and optical theorem
We consider an atom of mass m with three distinguishable states that we label 1, 2,
and 3 and refer to as spin states. We denote the scattering length of the pair i and
j by either aij = aji or ak, where (ijk) is a permutation of (123). The rate equations
for the number densities ni of atoms in the three spin states are
d
dtni = −K3n1n2n3. (2.1)
39
By the optical theorem, the event rate constant K3 in the low-temperature limit can
be expressed as twice the imaginary part of the forward T-matrix element for 3-atom
elastic scattering in the limit where the momenta of the atoms all go to 0. Using
diagrammatic methods, the T-matrix element for elastic scattering can be expressed
as the sum of 9 amplitudes corresponding to the 3 possible pairs that are the first
to scatter and the 3 possible pairs that are the last to scatter. For small collision
energies, the leading contributions to those amplitudes come from the S-wave terms,
which we denote by Aij(p, p′), where p (p′) is the relative momentum between the
pair that scatters first (last) and the third atom labelled i (j). The rate constant K3
in Eq. (4.1) is
K3 =32π2
m
∑
i,j
aiajImAij(0, 0), (2.2)
where the sums are over i, j = 1, 2, 3.
2.2 STM equations at threshold
The amplitudes Aij(p, p′) can be calculated in the zero-range limit by solving 9 cou-
pled integral equations that are generalizations of the Skorniakov–Ter-Martirosian
(STM) equation [46]. To determine ImAij(0, 0), it is sufficient to solve the 9 coupled
STM equations for Aij(p, 0):
Aij(p, 0) =1− δijp2
+2
π
∑
k
(1− δkj)
×∫ Λ
0
dq Q(q/p)Dk(q)Aik(q, 0), (2.3)
where
Q(x) =x
2log
1 + x+ x2
1− x+ x2, (2.4)
Dk(q) = (−1/ak +√
3q/2)−1, (2.5)
40
and Λ is an ultraviolet cutoff. The solutions to Eqs. (2.3) are singular as p → 0.
The singular terms, which are proportional to 1/p2, 1/p, and ln p, appear only in
ReAij(p, 0) for real p and can be derived by iterating the integral equations [50].
Since ImAij(p, 0) must be extrapolated to p = 0, it is useful to transform Eqs. (2.3)
into coupled STM equations for amplitudes Aij(p, 0) obtained by subtracting the
singular terms from Aij(p, 0). For p Λ, the solutions depend log-periodically on
Λ with a discrete scaling factor eπ/s0 ≈ 22.7, where s0 = 1.00624. The dependence
on the arbitrary cutoff Λ can be eliminated in favor of a physical 3-body parameter,
such as the Efimov parameter κ∗ defined by the spectrum of Efimov states in the
limit where all 3 scattering lengths are infinitely large [11]:
En = −(
e2π/s0
)−n h2κ2∗
m(a12 = a23 = a31 = ±∞). (2.6)
If we restrict Λ to a range that corresponds to a multiplicative factor of 22.7, then Λ
differs from κ∗ only by a multiplicative numerical constant. Thus we can also simply
take Λ as the 3-body parameter.
2.3 Recombination into deep dimer
If aij > 0, there is a contribution to K3 from 3-body recombination into the shallow
dimer whose constituents have spins i and j and whose binding energy is h2/(ma2ij). If
a12, a23, and a31 are all negative, there are no shallow dimers. The solutions Aij(p, 0)
to the coupled STM equations in Eq. (2.3) are all real-valued in this case, so the rate
constant K3 in Eq. (2.2) is predicted to be 0.
If there are deeply-bound diatomic molecules (deep dimers) in any of the three 2-
body channels, there are also contributions to K3 from 3-body recombination into the
deep dimers. If all 3 scattering lengths are negative, these are the only contributions to
K3. The coupled STM equations in Eq. (2.3) do not take into account contributions
41
from deep dimers. The inclusive effect of all the deep dimers can be taken into
account by analytically continuing the Efimov parameter κ∗ to a complex value [19]:
κ∗ → κ∗ exp(iη∗/s0), where η∗ is a positive real parameter. Making this substitution
in Eq. (2.6), we find that the Efimov states acquire nonzero decay widths determined
by η∗. If we use the ultraviolet cutoff Λ as the 3-body parameter, the inclusive effects
of deep dimers can be taken into account by changing the upper limit of the integral
in Eq. (2.3) to Λ exp(iη∗/s0), so the path of integration extends into the complex
plane. Having made this change, the solutions Aij(p, 0) are complex-valued even if
a12, a23 and a31 are all negative. The rate constant K3 in Eq. (2) is a function of
the scattering lengths a12, a23, and a31 and the 3-body parameters Λ and η∗ and it
vanishes as η∗ → 0. It gives the inclusive rate for 3-body recombination into all deep
dimers.
2.4 Equal negative scattering lengths
We focus our attention on cases in which all scattering lengths are negative, so the
only recombination channels are into deep dimers. We first consider the case of 3
equal scattering lengths: a12 = a23 = a13 = a < 0. In this case Eq. (2.3) reduces –
after summing over i and j – to the STM equation for identical bosons. In Ref. [19],
Braaten and Hammer deduced an analytic expression for the 3-body recombination
rate constant for identical bosons with a large negative scattering length a:
K3 =16π2C sinh(2η∗)
sin2[s0 ln(D|a|κ∗)] + sinh2 η∗
ha4
m, (2.7)
where s0 = 1.00624, C, andD are numerical constants. This formula exhibits resonant
enhancement for a near the values (eπ/s0)n(Dκ∗)−1 for which there is an Efimov state
at the 3-body threshold. Fitting our numerical results for K3/a4 as functions of aΛ
and η∗, we determine the numerical constants to be C = 29.62(1) and D = 0.6642(2).
42
10-2 10-1 100 101 102 103
a23/a10-2
10-1
100
101
102
103
104
C (a
23/a
)-2, D
Figure 2.1: The coefficients C scaled by (a23/a)−2 (upper curve) and D (lower curve)
in Eq. (3.16) as functions of a23/a for the case of two equal negative scattering lengthsa and a third negative scattering length a23.
These values are more accurate than previous results for identical bosons [11]. A
separate calculation of the spectrum of Efimov states in the limit a → ±∞ with
η∗ = 0 is necessary to determine the relation between the Efimov parameter and the
ultraviolet cutoff: κ∗ = 0.17609(5)Λ.
We next consider the case of 2 equal negative scattering lengths and a third that
vanishes: a12 = a13 = a < 0, a23 = 0. In this case with only two resonant scattering
channels, s0 = 0.413698 and the discrete scaling factor is eπ/s0 ≈ 1986. Eq. (3.16)
again gives an excellent fit to our numerical results and we determine the numerical
constants as C = 0.8410(6) and D = 0.3169(1).
43
We now consider the case of 2 equal negative scattering lengths and a third that
is unequal: a12 = a13 = a < 0, a23 < 0. Eq. (3.16) with s0 = 1.00624 continues
to provide an excellent fit to our numerical results. The fitted values of C and D
are shown as functions of a23/a in Fig. 2.1. For |a23| |a|, the coefficients seem to
have the limiting behaviors C ≈ 10.88(2) (a23/a)2 and D ≈ 1.30(1). Their limiting
behaviors for |a23| |a| do not seem to be simple power laws. This is not surprising,
because the discrete scaling factor 22.7 changes to 1986 when a23 = 0.
2.5 Unequal negative scattering lengths
Finally we consider the general problem of 3 different negative scattering lengths, for
which we can obtain numerical results for given values of a12, a23, and a13. We apply
our method to 6Li atoms in the three lowest hyperfine states. The 3 pair scattering
lengths a12, a23, and a13 are shown as functions of the magnetic field in Fig. 1.3 and
1.4. There are two regions of the magnetic field in which all 3 scattering lengths are
negative and satisfy |aij | > 2`vdW: a low-field region 122 G < B < 485 G and a
high-field region B > 834 G. In the low-field region, the smallest scattering length
is a12 and it achieves its largest value −290 a0 = −4.6 `vdW near 320 G. The zero-
range approximation may be reasonable near this value of B. In the high-field region,
the smallest scattering length is a13. It increases from −3285 a0 at B = 834 G to
−2328 a0 ≈ −37 `vdW at 1200 G. Thus the zero-range approximation should be very
accurate in this region. We emphasize that the 3-body parameters κ∗ and η∗ need
not be the same in the two universal regions, since there are zeroes of the scattering
lengths between them.
The 3-body recombination rate K3 for 6Li atoms in the three lowest hyperfine
states has recently been measured by the Heidelberg group [7] and by the Penn State
group [8]. Their results are shown in Figs. 2.2 and 2.3. In Ref. [7], K3 was measured
44
0 200 400 600B [G]
10-26
10-25
10-24
10-23
10-22
10-21
K 3 [cm
6/s
]
Figure 2.2: The 3-body recombination rate constant K3 as a function of the magneticfield B. The two vertical dotted lines mark the boundaries of the region in which|a12| > 2 `vdW. The solid squares and dots are data points from Refs. [7] and [8],respectively. The curve is a 2-parameter fit to the shape of the data from Ref. [7].
for each of the three spin states separately. Those results have been averaged to get
a single value of K3 at each value of B. Both groups observed dramatic variations
in K3 with B, including a narrow loss feature near 130 G and a broader loss feature
near 500 G.
The narrow loss feature and the broad loss feature observed in Refs. [7, 8] appear
near the boundaries of the low-field region in which all 3 scattering lengths satisfy
|aij | > 2`vdW. The zero-range approximation is questionable near the boundaries
of this region. We nevertheless fit the data for K3 in this region by calculating the
3-body recombination rate using the B-dependence of a12, a23, and a13 shown in
Fig. 1.3, while treating Λ and η∗ as fitting parameters. Since the systematic error in
the normalization of K3 was estimated to be 90% in Ref. [7] and 70% in Ref. [8], we
45
only fit the shape of the data and not its normalization. A 2-parameter fit to the data
from Ref. [7] in the region 122 G < B < 485 G gives Λ = 436 a−10 and η∗ = 0.11. The
fit to the shape of the narrow loss feature is excellent as shown in Fig. 2.2. Having
fit the position and width of the loss feature feature, the normalization of K3 is
determined. In the region of the narrow loss feature, the prediction for K3 lies below
the data of Ref. [7] by about a factor of 2, which is well within the systematic error
of 90%. The excellent fit to the shape of the narrow loss feature and the prediction of
the normalization of K3 consistent with the data suggests that this loss feature may
arise from an Efimov state near the threshold for atoms in spin states 1, 2, and 3. As
shown in Fig. 2.2, our fit predicts that K3 should be almost constant in the middle of
the low-field region and that there should be another narrow loss feature at its upper
end near 500 G. The data from both groups in Fig. 2.2 increases monotonically in the
middle of the low-field region and, instead of a narrow loss feature, there is a broad
loss feature near the upper end of this region. We are unable to get a good fit to the
slope of logK3 in the middle of the low-field region or to the shape of the broad loss
feature by adjusting Λ and η∗.
In Ref. [8], the 3-body recombination rate was also measured at higher values of
the magnetic field. They include three data points in the region B > 834 G, where all
3 scattering lengths are extremely large and negative. If the central values of the last
two data points are used to determine the 3-body parameters, we obtain Λ = 37.0 a−10
and η∗ = 2.9×10−4. As shown in Fig. 2.3, this fit predicts the resonant enhancement
of the 3-body recombination rate near 1160 G. If we allow for the systematic error
by increasing or decreasing both data points by 70%, the position of the resonance
does not change, but η∗ increases to 5×10−4 or decreases to 9×10−5, respectively. If
we take into account the statistical errors by increasing or decreasing the data points
by one standard deviation, the position of the resonance can be shifted downward to
46
600 800 1000 1200 1400B [G]
10-22
10-21
10-20
10-19
10-18
10-17
10-16
K 3 [cm
6/s
]
Figure 2.3: The 3-body recombination rate constant K3 as a function of the magneticfield B. The three vertical dashed lines mark the positions of the Feshbach resonances.The solid dots are data points from Ref. [8]. The curve is a 2-parameter fit to thelast two data points.
1109 G or upward to 1252 G. Thus it might be worthwhile to search for an Efimov
resonance in this region. If such a feature were observed, measurements of its position
and width would determine accurately the two 3-body parameters κ∗ and η∗. Our
equations could then be used to predict the total 3-body recombination rate in the
entire universal region B > 610 G, including the regions where 1, 2, or 3 of the
scattering lengths are positive. Note that the third-to-last data point in Fig. 2.3
shows no sign of the large increase in K3 near the Feshbach resonance at 834 G that
is predicted by our fit. However the measurement of K3 involves a model for the
heating of the system, and the failure of our fit at 835 G might be attributable to the
breakdown of that model near the Feshbach resonance.
