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Many- and few-body physics in low-dimensional resonantly-interacting Fermi
quantum gases
Kurlov, D.V.
Publication date 2020
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Kurlov, D. V. (2020). Many- and few-body physics in low-dimensional resonantly-interacting Fermi quantum gases.
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in low-dimensional
resonantly-interacting
Fermi quantum gases
Many- and few-body physics
in low-dimensional
resonantly-interacting
Fermi quantum gases
Many- and few-body physics
in low-dimensional
resonantly-interacting
Fermi quantum gases
Academisch Proefschrift
ter
verkrijging van de graad van doctor
aan
de Universiteit van Amsterdam
op
gezag van de Rector Magnificus
prof.
dr. ir. K.I.J. Maex
ten
overstaan van een door het College voor Promoties ingestelde
commissie,
in het openbaar te verdedigen
op woensdag 27 mei 2020, te 12:00 uur
door
Denis Viktorovich Kurlov
Promotor: Prof. dr. F. E. Schreck Universiteit van Amsterdam Co-promotor: Prof. dr. G. V. Shlyapnikov Universiteit van Amsterdam Overige leden: Prof. dr. J. T. M. Walraven Universiteit van Amsterdam Prof. dr. J. S. Caux Universiteit van Amsterdam Prof. dr. C. de Morais Smith Utrecht Universiteit Prof. dr. V. I. Yudson Russian Quantum Center Dr. V. Gritsev Universiteit van Amsterdam Dr. R. J. C. Spreeuw Universiteit van Amsterdam Faculteit der Natuurwetenschappen, Wiskunde en Informatica
The work described in this thesis has been supported by the European Research Council (ERC) and the Netherlands Organisation for Scientific Research (NWO), and was carried out in the group “Quantum Gases & Quantum Information”, at the Van der Waals-Zeeman Institute of the University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
Publications
This Thesis is based on the following publications:
• Y. Jiang, D. V. Kurlov, X.-W. Guan, F. Schreck, and G. V. Shlyapnikov, “Itinerant ferromagnetism in one-dimensional two-component Fermi gases”,
Phys. Rev. A 94, 011601(R) (2016). • D. V. Kurlov and G. V. Shlyapnikov,
“Two-body relaxation of spin-polarized fermions in reduced dimensionalities near a p-wave Feshbach resonance”,
Phys. Rev. A 95, 032710 (2017).
• D. V. Kurlov, S. I. Matveenko, V. Gritsev, and G. V. Shlyapnikov,
“One-dimensional two-component fermions with contact even-wave repul-sion and SU(2)-symmetry-breaking near-resonant odd-wave attraction”,
Phys. Rev. A 99, 043631 (2019).
Contents
1 Introduction 3
1.1 Background and motivation . . . 3
1.2 Outline. . . 6
2 Two-body relaxation of spin-polarized fermions in reduced di-mensionalities near a p-wave Feshbach resonance 11 2.1 Introduction . . . 11
2.2 Two-body inelastic collisions in 3D . . . 12
2.3 Two-body inelastic collisions in 2D . . . 16
2.4 Two-body inelastic collisions in 1D . . . 19
2.5 Two-body inelastic rate near the resonance in 2D and 1D: Conclu-sions . . . 23
Appendices 29 2A Spin-relaxation taking into account the doubling of the p-wave Fes-hbach resonance . . . 29
2B Derivation of the near-resonant inelastic rate constant . . . 36
2B.1 Zero temperature case . . . 36
2B.2 Thermal case . . . 38
3 Itinerant ferromagnetism in one-dimensional two-component Fermi gases 41 Appendices 53 3A Scattering amplitude . . . 53
3B Two-body and many-body contributions to the interaction energy 55
4 One-dimensional two-component fermions with contact even-wave
4.1 Introduction . . . 59
4.2 Interaction between particles and Hamiltonian of the system . . . 61
4.2.1 Even-wave interaction . . . 61
4.2.2 Odd-wave interaction . . . 61
4.2.3 Effective odd-wave interaction . . . 63
4.3 Bosonization Procedure. . . 64
4.3.1 Notations . . . 64
4.3.2 Even-wave interaction . . . 65
4.3.3 Odd-wave interaction . . . 66
4.3.4 Total bosonized action . . . 68
4.4 Renormalization Group Analysis. . . 69
4.5 Phase transition criterion. . . 73