In summary, we have calculated the recombination rate of three distinguishable
47
atoms with large negative pair scattering lengths in the zero-range limit. We have
provided simple semi-analytical expressions for the rate if 2 or 3 scattering lengths
are equal. Using our general result for 3 unequal scattering lengths, we showed that
the narrow 3-body loss feature for 6Li atoms with three spin states [7, 8] may be
attributed to an Efimov state near threshold. In next chapter, it will be clarified by
calculating the energy spectrum of the Efimov trimer.
2.6 Postscript
The analysis presented in this chapter was completed in November 2008. Shortly after
it appeared, similar analyses of the three-body recombination rate in the low-field
region were completed by a group at the University of Tokyo in November 2008 [51]
and a group at the University of Heidelberg in December 2008 [52]. The Heidelberg
group used a functional renormalization group method to calculate the three-body
recombination rate. The Tokyo group calculated the three-body recombination rate
by solving the free three-body Schrodinger equation with boundary conditions at
short distance. They also calculated the binding energy of the Efimov trimer. This
demonstrated explicitly that the narrow loss resonance near 120 G and the broad
loss resonance near 480 G are both due to the Efimov trimer crossing the three-
atom threshold. The different width of these two loss resonances was subsequently
explained by the experimental group at Heidelberg as being due to the changes with
the magnetic field of the binding energies of the deep dimers that are produced by
three-body recombination [53].
In the analysis presented above, we pointed out that there should also be an
Efimov loss resonance for 6Li atoms in the high-field region above 600 G. In August
2009, a narrow loss resonance was discovered near 900 G by the Penn State group [35]
and also by the Heidelberg group [54]. In Chapter 3, we use the measured position
48
and width of this feature to make universal predictions for the Efimov spectrum and
the three-body recombination rate for 6Li atoms in the high-field region.
49
Chapter 3
Efimov trimer spectrum and
three-body recombination
In this chapter, we calculate universal predictions for various aspects of Efimov physics
for the three lowest hyperfine spin states of 6Li atoms. In Section 3.1, we explain how
universal predictions for 3-body observables can be calculated efficiently by solving
coupled sets of integral equations. We apply these methods specifically to the 3-body
recombination rate and to the binding energies and widths of Efimov trimers. In
Section 3.2, we summarize previous experimental and theoretical work on 6Li atoms
in the universal region at low magnetic fields where all 3 scattering lengths are negative
and relatively large. In Section 3.3, we use the complex 3-body parameter determined
by the Penn State group to calculate universal predictions for the binding energies and
widths of Efimov trimers and for the three-body recombination rate in regions where
one or more of the scattering lengths are large and positive. We predict an atom-dimer
resonance at 672 ± 2 G where an Efimov trimer disappears through an atom-dimer
threshold. We predict an interference minimum in the 3-body recombination rate at
759± 1 G where the 3-spin mixture may be sufficiently stable to allow experimental
study of the many-body system. We also discuss the implications of our predictions
for the many-body physics of 6Li atoms. We summarize our results in Section 3.4.
The analysis presented in this chapter was carried out in collaboration with Eric
50
Braaten, Hans-Werner Hammer, and Lucas Platter and was published in Physical
Review A [36] in January 2010.
3.1 Theoretical formalism
3.1.1 Three-body recombination rates
Three-body recombination is a three-atom collision in which two of the atoms form a
dimer. In the case of three fermions in the same spin state, 3-body recombination is
strongly suppressed at low temperature, because each pair of atoms has only P-wave
interactions. In the case of two fermions of type i and a third atom of a different type
j, two of the pairs have S-wave interactions with scattering length aij . The rate for
3-body recombination in the zero-range limit still decreases to 0 as the energy E of
the atoms approaches the threshold, decreasing like E if aij > 0 [55] and like E3 if
aij < 0 [56]. In the case of three distinct spin states, all three pairs of atoms can have
S-wave interactions. There is no threshold suppression of three-body recombination
if at least two of the three scattering lengths are large. If aij is large and positive,
one of the recombination channels is into the (ij) dimer and a recoiling atom with
complimentary spin k. If there are deep dimers in any of the three 2-body channels,
they provide additional recombination channels. If all three scattering lengths are
negative, the only recombination channels are into deep dimers.
The rate equations for the number densities ni of atoms in the three spin states
are
d
dtni = −K3n1n2n3. (3.1)
The event rate constant K3 can be separated into the inclusive rate constant Kdeep3
for recombination into deep dimers and the exclusive rate constants K(ij)3 for recom-
51
bination into each of the three possible shallow dimers:
K3 = Kdeep3 +Kshallow
3 , (3.2)
Kshallow3 = K
(12)3 +K
(23)3 +K
(13)3 . (3.3)
The term K(ij)3 is nonzero only if aij > 0.
In the low-temperature limit, the rate constantK3 and the exclusive rate constants
K(ij)3 can be expressed in terms of T-matrix elements for processes in which the initial
state consists of three atoms in the spin states 1, 2, and 3 with momentum 0. By the
optical theorem, K3 is twice the imaginary part of the forward T-matrix element for
3-atom elastic scattering in the limit where the momenta of the atoms all go to 0:
K3 = 2 Im T (0,0,0;0,0,0). (3.4)
The T-matrix element is singular as all the momenta go to zero, but its imaginary
part is not. If aij > 0, the rate constant K(ij)3 for recombination into the (ij) dimer
is the square of the T-matrix element for three atoms with momentum 0 to scatter
into the dimer and a recoiling atom multiplied by the atom-dimer phase space:
K(ij)3 =
4m
3√
3πhak
|Tk(0,0,0; p,−p)|2∣
∣
∣
∣
|p|=2h/(√
3ak)
. (3.5)
The dimer and the recoiling atom with complementary spin k both have momentum
2h/(√
3ak). For convenience, we will switch to wavenumber variables in the remainder
of the paper.
The T-matrix elements in Eqs. (3.4) and (3.5) can be expressed in terms of am-
plitudes Aij(p, q;E) for the transition from an atom of type i and a complimentary
diatom pair into an atom of type j and a complimentary diatom pair, with the two
diatom pairs being the first to interact and the last to interact, respectively. The
projection onto S-waves reduces the amplitude to a function of three variables: the
52
relative wavenumber p of the incoming atom and diatom, the relative wavenumber q
of the outgoing atom and diatom, and the total energy E of either the incoming atom
and diatom or the outgoing atom and diatom. The rate constant K3 in Eq. (3.4) can
be expressed as
K3 =32π2h
m
∑
i,j
aiajImAij(0, 0; 0), (3.6)
where the sums are over i, j = 1, 2, 3. The exclusive rate constant in Eq. (3.5) for
3-body recombination into the (ij) dimer can be expressed as
K(ij)3 =
512π2h
3√
3ma2k
∣
∣
∣
∣
∣
∑
l
alAlk(0, 2/(√
3ak); 0)
∣
∣
∣
∣
∣
2
, (3.7)
where k is the complimentary spin to ij and the sum is over l = 1, 2, 3.
3.1.2 STM equations
The 9 amplitudes Aij(p, q;E) satisfy coupled integral equations in the variable q
that are generalizations of the Skorniakov–Ter-Martirosian (STM) equation [46]. To
determine the rate constants for 3-body recombination in Eqs. (3.6) and (3.7), it is
sufficient to set the relative wavenumber in the initial state to 0 and the total energy
to 0. The 9 coupled STM equations for Aij(0, p; 0) are [34]
Aij(0, p; 0) =1− δijp2
+2
π
∑
k
(1− δkj)
∫ Λ
0
dq Q(p, q; 0)Dk(3q2/4)Aik(0, q; 0),(3.8)
where
Q(p, q;E) =q
2plog
p2 + pq + q2 −mE/h2
p2 − pq + q2 −mE/h2 , (3.9)
Dk(p2) =
[
−1/ak +√
p2 − iε]−1
, (3.10)
and Λ is an ultraviolet cutoff that must be large compared to p, 1/|a1|, 1/|a2|, and
1/|a3|. Since the T-matrix elements in Eqs. (3.6) and (3.7) involve only the three
53
linear combinations∑
i aiAij(0, p; 0), the 9 coupled STM equations can be reduced to
3 coupled integral equations for these 3 linear combinations. If Λ is sufficiently large,
the solutions to the integral equations in Eqs. (3.8) depend only log-periodically on
Λ with a discrete scaling factor λ0 ≈ 22.7. The dependence on the arbitrary cutoff
Λ can be eliminated in favor of a physical 3-body parameter, such as the Efimov
parameter κ∗ defined by Eq. (1.26). If we restrict Λ to a range that corresponds to a
multiplicative factor of 22.7, then Λ differs from κ∗ only by a multiplicative numerical
constant. Thus, we can also simply use Λ as the 3-body parameter [57].
If Λ is real valued, the STM equations describe atoms that have no deep dimers.
The rate Kdeep3 for 3-body recombination into deep dimers, which can be obtained
by combining Eqs. (3.2), (3.3), (3.6), and (3.7), must therefore be zero. For atoms
that have deep dimers, the effects of the deep dimers can be described indirectly by
using a complex-valued 3-body parameter. If we use the ultraviolet cutoff Λ as the
3-body parameter, the inclusive effects of deep dimers can be taken into account by
analytically continuing the upper endpoint of the integral Λ in Eqs. (3.8) to a complex
value Λ exp(iη∗/s0). The path of integration in the variable q can be taken to run
along the real axis from 0 to Λ and then along the arc from Λ to Λ exp(iη∗/s0). This
path can be deformed to run along the straight line from 0 to Λ exp(iη∗/s0) provided
we add explicitly the contributions from any poles that are crossed as the contour is
deformed. These poles arise from the diatom propagator Dk(p2), which in the case
ak > 0 has a pole associated with the shallow dimer at p = 1/ak. For example, if
ak > 0, the integral in Eq. (3.8) with a complex ultraviolet cutoff Λeiη∗/s0 can be
written
(
∫ Λ
0
dq +
∫ Λeiη∗/s0
Λ
dq
)
Q(p, q; 0)Dk(3q2/4)Aik(0, q; 0)
= Q(p, qk; 0)4πi√
3Aik(0, qk; 0) +
∫ Λeiη∗/s0
0
dq Q(p, q; 0)Dk(3q2/4)Aik(0, q; 0),(3.11)
54
where qk = 2/(√
3ak). In the integral on the right side, the integration contour runs
along the straight line path from 0 to Λeiη∗/s0. With the ultraviolet cutoff replaced by
Λ exp(iη∗/s0), the rate Kdeep3 for 3-body recombination into deep dimers is nonzero.
The rate constant K3 in Eq. (3.6) requires the extrapolation of the solutions
Aij(0, p; 0) to the STM equations in Eqs. (3.8) to p = 0. These solutions are singular as
p→ 0. The singular terms, which are proportional to 1/p2, 1/p, and ln p, appear only
in ReAij(0, p; 0) for real p and can be derived by iterating the integral equations [50].
Since ImAij(0, p; 0) must be extrapolated to p = 0, it is useful to transform Eqs. (3.8)
into coupled STM equations for amplitudes Aij(0, p; 0) obtained by subtracting the
singular terms from Aij(0, p; 0):
Aij(0, p; 0) = Aij(0, p; 0) − 1− δijp2
+π
3p
∑
n
an(1− δin)(1− δnj)
− logp
Λ
√3
π
∑
n
a2n(1− δin)(1− δnj)
−2
3
∑
m,n
aman(1− δim)(1− δmn)(1− δnj)
.(3.12)
3.1.3 Three equal scattering lengths
We can obtain analytic results for the 3-body recombination rate in the case of three
equal scattering lengths: a12 = a23 = a13 = a. In this case, the 3-body recombination
rates in Eqs. (3.6) and (3.7) depend only on the combination∑
iAij(0, p; 0). One
can show that the solutions to the STM equation for∑
iAij(p, q;E) are the same
for j = 1, 2, 3 and are equal to the corresponding amplitude A(p, q;E) for identical
bosons with scattering length a:
∑
i
Aij(p, q;E) = A(p, q;E), j = 1, 2, 3. (3.13)
55
The STM equation analogous to Eq. (3.8) for identical bosons is1
A(0, p; 0) =2
p2+
4
π
∫ Λ
0
dq Q(p, q; 0)D(3q2/4)A(0, q; 0). (3.14)
If the low-temperature limit is taken with the number density for the identical bosons
much smaller than the critical density for Bose-Einstein condensation, the rate con-
stant for 3-body recombination of identical bosons into the shallow dimer is [11]
Kshallow3 =
512π2h√3m
∣
∣
∣A(0, 2/(
√3a); 0)
∣
∣
∣
2
. (3.15)
Upon making the substitution∑
l alAlk → aA in Eq. (3.7), we see that the re-
combination rate K(ij)3 for the three fermions into the (ij) dimer is exactly 1/3 of
the recombination rate for identical bosons in Eq. (3.15). Summing over the three
shallow dimers, we find that the expression for the recombination rate of the three
fermions into shallow dimers is identical to the expression for the recombination rate
of identical bosons into the single shallow dimer. Similarly, we find that the expres-
sion in Eq. (3.6) for the total recombination rate of the three fermions is identical to
that for the total recombination rate of identical bosons.
Braaten and Hammer deduced a semi-analytic expression for the rate constant
for 3-body recombination of identical bosons into deep dimers with a large negative
scattering length a [19]:
Kdeep3 =
16π2C sinh(2η∗)
sin2[s0 ln(a/a′∗)] + sinh2 η∗
ha4
m(a < 0), (3.16)
where s0 = 1.00624, a′∗ = −1/(Dκ∗), and C and D are numerical constants. The most
accurate values for the numerical constants are C = 29.62(1) and D = 0.6642(2)
[34, 58]. The relation between the Efimov parameter and the ultraviolet cutoff is
1The amplitude A(p, q; E) differs from the amplitude AS(p, q; E) in Ref. [11] by a multiplicativefactor of a/(8π).
56
κ∗ = 0.17609(5)Λ, modulo multiplication by an integer power of λ0 ≈ 22.7. The
expression for the recombination rate constant in Eq. (3.16) exhibits resonant en-
hancement for a near the values λn0a∗ for which there is an Efimov trimer at the
3-body threshold. The line shape in Eq. (3.16) for the 3-atom loss resonance as a
function of the scattering length played a key role in the discovery of an Efimov state
for 133Cs atoms in the lowest hyperfine state [27]. It applies equally well to three
fermions with equal negative pair scattering lengths.
Macek, Ovchinnikov, and Gasaneo [59] and Petrov [60] have deduced a completely
analytic expression for the 3-body recombination rate constant for identical bosons
with a large positive scattering length a in the case where there are no deep dimers.
Braaten and Hammer generalized their result to the case where there are deep dimers
by making the analytic continuation κ∗ → κ∗ exp(iη∗/s0) in the amplitude for this
process [12]. The resulting analytic expression for the recombination rate is [12]
Kshallow3 =
128π2(4π − 3√
3)(sin2[s0 ln(a/a∗0)] + sinh2 η∗)
sinh2(πs0 + η∗) + cos2[s0 ln(a/a∗0)]
ha4
m(a > 0) , (3.17)
where a∗0 ≈ 0.32κ−1∗ . This expression exhibits minima for a near the values λn
0a∗0
arising from destructive interference between two pathways for recombination. This
formula applies equally well to three fermions with equal positive pair scattering
lengths. It gives the recombination rate K shallow3 in Eq. (3.3), which is summed over
the 3 shallow dimers.
3.1.4 Homogeneous STM equations and Efimov trimers
The transition amplitudes Aij(p, q;E) have poles in the total energy E at the energies
E(n) of the Efimov trimers. Near the pole, the amplitudes factor:
Aij(p, q;E) −→ Bi(p)∗Bj(q)
E − E(n). (3.18)
57
The spectrum of Efimov trimers can be obtained by solving the three coupled homo-
geneous integral equations for Bj(q):
Bj(p) =2
π
∑
k
(1− δkj)
∫ Λ
0
dq Q(p, q;E)Dk(3q2/4−mE/h2)Bk(q), (3.19)
where the ultraviolet cutoff Λ must be much larger than p, |mE/h2|1/2, and all three
inverse scattering lengths 1/ai. The set of homogeneous STM equations in Eq. (3.19)
are nonlinear eigenvalue equations for E. If Λ is real valued, the eigenvalues are real
valued if the energy is below all the scattering thresholds. The Efimov trimers are
therefore sharp states with 0 widths. There is always a 3-atom scattering threshold
at E = 0. If aij > 0, the atom-dimer scattering threshold at E = −h2/(ma2ij) has
lower energy.
If there are deep dimers, the Efimov trimers can decay into a deep dimer and a
recoiling atom. Their widths can be calculated by analytically continuing the upper
endpoint of the integral Λ in Eqs. (3.19) to a complex value Λ exp(iη∗/s0). The
complex energy eigenvalue for the trimer can be expressed as
E(n) = −E(n)T − iΓ
(n)T /2. (3.20)
If Γ(n)T is small compared to the difference between E
(n)T and the nearest scattering
threshold, then E(n)T and Γ
(n)T can be interpreted as the binding energy and the width
of the trimer, respectively. If Γ(n)T is not small, they do not have such precise inter-
pretations.
If η∗ 1, the binding energies and widths of the Efimov trimers can be calculated
approximately by solving the STM equation with a real-valued cutoff Λ. The complex
energy E(n) in Eq. (3.20) is a function of the complex parameter Λ exp(iη∗/s0) that
must be real valued in the limit η∗ → 0. Expanding that function in powers of η∗, we
58
find that the leading approximations to the binding energy and the width are
E(n)T ≈ −E(n)
∣
∣
η∗=0, (3.21)
Γ(n)T ≈ −2η∗
s0
Λ∂
∂ΛE(n)
∣
∣
∣
∣
η∗=0
. (3.22)
The derivative with respect to Λ in Eq. (3.22) is calculated with the scattering lengths
aij held fixed. The leading corrections to Eqs. (3.21) and (3.22) are suppressed by a
factor of η2∗.
3.1.5 Dimer relaxation
If our system of fermionic atoms is a mixture of (jk) dimers and atoms of type i,
another 3-atom loss process is dimer relaxation: the inelastic scattering of the atom
and the (jk) dimer into an atom and a dimer with a larger binding energy. If i
coincides with j so there are only two distinct spin states, the dimer relaxation rate
decreases as a−3.33ik as the scattering length increases [61]. The atom-dimer mixture
is therefore remarkably stable when the scattering length is very large. This stability
was verified in the recent experiments with 6Li atoms [7, 8]. If i is distinct from
both j and k so there are three distinct spin states, there is no Pauli suppression
of the atom-dimer relaxation rate. The rate increases as the scattering lengths are
increased. On top of that, there can also be resonant enhancement associated with
Efimov physics.
To be definite, we consider a mixture of (23) dimers and atoms of type 1. We
denote the number densities of atoms and (23) dimers by n1 and n(23), respectively.
The loss rate from atom-dimer relaxation can be expressed in terms of a rate constant
β1(23):
d
dtn1 =
d
dtn(23) = −β1(23)n1n(23). (3.23)
Atom-dimer relaxation channels are also inelastic atom-dimer scattering channels. So
59
if β1(23) > 0, the atom-dimer scattering length a1(23) must have a negative imaginary
part. These two quantities are related by the optical theorem:
β1(23) = −6πh
mIm(a1(23)). (3.24)
If the scattering lengths are all large, then by dimensional analysis, β1(23) must be
ha23/m multiplied by a dimensionless coefficient that depends on the ratios of scatter-
ing lengths a12/a23 and a13/a23 and also on a23κ∗, where κ∗ is the Efimov parameter.
The dependence on a23κ∗ is required to be log-periodic with discrete scaling factor
λ0 ≈ 22.7. The universal predictions for dimer relaxation rates can be calculated by
solving appropriate sets of coupled STM equations. The dimensionless coefficient of
ha23/m in the relaxation rate constant β1(23) can be especially large if there is an
Efimov trimer close to the 1+(23) atom-dimer threshold. In this case, there is res-
onant enhancement of the dimer relaxation rate. The resulting loss feature is called
an atom-dimer loss resonance.
3.2 Low-field universal region
In this section, we apply our formalism to the lowest three hyperfine spin states of
6Li atoms in the region of low magnetic field from 0 to 600 G.
3.2.1 Three-body recombination revisited
The first measurements of the 3-body recombination rate K3 for the three lowest
hyperfine spin states of 6Li atoms were carried out by the Heidelberg group [7] and
by the Penn State group [8]. Their results for magnetic field in the region from 0 to
600 G are shown in Fig. 3.1. The results for K3 in Ref. [7] from measurements of the
loss rates of the three individual spin states have been averaged to get a single value
of K3 at each value of B. Both groups observed dramatic variations in K3 with B,
60
0 200 400 600B [G]
10-26
10-25
10-24
10-23
10-22
10-21
K 3 [cm
6/s
]
Figure 3.1: Three-body recombination rate constant K3 as a function of the magneticfield B from 0 to 600 G. The solid squares and dots are data points from Refs. [7]and [8], respectively. The curve is the absolutely normalized result for Kdeep
3 withκ∗ = 76.8 a−1
0 and η∗ = 0.11. The two vertical dotted lines mark the boundaries ofthe region in which the absolute values of all three scattering lengths are greater than2 `vdW.
including a narrow loss feature near 130 G and a broader loss feature near 500 G.
The narrow loss feature and the broad loss feature in the measurements of Refs. [7,
8] both appear near the boundaries of the region in which all three scattering lengths
satisfy |aij | > 2`vdW. The effects of the finite range of the interaction may be sig-
nificant near the boundaries of this region. In Ref. [34], we fit the data for K3 in
this region by calculating the 3-body recombination rate Kdeep3 in Eq. (3.6), using
the magnetic field dependence of the three scattering lengths shown in Fig. 1.3 and
treating Λ and η∗ as fitting parameters. Since the systematic error in the normal-
ization of K3 was estimated to be 90% in Ref. [7] and 70% in Ref. [8], we only fit
the shape of the data and not its normalization. A 3-parameter fit to the data from
Ref. [7] in the region 122 G < B < 485 G with an adjustable normalization factor
61
determines the 3-body parameters Λ = 436 a−10 and η∗ = 0.11. This value of the
cutoff is equivalent to κ∗ = 76.8 a−10 . These parameters κ∗ and η∗ determine the
normalization of Kdeep3 , so the theoretical curve in Fig. 3.1 is absolutely normalized.
The fit to the shape of the narrow loss feature is excellent. The normalization is also
correct to within the systematic error in the data. However the fit predicts that K3
should be almost constant in the middle of the low-field region and that there should
be another narrow loss feature at its upper end near 500 G. These predictions are not
consistent with the data in Fig. 3.1, which increases monotonically in the middle of
the low-field region and has a broad loss feature near the upper end of this region.
Similar results for the 3-body recombination rate of 6Li atoms were obtained
subsequently by two other groups [51, 52]. Naidon and Ueda used hyperspherical
methods to calculate the recombination rate [51]. For their 3-body parameters, they
used the real and imaginary parts of the logarithmic derivative of the hyperradial
wavefunction. Schmidt, Floerchinger, and Wetterich used functional renormalization
methods to calculate the recombination rate [52]. For their 3-body parameters, they
used the real and imaginary parts of the detuning energy of a triatomic molecule.
The results of both groups for K3 as a function of the magnetic field are similar to
our results in Fig. 3.1. One difference is that in Refs. [51, 52] the recombination rate
was calculated only up to an overall normalization constant that was determined by
fitting the data. In our calculation, the absolute normalization is determined by the
3-body parameters κ∗ and η∗.
In Ref. [53], the Heidelberg group provided an explanation for the loss feature
near 500 G being much broader than predicted in Refs. [34, 51, 52]. They pointed
out that there are deep dimers whose binding energies vary significantly over the
low-field region. These dimers are those responsible for the Feshbach resonances
690 G, 811 G, and 834 G. In the high-field region, they are shallow dimers, but they
62
become deep dimers in the low-field region. Their binding energies change over the
low-field universal region by as much as a factor of 6. The Heidelberg group assumed
that contributions to the 3-body parameter η∗ scale like the inverse of the binding
energy of the deep dimer. They used the coefficient in this scaling relation as a fitting
parameter along with the 3-body parameter equivalent to κ∗. They obtained an
excellent fit to K3 over the entire low-field region, including the narrow loss feature,
the broad loss feature, and the monotonic rise in between. Their assumption for the
scaling of η∗ with the binding energy of the deep dimer can be justified by an explicit
calculation in a two-channel model [62].
In Ref. [53], the Heidelberg group proposed a simple analytic approximation for
the 3-body recombination rate in regions where all 3 scattering lengths are negative.
Their approximation is the analytic result for equal scattering lengths in Eq. (3.16),
with a replaced by an effective scattering length am given by
am = −[
(a21a
22 + a2
2a23 + a2
3a21)/3
]1/4. (3.25)
Another possible choice for an effective scattering length is the geometric mean of the
three scattering lengths:
ag = −|a1a2a3|1/3. (3.26)
We can use our universal results to test the accuracy of that approximation. We find
that the analytic result in Eq. (3.16) with a replaced by the geometric mean ag in
Eq. (3.26) is a significantly more accurate approximation.