4.6 Discussion and Conclusions . . . 75
Appendices 79 4A The integral in Eq. (4.25) . . . 79
4B Bosonized odd-wave interaction . . . 81
4C Effective Lagrangian for the spin fields . . . 81
4C.1 Total bosonized Lagrangian . . . 82
4C.2 Integration over the charge degrees of freedom . . . 82
4C.3 Derivation of Eq. (4.89) . . . 85
4C.4 Effective spin Lagrangian in the Φσ-representation . . . 86
4D Momentum-shell Renormalization Group . . . 87
4D.1 First order correction . . . 88
4D.2 Second order correction. . . 89
4D.3 Renormalized action and RG equations . . . 91
Bibliography 95
Summary 105
Samenvatting 107
Chapter 1
Introduction
1.1
Background and motivation
Over the past few decades the field of ultracold quantum gases has developed into a mature and broad research area that interconnects atomic physics, quantum optics, condensed matter physics, metrology, and quantum information. Studies of ultracold quantum gases also provide insights into such seemingly unrelated fields as nuclear physics [1], high energy physics [2], and astrophysics [3]. This remarkable progress has become possible due to tremendous advances in the atom trapping and cooling techniques [4–6]. Let us begin with a brief overview of the major historical achievements that have led to the present state of the field, and introduce the key physical concepts along the way.
One of the first important milestones in the development of the field was achieved with the creation of a dilute ultracold gas of spin-polarized hydrogen atoms in 1979 by the group of I. F. Silvera and J. T. M. Walraven in Amster-dam [7]. The gas sample had temperatures of the order of hundreds of millikelvins and densities ∼ 1016 cm−3. In a dilute gas the mean interparticle separation greatly exceeds the characteristic radius of the interaction1, R
e, which typically
ranges from a few dozens to a hundred of angstroms. An important length scale in a gas is the atomic thermal de Brogile wavelength λT = ~/
√
2πmkBT , where
m is the atom mass, T is the temperature, and kB is the Boltzmann constant.
Once λT also becomes greater than Re, the gas enters the ultracold regime.
How-ever, if λT remains smaller than the interparticle separation, the statistics is still
classical.
It took another sixteen years to reach the limit of quantum degeneracy, the regime in which λT becomes comparable to the mean interparticle separation and
quantum statistics comes into play. In order to enter this regime one has to cool
1The quantity R
eis defined as the interparticle separation at which the characteristic kinetic
energy of the relative motion is of the same order as the potential energy of the interaction. For neutral atoms this gives Re∼ (m CvdW/~2)1/4, where CvdWis the van der Waals constant.
the gas down to the temperature of quantum degeneracy, kBTd∼ ~2n2/3/m, where
n is the gas density. With the help of optical and evaporative cooling techniques this was achieved in 1995, when the first Bose-Einstein condensation (BEC) of alkali atoms was created by the groups of E. Cornell and C. Wieman at JILA [8], W. Ketterle at MIT [9], and R. Hulet at RICE [10]. The temperatures in these experiments were ranging from hundreds of nanokelvins to a few microkelvins, and the densities were up to ∼ 1014 cm−3.
Soon after, in 1999, the first quantum degenerate atomic Fermi gas was ob-tained in the group of D. Jin at JILA [11]. In that experiment the two-component gas of 40K atoms was held in a magnetic trap and cooled down to a fraction of
the Fermi energy. In a later stage, in 2002, the groups in the Duke University [12] and at JILA [13] obtained the first quantum degenerate atomic Fermi gas in an optical dipole trap. This was a crucial development because it allowed one to use magnetic fields for purposes other than trapping, namely for manipulating the interactions between atoms using the so-called Feshbach resonances [14].