3.2.2 Efimov trimers
Naidon and Ueda [51] and Schmidt, Floerchinger, and Wetterich [52] calculated the
binding energy of the Efimov trimer that is responsible for the loss features in Fig. 3.1.
Its binding energy goes to 0 near both the observed narrow loss feature near 130 G
63
100 200 300 400 500B [G]
0
5
10
15
E T/h [M
Hz]
Figure 3.2: Energy of the Efimov trimer as a function of the magnetic field B in thelow-field region. The binding frequency E
(1)T /(2πh) (dark solid line) and the frequency
Γ(1)T /(2πh) associated with the width (difference between dark dashed lines) were
obtained from the complex energy eigenvalue calculated using the parameters κ∗ =76.8 a−1
0 and η∗ = 0.11. Also shown for comparison are the small-η∗ approximation inEqs. (3.21) and (3.22) for the binding frequency (light solid line) and for the frequencyassociated with the width (difference between light dashed lines). The two verticaldotted lines mark the boundaries of the region in which the absolute values of allthree scattering lengths are greater than 2 `vdW.
and the predicted narrow loss feature near 500 G. This demonstrates explicitly that
both loss features arise from the same Efimov trimer crossing the 3-atom threshold.
We calculate the spectrum of Efimov trimers by solving for the complex energy
eigenvalues of the homogeneous STM equations in Eq. (3.19). The energies and widths
of the trimers are obtained by expressing the complex energies in the form −E (n)T −
iΓ(n)T /2. We choose to label the Efimov trimer responsible for the narrow loss feature
near 135 G by the integer n = 1. Our predictions for the binding energy E(1)T of the
shallowest Efimov state are shown in Fig. 3.2 as a function of the magnetic field. The
binding frequency E(1)T /(2πh) increases from 0 at 134 G to a maximum of 9.56 MHz
64
at 334 G and then decreases to 0 at 494 G. Our maximum binding frequency agrees
well with the maximum frequencies of about 10 MHz and about 11 MHz obtained in
Refs. [51] and [52], respectively. The binding frequency E(0)T /(2πh) associated with the
next deeper Efimov trimer is predicted to change gradually from 12.1 GHz at 134 G
to 12.5 GHz at 334 G and then to 12.1 GHz at 494 G. Since this binding energy is
much larger than the van der Waals frequency νvdW = 154 MHz, this Efimov trimer
and all the deeper ones predicted by the STM equations are artifacts of the zero-
range approximation. Our value of κ∗ can be interpreted as the binding wavenumber
(mE(−2)T /h2)1/2 of the fictitious Efimov trimer labelled by n = −2. It corresponds to
the choice n∗ = −2 in Eq. (1.26).
We note that the finite range corrections to the binding energies of the deeper
trimers can be substantial if their energy becomes comparable to the van der Waals
energy even if all scattering lengths are large. The leading corrections to the binding
energy for the deep trimers are of order `vdW (mE(n)T /h2)1/2. For the trimer labeled
n = 1, these corrections are 25% for the largest value of the binding energy.
In Ref. [51], Naidon and Ueda also calculated the width of the Efimov trimer.
In Fig. 3.2, our results for the width Γ(1)T are illustrated by plotting the frequencies
associated with the energies E(1)T ± Γ
(1)T /2 as functions of the magnetic field. The
frequency Γ(1)T /(2πh) increases from 2.73 MHz at 134 G to a maximum of 7.14 MHz
at 332 G and then decreases to 2.56 MHz at 494 G. Our maximum width is more than
twice as large as the maximum width of about (3 MHz)×2πh obtained in Ref. [51].
Since our result for Γ(1)T is always comparable to or larger than E
(1)T , the interpretations
of E(1)T and Γ
(1)T as the binding energy and width of the trimer should be viewed with
caution.
Since η∗ = 0.11 is relatively small, we can also use the small-η∗ approximations to
the binding energy and the width given in Eqs. (3.22) and (3.21). These approxima-
65
tions, which are calculated using κ∗ = 76.8 a−10 , are shown in Fig. 3.2 for comparison.
The maximum binding frequency E(1)T /(2πh) is 10.1 MHz at 334 G, which is larger
than the result from the complex energy by about 6%. The frequency Γ(1)T /(2πh)
increases from 2.76 MHz at 134 G to a maximum of 7.23 MHz at 332 G and then
decreases to 2.59 MHz at 494 G. These values are larger than the results from the
complex energy by only about 1%.
3.3 High-field universal region
In this section, we apply our formalism to the lowest three hyperfine spin states of
6Li atoms in the region of high magnetic field from 600 to 1200 G.
3.3.1 Measurements of three-body recombination
In Ref. [8], the Penn State group presented measurements of the 3-body recombi-
nation rate at 6 values of the magnetic field in the range 600 G to 1000 G. The
low-temperature limit of the recombination rate could not be measured, because the
temperature was not low enough and because of heating associated with the Fesh-
bach resonance. In Ref. [34], we showed that a naive fit of the last two data points at
894 G and 953 G using our zero-temperature calculations indicated an Efimov reso-
nance near 1200 G. We suggested that it might be worthwhile to search the high-field
region for an Efimov resonance.
A narrow 3-atom loss resonance in the high-field region near 895 G was recently
discovered by the Penn State group [35] and by the Heidelberg group [54]. The Penn
State group measured the 3-body recombination rate at magnetic fields from 842 G
up to 1500 G at temperatures lower than 180 nK and from 834 G up to 955 G at
temperatures lower than 30 nK. Their data is shown in Fig. 3.3. By fitting their
measurements using our formalism, they obtained the 3-body parameters κ∗ = (6.9±
66
600 700 800 900 1000 1100 1200B [G]
10-25
10-23
10-21
10-19
10-17
10-15
K 3 [cm
6 /s]
totaldeep
Figure 3.3: Three-body recombination rate constant K3 as a function of the magneticfield B from 600 G to 1200 G. The solid dots and triangles are data points fromRef. [8] at temperatures less than 180 nK and less than 30 nK, respectively. Thecurve between 834 G and 1200 G is a fit to that data. In the region from 600 Gand 834 G, the curves are our predictions for the total 3-body recombination rate(thick line) and the contribution from recombination into deep dimers (thin line) forκ∗ = 80.7 a−1
0 and η∗ = 0.016. The three vertical lines mark the positions of theFeshbach resonances.
0.2)×10−3 a−10 and η∗ = 0.016+0.006
−0.010 [35]. Since κ∗ is only defined up to multiplication
by integer powers of λ0 ≈ 22.7, an equivalent value is κ∗ = 80.7± 2.3 a−10 . Their fit
for the central values of κ∗ and η∗ is shown in Fig. 3.3 as the thick solid line between
834 G and 1200 G. The fitted value for the position of the 3-atom loss resonance is
895+4−5 G. The peak value of the 3-body recombination rate at zero temperature is
predicted to be (4.1+8.5−1.5)× 10−17 cm6/s.
It is interesting to compare the values of the 3-body parameters κ∗ and η∗ in
the high-field region with those in the low-field region. The values in the low-field
67
region obtained by fitting the measurements of the narrow 3-body recombination loss
feature by the Heidelberg group [7] were κ∗ ≈ 76.8 a−10 and η∗ ≈ 0.11. It is difficult to
quantify the errors in these parameters, because the narrow loss feature is at the edge
of the universal region where range corrections may be significant. The values of κ∗ in
the high-field region and the low-field region are consistent within errors. This could
be just a coincidence, but it suggests that the phase in the 3-body wavefunction at
short distances that controls Efimov physics is insensitive to the Feshbach resonances
that change the scattering lengths. The value of η∗ in the high-field region is about an
order-of-magnitude smaller than in the low-field region. This demonstrates that the
3-body parameters need not be equal in the two universal regions. The Heidelberg
group proposed a mechanism for variations in η∗, namely significant changes in the
binding energies of deep dimers with the magnetic field [53]. This mechanism might
also explain the order-of-magnitude difference in η∗ between the two regions.
Using the values of κ∗ and η∗ determined by the Penn State group [35], we can
calculate the universal predictions for other aspects of Efimov physics for 6Li atoms
in the high-field universal region. The error bars on the values of κ∗ and η∗ can be
used to give error bars on the predictions. In the next subsections, we give universal
predictions for the binding energies and the widths of the Efimov trimers and for the
3-body recombination rate.
3.3.2 Efimov trimers
We calculate the binding energies and the widths of the Efimov trimers by solving for
the complex energy eigenvalues of the homogeneous STM equations in Eq. (3.19). We
choose to label the Efimov trimer responsible for the narrow loss feature near 895 G
by the integer n = 1. In Fig. 3.4, the predicted binding energies E(0)T and E
(1)T of
the two shallowest Efimov trimers are shown as functions of the magnetic field. The
68
600 700 800 900 1000B [G]
10-3
10-2
10-1
100
101
102
E T/h
[MHz
]
13 23 12
Figure 3.4: Energies of the Efimov trimers as functions of the magnetic field B in thehigh-field region. The solid curves are the predicted binding frequencies E
(n)T /(2πh)
for κ∗ = 80.7 a−10 and η∗ = 0.016. The dashed curves are the upper and lower
error bars obtained by varying κ∗. The curves labelled 12, 23, and 13 are the atom-dimer thresholds. The dots indicate the points where the trimers disappear throughthe 1+(23) atom-dimer threshold. The horizontal dotted line is the van der Waalsfrequency 154 MHz.
Efimov trimer responsible for the narrow loss feature has a binding energy E(1)T that
vanishes at 895+4−5 G. As the magnetic field decreases, E
(1)T increases monotonically
until the critical value B∗ = 672±2 G where the Efimov trimer disappears through the
1+(23) atom-dimer threshold. When it disappears, its binding energy E(1)T relative
to the 3-atom threshold is 871+43−68 kHz, which is also the binding energy h2/(ma2
23) of
the (23) dimer. The next deeper Efimov trimer has a binding frequency E(0)T /(2πh)
that increases monotonically from 26.4+1.7−1.6 MHz at 895 G to 34.9+1.9
−1.9 MHz at B∗.
Our solutions to the STM equations predict that it crosses the 1+(23) atom-dimer
threshold at 597 G, where its binding frequency relative to the 3-atom threshold
is 203 MHz. This frequency is larger than the van der Waals frequency 154 MHz.
69
600 700 800 900 1000B [G]
10-3
10-2
10-1
100Γ T/h
[MHz
]
Figure 3.5: Widths of the Efimov trimers as functions of the magnetic field B in thehigh-field region. The solid curves are the frequencies Γ
(n)T /(2πh) associated with the
widths for κ∗ = 80.7 a−10 and η∗ = 0.016. The dashed curves are the upper and lower
error bars obtained by varying η∗.
Moreover the smallest scattering length at this point is only 54 a0, which is smaller
than the van der Waals length 65 a0. The zero-range predictions are not expected
to be accurate for such large energies and for such small scattering lengths. All the
deeper Efimov trimers have binding energies that are much larger than the van der
Waals energy and are therefore artifacts of the zero-range approximation. Our value
of κ∗ can be interpreted as the binding wavenumber (mE(−2)T /h2)1/2 of the fictitious
Efimov trimer labelled by n = −2. It corresponds to the choice n∗ = −2 in Eq. (??).
As discussed in Sec. 3.2.2, the leading range corrections to the binding energy
for the deep trimers are estimated to be of order `vdW (mE(n)T /h2)1/2. For the trimer
labeled n = 1, these corrections are 8% where it crosses the 1+(23) atom-dimer
threshold and become significantly smaller at larger values of the magnetic field. For
the trimer labelled n = 0 these corrections are about 40% at B∗ and larger values of
the magnetic field.
In Fig. 3.5, the predicted widths Γ(n)T of the Efimov trimers are shown as functions
of the magnetic field. The frequency Γ(1)T /(2πh) associated with the width of the
70
shallower trimer increases from 0.706 kHz at 895 G to a maximum of 7.98 kHz at
692 G and then decreases to 0.940 kHz at B∗. The frequency Γ(0)T /(2πh) associated
with the width of the deeper trimer increases from 1.74 MHz at 895 G to 2.00 MHz
at B∗.
3.3.3 Predictions for three-body recombination
Our predictions for the 3-body recombination rate in the region 600 G to 834 G,
where one or more of the scattering lengths are positive, are shown in Fig. 3.3. The
upper curve is the total recombination rate K3, while the lower one is the contri-
bution Kdeep3 from deep dimers. Besides the Efimov resonance at 895 G, the only
other peaks are at the Feshbach resonances at 690 G, 811 G, and 834 G. They arise
simply from the scaling of K3 as a4 up to logarithms, where a is some appropri-
ate mean of the scattering lengths a12, a23, and a13. The total recombination rate
K3 is predicted to have three local minima. There is a broad local minimum at
759± 1 G, where the minimum recombination rate is (2.5+0.5−0.5)× 10−22 cm6/s. There
is a narrow local minimum at 829 ± 1 G, where the minimum recombination rate
is (2.2+0.9−1.3) × 10−18 cm6/s. Finally there is a local minimum at 861 ± 2 G between
the last Feshbach resonance and the 3-atom Efimov resonance, where the minimum
recombination rate is (5.8+3.8−4.0)× 10−19 cm6/s.