The mechanism of the Feshbach resonance in a pair atomic collision can be explained as follows. The interaction between colliding atoms depends on their internal (hyperfine) states, and different scattering channels are coupled to each other by the hyperfine interaction. In the ultracold limit the scattering state of colliding atoms (open channel) lies just above the continuum threshold in their interaction potential. The latter is determined by the hyperfine energy of the two atoms at asymptotically large distances. Any channel with a different hyperfine energy is referred to as a closed channel. The Feshbach resonance occurs when there is a weakly bound state in the closed-channel interaction potential with the binding energy very close to the continuum threshold in the open channel. In this case, during the collision the atoms may undergo a virtual transition to the weakly-bound molecular state. This situation corresponds to the resonant scat-tering on the (quasi)discrete level embedded into continuum, well known from the general scattering theory [15]. The open and the closed channels belong to differ-ent hyperfine domains, and the difference between their interaction potdiffer-entials at asymptotically large distances depends on the magnetic field. Thus, an external magnetic field shifts the closed-channel bound state with respect to the open-channel continuum threshold, allowing one to modify the interaction strength in a broad range and even change the sign of the interaction. This fascinating de-gree of control opened new prospects for investigating a large variety of strongly interacting and strongly correlated systems, which previously appeared mostly in the context of condensed matter or nuclear physics.
Another example of a remarkable tunability provided by ultracold quantum gases is the possibility to change the dimensionality of a system with the aid of highly anisotropic magnetic [16–21] or optical trapping potentials [22,23]. By making the harmonic confinement frequency in one or two directions sufficiently large, one may reduce the atom motion to zero point oscillations. In this case, kinematically the gas becomes, correspondingly, two- or one-dimensional (2D or
1.1. Background and motivation 5 1D), and it differs from purely low-dimensional gases only by the values of the interaction parameters that now depend on the confinement frequency (note that this dependence may have a resonant character [24,25]). Starting from 2001, the 2D and 1D regimes were realized for trapped atomic BEC at MIT [26], LENS [27], Innsbruck [28], and JILA [29]. This was done by a tight optical or magnetic confinement [26,27], or by a fast rotation of the cloud that led to an increase in the size of the sample in two directions [29]. Subsequent experimental progress resulted in successful observations of low-dimensional quantum degenerate Fermi gases [30–33] and Bose-Fermi mixtures [34]. To a large extent this became possible owing to the invention of optical lattices [35–39]. Their physical mechanism is the following. Like in all optical traps for neutral atoms, the trapping potential originates from a spatially inhomogeneous ac Stark shift experienced by the atoms in an off-resonant laser field [40]. The potential strength is proportional to the atomic polarizability and the laser intensity. In the case of optical lattices, one creates a periodic potential by overlapping counter-propagating laser beams, such that the light field has a standing wave configuration. Depending on the sign of the polarizability, the atoms are attracted either to the nodes or the antinodes of the lattice. Ever since their appearance, optical lattices have proved themselves as an invaluable tool for exploring many-body physics with ultracold atoms. Not only one can construct lattices of different geometry and dimensionality, but also make them species-selective [41]. Moreover, one can confine the atoms into an array of quasi2D pancake-shaped clouds or quasi1D tubes [38]. The tunneling rate between these low-dimensional clouds can be tuned at will, allowing one to have them either coupled or isolated.
This unprecedented control over the interactions, dimensionality, and other relevant parameters makes ultracold quantum gases an ideal platform for simu-lating a large variety of strongly correlated quantum many-body systems that are of great interest for condensed matter physics and other research areas. Numerous experiments were dedicated to realize novel many-body phases with bosons and two-component fermions. In the former case examples include the observation of the superfluid to Mott-insulator transition in optical lattices [42], Berezinskii-Kosterlitz-Thouless transition in a two-dimensional Bose gas [43,44], Tonks-Girardeau gas of hard-core bosons [45,46], and many others. Experiments with strongly interacting two-component Fermi gases explored the BCS-BEC crossover [47–54], superfluidity in population-imbalanced mixtures [55,56], and the paradigmatic Hubbard model [57].