In Fig. 3.6, we show the predictions for the 3-body recombination rate in the
region from 650 G to 850 G in more detail. In addition to K3 and Kdeep3 , we show the
contributions from recombination into the (12), (23), and (13) dimers. This figure
reveals that the local minima at 759 G and 829 G are associated with interference in
the recombination into shallow dimers. The narrow minimum at 829 G arises from the
combination of a rapidly increasing rate into deep dimers and a rapidly decreasing
rate into (12) dimers, which has its minimum at 830 G. The broad minimum at
71
650 700 750 800 850B [G]
10-25
10-23
10-21
10-19
10-17
10-15
K 3 [cm
6 /s]
totaldeep122313
Figure 3.6: Three-body recombination rate constant K3 as a function of the magneticfield B from 650 G to 850 G. The curves are our predictions for the total 3-bodyrecombination rate (thick solid line), the contribution from recombination into deepdimers (thin solid line), and the contributions from recombination into the (12) dimer(short-dashed line), the (23) dimer (dash-dotted line), and the (13) dimer (long-dashed line) for κ∗ = 80.7 a−1
0 and η∗ = 0.016.
759 G arises from minima in the rates into (12) and (23) dimers at 757 G and 765 G,
respectively. There are also minima in the rates into the (13) dimer at 672 G and
the (23) dimer at 600 G, but their effects are not visible in the total recombination
rate. We have verified that these minima in the recombination rates into individual
shallow dimers arise from interference effects by showing that they become zeroes as
η∗ is decreased to 0. Thus these minima are the result of destructive interference
between two recombination pathways. In the case of three equal positive scattering
lengths, a similar interference effect is evident in the analytic expression for the 3-body
72
recombination rate into shallow dimers in Eq. (3.17).
3.3.4 Atom-dimer resonance
An atom-dimer loss resonance can appear at a value of the scattering length for which
an Efimov trimer crosses the atom-dimer threshold. From the Efimov trimer spectrum
in Fig. 3.4, one can see that atom-dimer resonances are predicted at the two values of
the magnetic field where the Efimov trimers cross the 1+(23) atom-dimer threshold.
The atom-dimer resonance associated with the shallower of the two Efimov trimers
is predicted to occur at B∗ = 672 ± 2 G. The universal predictions for the dimer
relaxation rate near this resonance can be calculated by solving appropriate sets of
coupled STM equations. The atom-dimer resonance associated with the deeper of
the two Efimov trimers is predicted to occur at B ′∗ = 597 G. This resonance occurs
slightly outside the universal region, so universal predictions for the position of the
resonance and for the dimer relaxation rate are not expected to be accurate.
If we restrict our attention to magnetic fields very near the atom-dimer resonance
at B∗, we can get an approximation to the universal predictions for the dimer re-
laxation rate β1(23) without actually solving the STM equations. We take advantage
of the fact that the atom-dimer scattering length a1(23) diverges at B∗. The 3-atom
problem in this region therefore reduces to a universal 2-body problem for the atom
and the (12) dimer. The universal properties are determined by the large scattering
length a1(23). For example, in the region B > B∗ but very close to B∗, the binding
energy of the Efimov trimer is well approximated by the sum of the binding energy of
the (23) dimer and a universal term determined by the atom-dimer scattering length:
E(1)T ≈ −
(
h2
ma223
+3h2
4ma21(23)
)
. (3.27)
This should be a good approximation as long as the second term is much smaller
73
than the first term. In the region near B∗, the atom-dimer scattering length can
be approximated by an expression that has the same form as the analytic result for
identical bosons but with different numerical coefficients:
a1(23) ≈ (C1 cot[s0 ln(a23/a∗) + iη∗] + C2) a23, (3.28)
where a∗ = a23(B∗). The coefficients C1 and C2 can in principle depend on ratios of the
scattering lengths, but a13 is an order of magnitude larger so it essentially decouples
and a13 and a23 are both increasing with B so their ratio does not change rapidly
near B∗. Thus we can treat C1 and C2 as numerical constants. These constants can
be determined by using Eq. (3.27) to extract a1(23) from the results for the real part
of the trimer binding energy shown in Fig. 3.4 and then fitting those results to the
expression for the atom-dimer scattering length in Eq. (4.10). The resulting constants
are C1 = 0.67 and C2 = 0.65.
With the constant C1 in hand, we can obtain an approximation for the dimer
relaxation rate constant β1(23) for B near B∗ simply by inserting the approximation
for a1(23) in Eq. (4.10) into the optical theorem relation in Eq. (3.24). The resulting
expression is
β1(23) ≈9.44C1 sinh(2η∗)
sin2[s0 ln(a23/a∗)] + sinh2 η∗
ha23
m. (3.29)
If B∗ = 672 G, the value of a∗ is 835 a0. The only dependence on the magnetic
field is through the B-dependence of a23. This approximation for β1(23) is shown as
a function of B − B∗ in Fig. 3.7. The maximum value of β1(23) at the peak of the
resonance is 4× 10−7 cm3/s for η∗ = 0.016.
Since the atom-dimer resonance associated with the deeper of the two Efimov
trimers occurs slightly outside the universal region, universal predictions for the dimer
relaxation rate are not expected to be accurate. They might however be useful for
making order-of-magnitude estimates. If we apply the analysis described above to
74
-15 -10 -5 0 5 10 15B-B* [G]
10-9
10-8
10-7
10-6
β [c
m3 /s
]
η* = 0.022η* = 0.016η* = 0.006
Figure 3.7: Dimer relaxation rate constant β1(23) for (23) dimers and atoms of type1 as a function of the magnetic field B. The magnetic field is measured relative tothe position B∗ of the atom-dimer resonance. The curves are predictions using theapproximation in Eq. (3.29) for three values of η∗: 0.006, 0.016, and 0.022.
the binding energy of the deeper Efimov trimer, we get the constants C1 = 0.7 and
C2 = 0.3. The value of a∗ is approximately 54 a0. If we use Eq. (3.29) to estimate
the dimer relaxation rate, we obtain a maximum value of β1(23) at the peak of the
resonance of 2.4× 10−8 cm3/s for η∗ = 0.016.
3.3.5 Many-body physics
One of the most important motivations for understanding few-body physics is that
there are aspects of many-body physics that are controlled by few-body observables.
Our universal results for the few-body physics of 6Li atoms with 3 spin states provides
useful information about many-body systems of 6Li atoms with sufficiently low tem-
perature and sufficiently low number densities. If the atoms are in thermal equilibrium
75
at temperature T , an important scale is the thermal length λT = (2πmkBT/h2)−1/2.
If the atoms of type i have number density ni, another important scale is the Fermi
wavenumber kFi = (3π2ni)1/3. In our calculation of the 3-body recombination rate,
we set the wavenumbers of the incoming atoms to 0. In order for our result to apply
quantitatively to all atoms in the system, it is necessary that λ−1T , kF1, kF2, and kF3
all be small compared to the wavenumber scale (1/a21 + 1/a2
2 + 1/a23)
1/2 set by the
interactions. Even if this condition is not well satisfied, our result can be used to
obtain order-of-magnitude estimates of the relevant time scales.
There are several relevant time scales that we can extract from our few-body
calculations. If the system contains Efimov trimers, an important time scale is the
lifetime h/Γ(n)T of the Efimov trimer, where Γ
(n)T is its width. The universal predic-
tions for the widths of the Efimov trimers are shown in Fig. 3.5. If the system can
be approximated by a gas of individual low-energy atoms, the time scale for disap-
pearance of a significant fraction of the atoms is set by the 3-body recombination
rate constant K3. From the rate equation in Eq. (4.1), we see that the time scale
for loss of a significant fraction of the atoms of type i is (K3njnk)−1, where j and
k are the two complimentary spin states. The universal predictions for the 3-body
recombination rate are shown in Figs. 3.3 and 3.6. Finally if the system contains
low-energy dimers and low-energy atoms in the complimentary spin state, the time
scale for disappearance of a significant fraction of the atoms or dimers is set by the
appropriate dimer relaxation rate constant. From the rate equation in Eq. (3.23), we
see that the time scales for loss of significant fractions of (23) dimers and of atoms of
type 1 are (β1(23)n1)−1 and (β1(23)n(23))
−1, respectively. The only universal informa-
tion we have about dimer relaxation rates is the approximation for β1(23) in Eq. (3.29),
which should be accurate for magnetic fields within about 10 G of B∗ ≈ 672 G. For
magnetic fields further from this critical value but still within the region where a23 is
76
positive, one may be able to use the extrapolation of this expression as an estimate
for the dimer relaxation rate.
If the system contains a Bose-Einstein condensate of low-energy dimers as well
as low-energy atoms in the complimentary spin state, another important few-body
observable is the atom-dimer scattering length for that particular atom and dimer.
The real part of the atom-dimer scattering length determines the mean-field en-
ergy of the atom in the dimer condensate. The mean-field energy of an atom of
type 1 in the (23) dimer condensate with sufficiently low number density n(23) is
(3πh2/m) Re(a1(23))n(23). The approximation for a1(23) in Eq. (4.10) should be accu-
rate for magnetic fields within about 10 G of B∗ ≈ 672 G.
We will illustrate the relevance of our universal result to many-body physics by
applying them to two specific values of the scattering length that are interesting from
a symmetry perspective. If the pair scattering lengths aij and aik are equal, atoms of
types j and k are related by an SU(2) symmetry. If all three scattering lengths are
equal, the three spin states are related by an SU(3) symmetry. There is an SU(2)
symmetry point at 731 G, where the scattering lengths are a12 = a23 ≈ +2500 a0 and
a13 ≈ −7100 a0. An SU(3) symmetry point can be approached by going to very high
magnetic fields, where all three scattering lengths approach the spin-triplet scattering
length −2140 a0 [17].
A quantum degenerate Fermi gas of 6Li atoms with approximately equal popu-
lations of the three lowest hyperfine spin states at 1500 G has been realized by the
Penn State group [35]. The scattering lengths are a12 ≈ −2460 a0, a23 ≈ −2360 a0,
and a13 ≈ −2240 a0, so there is an approximate SU(3) symmetry. One candidate
for a metastable ground state is a state that can be approximated by filled Fermi
spheres for all three spin states, with Cooper pairing that breaks the SU(3) symme-
try down to a U(1) subgroup. The lifetime for such a system is determined by the
77
3-body recombination rate K3, which is predicted to be about 8 × 10−22 cm6/s at
1500 G. Another candidate for a metastable ground state is a filled Fermi sphere of
Efimov trimers. The binding energy E(0)T of the Efimov trimer is predicted to be about
25 MHz×(2πh) at 1500 G. Its width Γ(0)T is predicted to be about 1.7 MHz×(2πh),
which corresponds to a lifetime h/Γ(0)T of about 9× 10−8 s.
We now consider a many-body system at 731 G, where there is an SU(2) symmetry
relating the atoms of types 1 and 3. For a state that can be approximated by filled
Fermi spheres for all three spin states, the lifetime is determined by the the 3-body
recombination rate K3, which is predicted to be about 7× 10−22 cm6/s at 731 G. A
better candidate for a metastable ground state is a state containing a Bose-Einstein
condensate of dimers, which breaks the SU(2) symmetry down to a U(1) subgroup,
and a filled Fermi sphere of the complimentary atoms. The lifetime of the state is
determined by the dimer relaxation rates β1(23) and β3(12), which are equal by the
SU(2) symmetry. We can estimate β1(23) by extrapolating the expression for the
dimer relaxation rate β1(23) in Eq. (3.29) to 731 G, which gives 3 × 10−10 cm3/s.
We can also estimate the mean-field energy of an atom of type 1 in a (23) dimer
condensate with number density n(23) by extrapolating the expression for the atom-
dimer scattering length a1(23) in Eq. (4.10) to 731 G. The resulting estimate is 2 ×
10−9 Hz cm3 × (2πh n(23)). The positive sign of the real part of a1(23) implies that
the atoms of type 1 are repelled by the (23) dimer condensate. A state in which
the dimer condensate and the atoms are spatially separated is therefore energetically
favored over a homogeneous state. A final candidate for a metastable ground state
is a filled Fermi sphere of Efimov trimers. The binding energy E(1)T of the shallower
Efimov trimer is predicted to be about 190 kHz×(2πh) at 731 G. Its width Γ(1)T is
predicted to be about 6 kHz×(2πh) at 731 G, which corresponds to a lifetime h/Γ(1)T
of about 3× 10−5 s.