In all examples above the interactions between atoms predominantly occur in the s-wave scattering channel. In ultracold Bose or two-component Fermi gases this is the leading scattering channel because all partial scattering amplitudes in the channels with higher values of the orbital angular momentum l are strongly suppressed. On the contrary, in single-component Fermi gases, the symmetry only allows scattering with odd l and the dominant contribution comes from the p-wave channel (l = 1). In the ultracold regime the p-p-wave interactions are usually
much weaker than the s-wave interactions, because the former are proportional to the square of the relative momenta of colliding particles. However, they can be significantly enhanced with the help of a p-wave Feshbach resonance [58]. Ul-tracold quantum gases with strong p-wave interactions offer prospects to study far richer physics than their s-wave interacting counterparts. For instance, there have been a number of theoretical proposals on observing unconventional px+ ipy
superfluid states in 2D [59–62]. In such superfluids, the orbital momentum of a Cooper pair is unity [60], and quantized vortices carry zero-energy non-Abelian Majorana modes on their cores [63]. The latter makes these states extremely attractive for topological quantum computing [64–69]. Other proposals [70,71] suggested realization of the prototypical system with topological order – the Ki-taev model [65]. Many interesting phenomena are offered by combining strong interactions in both s- and p-wave channels (even- and odd-wave in 1D, respec-tively). These include, e.g., exotic superfluid states with hybridized pairing [72] and magnetically ordered phases [73,74]. It has also been proposed to study a quantum phase transition from an integer to fractional quantum Hall states with Fermi gases near the p-wave Feshbach resonance [75].
Despite significant efforts, at present the aforementioned systems with strong p-wave interactions have not been realized in experiments with cold atoms. The reason is that on approach to the p-wave Feshbach resonance the rate of inelastic collisional losses usually becomes very large [58,76–80]. However, the results presented in Chapter2of this Thesis (see also Ref. [81]) and confirmed by recent experimental findings [82] show, that an increase of the rate of inelastic collisions near the p-wave resonance can be highly suppressed in low dimensional Fermi gases, especially in the 1D case. Presently, there is a renewed interest in the physics of resonantly-interacting Fermi gases, and recent experiments have further studied inelastic losses in the vicinity of a p-wave Feshbach resonance [83,84]. This offers a new hope for observing a variety of novel strongly-correlated many-body phases.
1.2
Outline
This Thesis is devoted to theoretical investigation of low-dimensional quantum gases of atomic fermions. The main emphasis is put on the role of resonant interparticle interactions. The results of the Thesis are strongly linked to the ongoing and future experimental studies.
The Thesis is organized as follows. In Chapter 2 we address the issue of in-elastic pair collisions in a spin-polarized Fermi gas. In three dimensions, in the vicinity of the Feshbach resonance, inelastic collisional processes are highly en-hanced compared to the off-resonant regime. This leads to a rapid decay of the gas and makes it very hard or even impossible to perform experimental studies of systems with p-wave interactions. Nevertheless, we demonstrate that in reduced
1.2. Outline 7 dimensionalities the situation is much more promising. We study two-body in-elastic collisional processes (two-body relaxation) in a spin-polarized ultracold Fermi gas and show that in reduced spatial dimensions the enhancement of the inelastic rate constant close to the p-wave Feshbach resonance is much weaker than in three dimensions. We also illustrate this with an important example of
40K atoms in the 92, −72
state. The effect of the suppressed enhancement of the inelastic rate is especially pronounced in one spacial dimension, and it opens prospects for realizing novel and exotic many-body phases of strongly interacting fermions. The results presented in this Chapter are published in Ref. [81] and have been confirmed experimentally [82].