78
3.4 Summary and discussion
Systems consisting of 6Li atoms with 3 spin states provide a rich playground for
the interplay between few-body physics and many-body physics. The experimental
study of many-body physics is only possible if the loss rates of atoms from few-
body processes are sufficiently low. Measurements of the position and width of a
single Efimov loss feature can be used to determine the 3-body parameters κ∗ and η∗.
Calculations in the zero-range limit can then be used to predict few-body reaction
rates in the entire universal region.
In the low-field region for 6Li atoms, the universal predictions were only qualita-
tively successful. A fit to the measurements of the 3-body recombination rate by the
Heidelberg group gives the 3-body parameters κ∗ ≈ 77 a−10 and η∗ ≈ 0.11. With these
parameters, the universal results for the 3-body recombination rate as a function of
the magnetic field give a good fit to the narrow loss feature near 130 G but do not
agree well with measurements in the upper half of the low-field region [34, 51, 52]. A
reasonable explanation was proposed by the Heidelberg group [53]: η∗ is particularly
sensitive to the binding energies of the shallowest of the deep dimers and there are 6Li
dimers whose binding energies change dramatically across the low-field region. As-
suming the scaling behavior η∗ ∼ E−1deep, they obtained a good fit to the recombination
rate in the low-field region.
In the high-field region for 6Li atoms, the scattering lengths are much larger so the
universal predictions should be much more accurate. A narrow 3-atom loss feature
near 895 G was discovered by the Penn State group [35] and by the Heidelberg group
[54]. By fitting their measurements of the 3-body recombination rate, the Penn
State group determined the 3-body parameters associated with Efimov physics to be
κ∗ = 80.7 ± 2.3 a−10 and η∗ = 0.016+0.006
−0.010. We used those parameters to calculate
the universal predictions for the binding energies and widths of the Efimov trimers
79
shown in Fig. 3.4 and 3.5. The Efimov trimer responsible for the narrow loss feature
is predicted to disappear through the 1+(23) atom-dimer threshold at 672 ± 2 G,
producing a spectacular atom-dimer loss resonance. There is also a deeper Efimov
trimer whose binding frequency and width in the universal region are approximately
30 MHz and 2 MHz×(2πh), respectively. This trimer is also predicted to disappear
through the 1+(23) atom-dimer threshold, but this happens outside the universal
region. We also used the 3-body parameters determined by the Penn State group
to calculate the universal predictions for the 3-body recombination rate, which are
shown in Figs. 3.3 and 3.6. Local minima in K3 are predicted at 759±1 G, 829±1 G,
and 861±1 G. Finally an approximate calculation of the dimer relaxation rate in the
region of the atom-dimer resonance is presented in Fig. 3.7. We look forward to the
experimental verification of these predictions.
In order to understand the behavior of atom-dimer mixtures at low temperatures,
it would be useful to have universal predictions for other 3-atom observables. They
include the atom-dimer scattering lengths, which are in general complex. The real
part of an atom-dimer scattering length determines the mean-field shifts of the atom in
the dimer condensate. Its imaginary part is proportional to the dimer relaxation rate.
For the shallow dimer with the largest binding energy, which is the (23) dimer for B <
730 G and the (12) dimer for 730 G < B < 834 G, the only relaxation channels are into
deep dimers. For shallow dimers with smaller binding energy, there are also relaxation
channels into other shallow dimers. It would be especially useful to have definitive
universal predictions for the relaxation rate constant β1(23) near the predicted atom-
dimer threshold at 672±2 G. It would improve upon the approximation illustrated in
Fig. 3.7 by taking into account the B-dependence of the ratios of scattering lengths.
There have not yet been any direct observations of Efimov trimers in ultracold
atoms. They have only been observed indirectly through the resonant enhancement
80
of 3-body recombination and through the resonant enhancement of atom-dimer re-
laxation provided by virtual Efimov trimers. The direct production of Efimov trimers
would be another milestone in the study of Efimov physics in ultracold atoms. Of
course, once they are produced, they would decay quickly. In the high-field universal
region for 6Li atoms, the deeper Efimov trimer is predicted to have a very short life-
time of about 10−7 s. The shallower Efimov trimer is predicted to have a lifetime of
10−4 s to 10−5 s. Our universal predictions for the binding energies of these Efimov
trimers should be useful in devising experimental strategies for producing them.
3.5 Postscript
The analysis presented in this chapter was completed in August 2009. The prediction
of two atom-dimer loss resonances in the high-field region was verified in experiments
by the Heidelberg group [37] and by the Tokyo group [38] in March 2010. The
measured position of the narrow resonance was 685±2 G, which is about 15 G above
the universal prediction of 672± 2 G. The position of the second wider resonance is
near 603 G. The scattering lengths at this magnetic field are not large enough for
universal predictions to be reliable. The Tokyo group also calculated the universal
predictions for the atom-dimer relaxation rate [38], with results consistent with the
analysis described above. To resolve the discrepancy between the measured position
of the narrow resonance and the universal prediction, the Tokyo group proposed a
model in which the three-body parameter varies with the magnetic field [38].
The Heidelberg group also reported two local minima in the atom-dimer relaxation
rate [37]. These loss features will be discussed in Chapter 4, in which we calculate
the dimer relaxation rate for fermions with three spin states.
81
Chapter 4
Dimer relaxation
In this chapter, we study Efimov physics in the atom-dimer relaxation rate for 6Li
atoms in the three lowest hyperfine states. We perform a complete zero-range cal-
culation at zero temperature and estimate the finite range corrections and the finite
temperature corrections. Our zero-range calculation predicts the two resonances and
the two minima that have been observed in experiment. Quantitatively, our zero
temperature results show deviations from the data. An approximate finite temper-
ature calculation near one of the resonances improves the description of the data
considerably and shows that finite temperature effects account for about 25% of the
discrepancy. In Sections 4.1 and 4.2, we explain our theoretical framework. The
numerical results at zero temperature are displayed and compared with the experi-
mental data in Sec. 4.3. In Sec. 4.4, we carry out an approximate finite temperature
calculation near the resonance at B = 685 G. We summarize our results and conclude
in Sec. 4.5. The analysis presented in this chapter was carried out in collaboration
with Hans-Werner Hammer and Lucas Platter and was published in Physical Review
A [39] in August 2010.
82
4.1 Dimer relaxation
In this section we discuss the losses of atoms and dimers through inelastic scattering
processes and present expressions for the relaxation rate constants.
Before proceeding with our discussion of atom-dimer scattering, we explain our
notation and terminology. We label an atom in one of the three hyperfine states
of the 6Li atoms with an index i where i = 1, 2 or 3. The S-wave scattering length
between atoms in states i and j is denoted either as aij or as ak where k 6= i 6= j. Two
atoms in the same state can not scatter in an S-wave because of the Pauli principle.
If the scattering length aij is positive and much larger than the van der Waals length
`vdW ≈ 65a0, the atoms i and j can form a dimer with binding energy h2/(ma2ij).
We call this dimer the (shallow) ij-dimer or simply a shallow dimer. Shallow dimers
have to be distinguished from deep dimers with binding energy of order h2/(m`2vdW)
or larger.
In a gas of atoms i and jk-dimers, the atoms and dimers can undergo inelastic
collisions into atoms and deeply bound dimers with a binding energy larger than that
of the jk-dimer. This inelastic process is called dimer relaxation. The difference in
the binding energies of the initial and final state dimers is released as kinetic energy
and the atom and dimer in the final state recoil from each other. If their kinetic
energies are larger than the trapping potential, they escape the trap. The loss rate
for the number density ni of atoms i and number density njk of jk-dimers is
d
dtni =
d
dtnjk = −βi ninjk, (4.1)
where the coefficient βi is the relaxation rate constant for the jk-dimer and atom i.
In the case of identical bosons, dimer relaxation is possible only if the final state
consists of an atom and a deep dimer. However, in the three-fermion system relaxation
can also occur into into shallow dimers. For example, in a scattering process of an
83
atom i and a jk-dimer, relaxation can proceed into the ij-dimer provided the binding
energy of the ij-dimer is larger than that of the jk-dimer. Therefore, the total rate
βi is the sum of all relaxation rates into atoms plus shallow dimers and atoms plus
deep dimers
βi =∑
j 6=i
βshi→j + βdeep
i , (4.2)
where the index i implies atom i plus jk-dimer in the initial state and the j implies
atom j plus ik-dimer in the final state. For brevity, we refer to βshi→j as the rate into
the shallow dimer or the rate into the (final) ik-dimer and to βdeepi as the rate into
the deep dimers.
The relaxation rate can be calculated from the T-matrix element for atom-dimer
scattering T ADij (k, p;E) where k and p are the relative wave numbers of the atom and
dimer in initial and final state, respectively, and E is the total energy. By using the
optical theorem, we can calculate the total rate for atom-dimer scattering which is the
sum of elastic and inelastic rates. However, in the low energy limit k → 0, the elastic
rate scales as k because T ADii (k, k;E) is constant and the two-body phase space gives
one power of k [11]. The elastic rate therefore vanishes at zero energy and the optical
theorem gives the total relaxation rate βi
βi =2h
mImT AD
ii (0, 0,−1/(ma2i )). (4.3)
The rate into the shallow dimer βshi→j is determined by the square of the T-matrix
element multiplied by the two-body phase space:
βshi→j =
2ph
3πm
∣
∣T ADij (0, p,−1/(ma2
i ))∣
∣
2θ(ai − aj), (4.4)
where the wave number p = (2/√
3)√
a−2j − a−2
i and the θ-function is inserted because
the relaxation is allowed only when the initial dimer binding energy is smaller than
the final one. The prefactor 2p/(3π) comes from the two-body phase space integral.
84
From Eq. (4.4), one can deduce that βshi→j vanishes as
√ai − aj near the crossing of
the ik- and jk-dimers where ai approaches aj .
The relaxation rate into deep dimers βdeepi could also be calculated by using the
T-matrix in a similar way to Eq. (4.4) if the theory described deep dimers explicitly.
Alternatively, the effects of deep dimers can be taken into account indirectly by
using an analytic continuation of the three-body parameter into the complex plane
as introduced in Ref. [63]. Then, the partial rate βdeepi can be obtained by using the
relation in Eq. (4.2).
D’Incao and Esry have calculated the scattering length dependence of ultracold
three-body collisions near overlapping Feshbach resonances for a variety of cases [26].
For the relaxation rate into the shallow dimer βshi→j, they find that it scales like a2
j/ai
times a log-periodic function of aj if ai aj while ak is non-resonant. The rate into
deep dimers βdeepi scales like a2
j/ai times a prefactor that is a constant for positive aj
and a log-periodic function of aj for negative aj.
4.2 STM equations for non-zero energy
The Skorniakov–Ter-Martirosian (STM) equation [46] is an integral equation that
describes three-atom scattering interacting through zero-range interactions. In this
section, we discuss the STM equation and relate the T-matrix element T ADij to the
amplitude Aij that is the solution to the STM equation.
We consider only the S-wave contribution and assume that higher partial wave
contributions are suppressed at low temperature. For the three-fermion system, the
STM equation forms 9 coupled equations for the amplitudes Aij(k, p;E) [34, 36]. For
85
non-zero energy, the equation is given by
Aij(k, p;E) = (1− δij)Q(k, p;E)
+2
π
∑
k
(1− δkj)
∫ Λ
0
dqq2Q(q, p;E)Dk(q;E)Aik(k, q;E), (4.5)
where Aij(k, p;E) is the amplitude for an atom i and a complementary pair of atoms
to scatter into an atom j and a complementary pair and Λ is an ultraviolet cutoff.
The function Q(k, p;E) and the 2-atom propagator Dk(q;E) are given by
Q(k, p;E) =1
2kpln
[
k2 + kp+ p2 −mE − iεk2 − kp+ p2 −mE − iε
]
, (4.6)
Dk(q;E) =1
−1/ak +√
34q2 −mE − iε
. (4.7)
The solutions of the STM equation (4.5) depend log-periodically on Λ with a
discrete scaling factor eπ/s0 ≈ 22.7, where s0 ≈ 1.00624. The dependence on the
arbitrary cutoff Λ can be eliminated in favor of a physical 3-body parameter such as
the binding wave number of an Efimov trimer in the unitary limit. For convenience,
we choose to work directly with the wave number cutoff Λ in our calculations. If
deep dimers are present, the trimer has a finite width that allows it to decay into
an atom and a deep dimer. The effects of deep dimers can be taken into account by
analytically continuing the cutoff Λ into the complex plane [63]
Λ→ Λeiη∗/s0 , (4.8)
where η∗ is a width parameter associated with the effects of the deep dimers. Λ and
η∗ are determined from experimental measurements of three-body recombination in
the 6Li system [35] and their numerical values are Λ = 456 a−10 and η∗ = 0.016 [36]
where a0 is the Bohr radius.
In order to obtain the T-matrix element, the amplitude Aij must be multiplied
86
with the dimer-wavefunction renormalization factors√
ZiZj, where Zi = 8π/ai,
T ADij (k, p;E) =
8π√aiaj
Aij(k, p;E). (4.9)
By solving the STM equation (4.5) and using the relation in Eq. (4.9), we can calculate
the relaxation rates in Eqs. (4.3) and (4.4).
4.3 Zero temperature results
In this section we present our numerical results for the dimer relaxation rate constants
βi at zero temperature and compare them with recent measurements in Refs. [37, 38].
We solve the STM equation in Eq. (4.5) numerically with 5 input parameters: the
three pair scattering lengths ai, i = 1, 2, 3, the cutoff Λ and the width parameter
η∗. We use the values Λ = 456 a−10 and η∗ = 0.016 that have been determined from
measurements of the three-body recombination rate in Ref. [35] and have been used
in Ref. [36] to predict the atom-dimer relaxation rate.
Fig. 1.4 shows the scattering lengths for the 3 lowest hyperfine state of 6Li as a
function of the magnetic field. The atom-dimer relaxation rate βi is non-zero in the
region of positive scattering length ai. The upper limit of the region is set by the
Feshbach resonances at 811 G, 690 G, and 834 G for a1, a2, and a3, respectively. If
the scattering length is much larger than the van der Waals length `vdW ≈ 65a0, the
universal theory is valid. We denote this region as the universal region. Corrections
due to the finite range of the interaction should be small in the universal region. In
practice, we apply the universal zero-range theory when all scattering lengths are at
least two times larger than `vdW corresponding to magnetic fields B > 608 G. The
expected error due to finite range corrections is `vdW/a and therefore smaller than
50% in this region.
In Ref. [36], two crossings of Efimov trimers with the atom-dimer threshold have
87
been predicted. Both are located at the 1(23)-threshold: at B∗ = 672 G and at
B′∗ ≈ 597 G. Here the index 1 denotes the atom and the index (23) the dimer. Since
the atom-dimer relaxation is resonant when the trimer appears near the threshold,
two resonances are expected in the rate β1 near B∗ and B′∗. The resonance at B∗ is
well in the universal region where all scattering lengths are much larger than `vdW
while B′∗ is slightly outside. Therefore, the resonance position B∗ should be accurately
determined with corrections of order `vdW/a23 ≈ 10% where the value of a23 at the
resonance was used. The position B ′∗ is outside the universal region and can receive
large non-universal corrections of order 100%. These error estimates are accurate up
to a prefactor of order one. The exact value of this prefactor can only be obtained
from an explicit calculation of the range corrections. Note also that these percentage
errors apply to the positions in terms of the scattering length. To obtain the errors
for the corresponding magnetic field they have to be converted using Fig. 1.4.
In Fig. 4.1, we show our numerical results for β1 and compare them with the
recent measurements of Refs. [37, 38]. We give the full relaxation rate as well the
individual contributions from shallow and deep dimers. In the magnetic field region
from 590 G to 730 G, only the relaxation into deep dimers contributes to β1 since the
energy of the 23-dimer is larger than the energy of the other shallow dimers. At a
magnetic field of 730 G, the 23-dimer crosses the 12-dimer and the relaxation channel
into 12-dimers opens up. As the crossing is approached from above, the rate vanishes
as√a1 − a3 in agreement with the analytical result from Sec. 4.1. Between 790 and
810 G, the condition a1 a3 with a2 non-resonant is satisfied approximately. The
rate into the shallow dimer in this region scales approximately as a23/a1 in agreement
with the prediction of D’Incao and Esry [26]. At 811 G, the 23-dimer disappears
through the three-atom threshold and the relaxation rate vanishes.
Our results show two resonances at B∗ = 672 G and B′∗ ≈ 597 G. There is also a
88
600 650 700 750 800B [G]
10-11
10-10
10-9
10-8
10-7
10-6
β 1 [cm
3 /s]
β1
β1->3sh
β1deep
Figure 4.1: The relaxation rate constant for the 23-dimer and atom 1 as a functionof the magnetic field B. The squares and circles are data points from Ref. [37] andRef. [38], respectively. The curves are our results for the total rate β1 (solid line),the partial rate into atom 3 and the 12-dimer βsh
1→3 (dashed line), and the rate intoan atom and a deep dimer βdeep
1 (dashed-dotted line). The light solid line gives thetotal rate β1 for the parameters obtained in Ref. [37] while the vertical line marks theboundary of the universal region.
dramatic change in the relaxation rate at 730 G because the relaxation channel into
12-dimers opens and the corresponding rate into the 12-dimer increases rapidly. Our
results describe the resonances in the experimental data qualitatively. The second
resonance has been measured at Bexp∗ = 685 G in Ref. [37, 38]. This value is 13 G
away from the theoretical prediction B∗ = 672 G. In terms of the scattering length,
this corresponds to a23(Bexp∗ )/a23(B∗) = 1076/835 ≈ 1.3, leading to a 30% shift in the
resonance position. This shift is a factor three larger than the naive error estimate
of 10%. Taking into account the unknown prefactor of order one in the estimate,
89
however, the two values are consistent. Except near the resonance, the experimental
data are generally above our results.
The Heidelberg group [37] analyzed their data near the resonance at B∗ using an
approximate analytic expression from [36] and extracted the resonance position and
the width parameter η∗. They found the value η∗ = 0.34 which is more than an order
of magnitude larger than the value 0.016 extracted from three-body recombination.
Moreover, the normalization of the relaxation rate was adjusted to describe the data.
The light solid curve in Fig. 4.1 gives our universal result for the parameters Λ =
329 a−10 (which reproduces the resonance position Bexp
∗ = 685 G) and η∗ = 0.34.
These parameters give a much better description of the data in Refs. [37, 38] but
are generally a factor 2-3 above the experimental data. However, one should keep
in mind that the experimental data are only a factor three lower than the unitarity
bound and finite temperature effects are likely important. We will come back to this
issue in the next section.
The reason for the considerably larger value of η∗ extracted in [37] compared to the
value from recombination data is not understood. However, we note that a similar
discrepancy between the values of η∗ from atom-dimer relaxation and three-body
recombination occurs in the bosonic system of 133Cs atoms [28, 64].
The Tokyo group [38] performed a numerical analysis for β1 based on the uni-
versal theory and on a two-channel model. Their results obtained with the universal
theory agree with our calculation. Within the two-channel model they derived energy-
dependent scattering lengths which introduce non-universal effects in the two-body
amplitudes. Because this model cannot resolve the discrepancy between the universal
results and the measurements they concluded that the three-body parameters Λ and
η∗ depend on the magnetic field.
In Fig. 4.2, we show the experimental data from Ref. [37] and our numerical
90
600 650 700B [G]
10-12
10-11
10-10
10-9
10-8
β 2 [cm
3 /s]
β2
β2->3sh
β2->1sh
β2deep
Figure 4.2: The relaxation rate constant for the 13-dimer and atom 2 as a functionof the magnetic field B. The squares are data points from Ref. [37]. The curves areour results for the total rate β2 (solid line) and the partial rates into an atom 3 anda 12-dimer βsh
2→3 (dashed line), into an atom 1 and a 23-dimer βsh2→1 (dashed-dotted
line), and into an atom and a deep dimer βdeep2 (dotted line). The light solid line
gives the total rate β1 for the parameters obtained in Ref. [37] while the vertical linemarks the boundary of the universal region.
results for β2. The total rates, both of the data and the numerical results, show no
pronounced structure. However, there is a local minimum in the partial rate βsh2→1 near
600 G that is outside the universal region. As discussed in Ref. [26], this minimum is
the effect of destructive interference between different recombination channels. This
interference pattern does not exist in systems with identical bosons where relaxation
can occur only into deep dimers. In the total rate, the interference is hidden by the
dominant process βsh2→3. The light solid curve again gives our universal result for the
parameters Λ = 329 a−10 and η∗ = 0.34 obtained in Ref. [37]. The difference between
91
600 650 700 750 800B [G]
10-12
10-11
10-10
10-9
10-8
10-7
β 3 [cm
3 /s]
β3
β3->1sh
β3deep
Figure 4.3: The relaxation rate constant for the 12-dimer and atom 3 as a function ofthe magnetic field B. The squares are data points from Ref. [37]. The curves are ourresults for the total rate β3 (solid line), the partial rate into atom 1 and a 23-dimerβsh
3→1 (dashed line), and the rate into an atom and a deep dimer βdeep3 (dashed-dotted
line). The light solid line gives the total rate β1 for the parameters obtained inRef. [37] while the vertical line marks the boundary of the universal region.
the two parameter sets is very small for β2, but the alternative set gives a slightly
better description of the data. Between 670 and 690 G, a2 is much larger than a1 and
a3 which are approximately equal. The scaling of the calculated rates into the shallow
dimers is consistent with the prediction of D’Incao and Esry [26]: βsh2→3 ∼ a2
3/a2 and
βsh1→3 ∼ a2
1/a2.
Fig. 4.3 shows our results and the recent measurement [37] for β3. Between 590 G
and 730 G, the energy of the 12-dimer is smaller than the energy of the 23-dimer
and the relaxation channel into 23-dimers is open. After the 23-dimer crosses the
12-dimer at 730 G this relaxation channel is closed and only relaxation into deep
92
dimers is possible. As the crossing is approached, the rate vanishes as√a3 − a1 in
agreement with the analytical result from Sec. 4.1. Between 832 and 834 G, a1 is
negative and the condition a3 |a1| with a2 non-resonant is satisfied approximately.
The rate into deep dimers scales in this region approximately as a21/a3 consistent with
the prediction by D’Incao and Esry. Two interference minima have been observed
at 610 G and 695 G [37] while our results show two minima at 600 G and 715 G.
Above 700 G, the data are larger than our results by more than a factor 10. Using
the alternative parameters Λ = 329 a−10 and η∗ = 0.34 obtained in Ref. [37], we again
find a better agreement with the data. With these parameters the second minimum
in the rate into the shallow dimer disappears beyond 730 G. Hence, the partial rate
decreases monotonically and vanishes near 730 G. For smaller η∗, the position of the
minimum in the total rate is around 730 G. As η∗ increases, the position remains
almost the same and the depth of the minimum becomes shallower. The minimum is
not visible when η∗ = 0.34 because the rate into deep dimers is much larger than the
rate into the shallow dimer. Therefore, with these parameters that were fit to data
for β1 the position of the second minimum in the data for β3 cannot be explained
correctly.
If the rate into 23-dimers could be separated experimentally from the total rate,
it would clearly determine the positions of the local minima. This could be achieved
by tuning the depth of trapping potential such that it is much larger than the kinetic
energies of atoms and 23-dimers in the final state but much smaller than the energies
of atoms and deep dimers in the final state. The kinetic energies of an atom and a
deep dimer in the final state could be estimated from the binding energy of the deep
dimers. Their energies would be of the order of the van der Waals energy or larger:
EvdW/h ≈ 154 MHz, where EvdW = h2/(m`2vdW).2 The kinetic energies of an atom
2 A convenient conversion constant for 6Li atoms is given by h2/(ma2
0) = 600h GHz= 28.8kB K.
93
and a 23-dimer are given by the difference in binding energies between the 12-dimer
and the 23-dimer: h/(2πm)(a−223 − a−2
12 ) is about 1 MHz at 650 G and vanishes at
730 G. This way one may be able to measure the rate into deep dimers separately
and to extract the rate into the 23-dimer.
4.4 Finite temperature results
The results from the previous section suggest that finite temperature effects may play
an important role in understanding the atom-dimer relaxation data from Refs. [37, 38].
A full finite temperature calculation of the 6Li system is beyond the scope of this work.
Therefore, we perform an approximate calculation of the relaxation rate β1 near the
resonance at Bexp∗ = 685 G.
We start from the approximate expression for the scattering length between an
atom 1 and a 23-dimer near the resonance at B∗ that was extracted from a calculation
of the trimer binding energy in [36]:
a1(23) ≈ (C1 cot[s0 ln(a23/a∗) + iη∗] + C2) a23, (4.10)
where a∗ = a23(B∗) and the coefficients are C1 = 0.67 and C2 = 0.65. Using the
scattering length approximation for the S-wave atom-dimer scattering amplitude in
the 1(23)-channel,
f1(23)(k) =[
−1/a1(23) − ik]−1
, (4.11)
we can calculate the inelastic scattering cross section. At low temperatures, the
contributions from higher partial waves can be neglected. We subtract the elastic
cross section from the total cross section obtained via the optical theorem as in [65]
and find for the inelastic cross section:
σ(inelastic)1(23) (k) =
4π
k
−Im a1(23)
1− 2kIm a1(23) + k2|a1(23)|2. (4.12)
94
0.6 0.8 1 1.2 1.4a/a*
10-9
10-8
10-7
10-6
β 1 [cm
3 /s] η* = 0.016
η* = 0.34
Figure 4.4: The relaxation rate constant β1 for a 23-dimer and an atom 1 as a functionof a/a∗ near the resonance at Bexp
∗ = 685 G (a∗ = 1076 a0). Squares and circles aredata points from Ref. [37] and Ref. [38], respectively. The solid (dashed) curvescorrespond to η∗ = 0.016 (η∗ = 0.34). The upper curves give the zero temperatureresult, while the lower curves give the finite temperature result for T = 60 ± 15 nKwith the shaded area indicating the temperature uncertainty.
The total dimer-relaxation rate β1 can then be calculated by taking a Boltzmann
thermal average of the inelastic reaction rate vrel σ(inelastic)1(23) (k) where vrel = 3hk/(2m)
is the relative velocity of the atom and dimer in the initial state. This leads to the
expression
β1(T ) =3h
2m〈kσ(inelastic)
1(23) (k)〉 =3h
2m
3λ3T
4π2
√
3
2
∫ ∞
0
k2dk kσ(inelastic)1(23) (k) e
− 3h2k2
4mkBT , (4.13)
where λT =√
2πh2/(mkBT ) is the thermal de Brogile wavelength of the atoms.
Our results for the atom-dimer relaxation rate constant near the resonance at
95
Bexp∗ = 685 G are shown in Fig. 4.4. The results for the width parameters η∗ = 0.016
and η∗ = 0.34 are given by the solid and dashed curves, respectively. In each case the
upper curves give the zero temperature result while the lower curves give the finite
temperature result for T = 60±15 nK. Here, the shaded area indicates the uncertainty
from the temperature. The value of a∗ = 1076 a0 has been fixed to reproduce the
resonance position at Bexp∗ = 685 G. For η∗ = 0.016, the finite temperature effects
decrease the height of the peak by an order of magnitude but the predicted resonance
shape is still much narrower than the data. The finite temperature effects are much
less severe for η∗ = 0.34. They only lead to a reduction of β1 by about 25 % but clearly
improve the description of the data. We conclude that finite temperature effects can
not resolve the question of the different values for η∗ in the three-body recombination
and dimer-relaxation data. Moreover, while finite temperature effects are important,
a qualitative description of the data at T ≈ 60 nK can be already achieved with a
zero temperature calculation.
4.5 Summary and outlook
In this chapter, we have studied Efimov physics in atom-dimer relaxation of 6Li
atoms in the three lowest hyperfine states using the universal zero-range theory. Two
resonances were observed at magnetic fields 603 G and 685 G in the relaxation rate
β1 in recent experiments [37, 38]. These resonances are consequences of two Efimov
trimers close to the atom-dimer threshold. Their positions have been predicted by
Braaten et al. [36]. The measured position of the resonance at 685 G [37, 38], which
is well within the universal region, is larger than the prediction by about 30%. This is
consistent with an error of order 10% error due to effective range corrections. However,
the value η∗ = 0.016 extracted from the three-body recombination data is not able to
describe the atom-dimer relaxation data which require the larger value η∗ = 0.34 [37].
96
The reason for the larger value of η∗ in dimer relaxation compared to the value from
recombination data is not understood. However, we note that a similar discrepancy
between the values of η∗ from atom-dimer relaxation and three-body recombination
occurs in the bosonic systems of 133Cs atoms [28, 64].
Using the value η∗ = 0.34, our zero temperature calculation is able to describe
the data qualitatively. In the vicinity of this resonance, we have also performed an
approximate finite temperature calculation and find sizable temperature effects that
can suppress the relaxation rate by an order of magnitude if η∗ = 0.016. For the
larger value η∗ = 0.34, however, these effects lead to a moderate suppression of about
25%, such that zero temperature results are useful as a first approximation.
In Ref. [37], also two local minima at 610 G and 695 G are discovered in the rate
β3. Those minima can be associated with destructive interference between different
recombination channels [26]. Our numerical results show that the partial rate into
23-dimers is responsible for the minima but the positions of the minima are displaced
by -10 G and +20 G from the measurements. These displacements correspond to
a 30% shift in terms of the scattering length compared to our predictions. Since
the scattering length, a23, is about a factor two larger in this magnetic field region
than around the resonance in β1, one would expect smaller corrections here. The
observed shifts indicate that the destructive interference leading to the minima might
be dominated by wave numbers k larger than 1/a23 such that corrections of order
k`vdW are important. Moreover, finite temperature effects could fill the minima in an
asymmetric fashion. The total rate β2 shows no structure. However, the partial rate
into the 23-dimer shows a local minimum near 600 G. In the total rate this feature is
hidden by the dominant rate into the 12-dimer.
In order to better understand the discrepancy between the values of η∗ and the
resonance positions extracted from three-body recombination and atom-dimer relax-
97
ation two important improvements are required in a future analysis. First, a full finite
temperature calculation of atom-dimer relaxation should be carried out. This requires
calculating atom-dimer scattering above the dimer breakup thresholds and in higher
partial waves. Due to the different pair scattering lengths in the three channels, such
a calculation is considerably more complex than in the case of identical bosons [66].
However, it will allow to better distinguish effects from the resonance width param-
eter η∗ and from the finite temperature which can be partially traded for each other
[66, 64]. Second, an analysis of the effective range corrections should be performed
in order to describe the observed shifts in the resonance positions quantitatively. A
similar analysis for systems of identical bosons was carried out in Refs. [67, 68].
4.6 Postscript
The analysis presented in this chapter was completed in June 2010. As of June 2011,
there have been no new experimental measurements of the dimer relaxation rate in
6Li atoms.
The observation of a resonance loss feature associated with an Efimov trimer near
a threshold is an indirect observation of the trimer. The first direct observation of
an Efimov trimer was through the radio-frequency association of a 6Li atom and a
dimer by the Heidelberg group in June 2010 [40] and by the Tokyo group in October
2010 [41]. The measurements of the magnetic field at which the Efimov trimer crosses
the atom-dimer threshold were consistent with the position of the atom-dimer loss
resonance observed by the Heidelberg group and by the Tokyo group [40, 41]. These
measurements are in qualitative agreement with the universal prediction in Chapter
3, but there is a quantitative discrepancy between the theory and experiment.
98
Chapter 5
Outlook
To conclude this thesis, we summarize unresolved questions raised in the studies of
the Efimov physics in 6Li atoms and we suggest what can be done to answer these
questions. We also discuss possible future directions.
In Chapter 2, we calculated the three-body recombination rate at zero tempera-
ture for fermions with three spin states and large negative scattering lengths. Our
calculation of the rate for 6Li atoms gave an excellent fit to the narrow loss resonance
near 130 G and predicted an equally narrow loss resonance near 500 G. The observed
loss resonance near 500 G is much broader [7, 8]. The Heidelberg group pointed out
that the three shallowest of the deep dimers in this region of magnetic field are the
diatomic molecules that are responsible for the Feshbach resonances in the high-field
region [53]. The binding energies of these dimers are zero at the Feshbach resonances
at 690 G, 811 G, and 830 G. Their binding energies increase as the magnetic field
decreases, changing by about a factor of 7 from 500 G to 130 G. The Heidelberg group
proposed a simple empirical relation between the three-body width parameter η∗ and
the binding energy Edeep of a deep dimer that dominates the width : η∗ ∝ 1/Edeep.
Rittenhouse deduced a more complicated functional form for η∗(Edeep) by calculating
the recombination rate using a simple model with two hyperspherical channels [69]. In
this thesis, we have implicitly assumed that the three-body parameters κ∗ and η∗ are
99
constant with magnetic field. The difference between the width of the loss resonances
near 130 G and 500 G is a clear counterexample to this assumption. This raises the
question of what are the important physics processes that determine the three-body
parameters κ∗ and η∗. Under what condition will these parameters depend on the
magnetic field when we use a Feshbach resonance to control the scattering length a?
In Chapter 3, we calculated the energy spectrum of Efimov trimers in the high-field
region. The spectrum predicts an atom-dimer loss resonance at 672± 2 G, where an
Efimov trimer disappears through the atom-dimer threshold. The Heidelberg group
and the Tokyo group discovered the resonance at a different position 685±2 G [37, 38].
The predicted position of the resonance is about seven standard deviations away from
the measured position. The scattering lengths are large enough at this magnetic field
that the universal prediction should be very accurate. What is responsible for this
discrepancy between theory and experiment? The predicted resonance position is
most sensitive to the three-body parameter κ∗, which was determined by the three-
body loss resonance near 895 G obtained by the Penn State and Heidelberg groups
[35, 54]. One possibility is that there is a systematic error in the determination of the
position of the three-body loss resonance in Ref. [35]. The systematic error could arise
from a thermal shift in the position. The discrepancy between theory and experiment
in the position of the atom-dimer loss resonance can be accommodated by allowing
κ∗ to vary with the magnetic field, decreasing by 30% between 895 G and 685 G.
However no mechanism for such a variation has been identified.
There are significant discrepancies between the universal predictions of the trimer
binding energy in Chapter 3 and the measurements reported by the Tokyo group
[41]. The binding energies of the Tokyo group are about 30% larger than those of
the Heidelberg group [40]. The Tokyo group pointed out that the temperature in
the Heidelberg experiment was higher and the resulting thermal shift in the binding
100
energy could account for the discrepancy. The universal predictions are compatible
with the measurement by the Heidelberg group, but they are about four standard
deviations (30%) larger than those by the Tokyo group. Part of the discrepancy with
the Tokyo data could be explained by a systematic error in the prediction from the
determination of κ∗
In Chapter 4, we computed the dimer relaxation rates at zero temperature. Al-
though the numerical results for the relaxation rates agree qualitatively with the
measured rates by the Heidelberg group [37] and by the Tokyo group [38], there are
significant discrepancies quantitatively. The discrepancy was decreased by the ap-
proximate calculations at finite temperature near the resonance in Chapter 4. Part of
the remaining discrepancy could be due to effective range corrections, which become
more important at finite temperature and could shift the resonance position [67, 68].
The Tokyo group has carried out theoretical studies of the discrepancies in the
trimer binding energies and in the atom-dimer relaxation rates. [38, 70]. They allowed
the three-body parameters κ∗ and η∗ to change with the magnetic field B or with the
collision energy. By fitting κ∗ and η∗ to their data for dimer relaxation, they could
successfully predict the trimer binding energy. However they were unable to get a
good global description of all the data for 6Li atoms using three-body parameters κ∗
and η∗ that are smooth monotonic functions.
In this thesis, we have calculated the universal predictions for the three-body re-
combination rates and the dimer relaxation rates at zero temperature. These rates
were calculated at the threshold where the collision energy is zero. It would be useful
to extend these calculations to obtain the rates as functions of the collision energy.
The results could be used to calculate the universal predictions for the rates as func-
tions of the temperature. Other useful observables in the 3-atom sector that could be
calculated are the atom-dimer scattering lengths (aAD). They determine the mean-
101
field energy shift of an atom in a Bose-Einstein condensate of shallow dimers. For suf-
ficiently low dimer density such that nDa3AD 1, the energy shift is 3π3h aADnD /m.
Finally, universality in the 4-atom sector for fermions with three spin states is com-
pletely unexplored territory. It would be particularly valuable to have predictions of
the binding energies of universal tetramers.
In 2005, Tan derived some universal relations that hold for any system that con-
sists of fermion with two spin states that interact with a large scattering length
[71, 72, 73]. They relate various observables to an extensive property of the system
called the contact. The contact is a measure of the probability for two atoms to be
close together. The universal relations can be derived using the analytic solution
to the two-body problem. They therefore reveal aspects of many-body physics that
are controlled by two-body physics. Tan’s universal relations were rederived by us-
ing quantum field theory methods, including the operator product expansion [74, 75].
Many other universal relations involving the contact have been derived subsequently
[76]. This technique clearly shows the connection of few-body physics to many-body
physics. Recently, analogous universal relations for identical bosons were derived [77].
In addition to the two-body contact these relations involve a three-body contact, which
is a measure of the probability for three atom to be close together [77]. In some of the
relations, the coefficient of the three-body contact is log-periodic function, which is a
signature of Efimov physics. It would be useful to derive the analogous universal re-
lations for fermions with three spin states. These relations would involve a two-body
contact for each of the three pairs of spin states and the three-body contact, which
is a measure of the probability for three atoms in the three different spin states to be
close together. These universal relations are a promising new probe to study Efimov
physics in 6Li atoms.
102
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