We then proceed with an example of a many-body phase that can be obtained with the help of resonant p-wave interactions. This is the itinerant ferromagnetic state in a two-component Fermi gas. The concept of itinerant ferromagnetism arises when one considers a system of delocalized fermions which possess internal degrees of freedom. The exact nature of internal states is irrelevant: it can be spin or, as in the case of ultracold atomic gases, the hyperfine state of an atom. Usually the two hyperfine energy levels are chosen as pseudo-spin states and are denoted as spin-↑ and spin-↓. Itinerant ferromagnetism is a long-standing prob-lem in condensed matter physics. Despite enormous amount of work dedicated to it, many aspects, such as the character of the phase transition, remain dis-putable. Ultracold atomic gases attract a great attention for studying itinerant ferromagnetism because of their high degree of control. However, experimental efforts in obtaining a stable three-dimensional itinerant ferromagnetic state with cold atoms did not succeed2. In Chapter 3we show that it is promising to realize this state in the one-dimensional atomic Fermi gas. We consider the regime of an infinitely strong contact intercomponent repulsion and show that including a near-resonant attraction in the odd-wave interaction channel makes the energy of the ferromagnetic state lower than those of non-ferromagnetic states. Our find-ings can be observed in a remarkable system of ultracold 40K atoms, which is
characterized by the proximity of the s-wave (even-wave in 1D) and the p-wave (odd-wave in 1D) Feshbach resonances. In magnetic fields between the two res-onances one can simultaneously have a very strong or infinite intercomponent repulsion in the even-wave channel and a significant odd-wave attraction within one of the components. The results of this Chapter are published in Ref. [73].
In Chapter4we continue with the studies of itinerant two-component fermions in 1D, and consider the case of weak or intermediate contact repulsion strength between different spin components. It is shown that including the odd-wave attractive interaction within one of the spin components leads to the first-order transition to the so-called spin segregated phase. In this phase the gas becomes
2A metastable itinerant ferromagnetic state in a 3D degenerate gas of6Li atoms has been
re-cently realized at LENS. In this experiment different spin components were artificially separated into two domains, and the lifetime of such configuration was ∼ 10 milliseconds [85].
separated into domains where, locally, the magnetization is different from zero. The total magnetisation is fixed by the population of different spin components and, in the absence of inelastic processes, remains constant across the phase transition. The results are published in Ref. [86].
Chapter 2
Two-body relaxation of
spin-polarized fermions in reduced
dimension-alities near a p-wave Feshbach resonance
We study inelastic two-body relaxation in a spin-polarized ultracold Fermi gas in the presence of a p-wave Feshbach resonance. It is shown that in reduced dimensionalities, especially in the quasi-one-dimensional case, the enhancement of the inelastic rate constant on approach to the resonance is strongly suppressed compared to three dimensions. This may open promising paths for obtaining novel many-body states.
2.1
Introduction
Recent progress in the field of ultracold atomic quantum gases opened fascinating prospects to explore novel quantum phases in the systems of degenerate fermions with p-wave interactions, for instance two-dimensional (2D) unconventional su-perfluidity [60], non-Abelian Majorana modes [63,67], and itinerant ferromag-netism [73,74,87,88]. Even though the p-wave interactions between cold fermions are much weaker than the s-wave interactions, Feshbach resonances allow one to tune the strength and the character of the interactions. However, in the vicinity of such resonances various inelastic collisional processes play a crucial role, re-sulting in a lifetime of the order of milliseconds at common densities. These are three-body recombination and, for fermionic atoms in an excited hyperfine state, two-body relaxation [58,61,76,77,80,89].
In this Chapter we show that in the quasi-2D and quasi-one-dimensional (quasi-1D) geometries the enhancement of two-body inelastic relaxation on ap-proach to the p-wave Feshbach resonance is suppressed compared to the three-dimensional (3D) case. This effect is mostly related to a much weaker enhance-ment of the relative wavefunction near the resonance in reduced dimensionalities. We then demonstrate this for the case of 40K atoms in the |F, m
Fi =
92, −72
state. A number of experiments were dedicated to the study of atomic fermions in the presence of a p-wave Feshbach resonance in quasi-2D and quasi-1D ge-ometries [30,79,90,91]. The atom loss rate has been measured in Ref. [30], and already from this experiment one can see that in reduced dimensionalities the enhancement of the losses near the resonance is reduced compared to the 3D case.
2.2
Two-body inelastic collisions in 3D
Let us consider two colliding identical fermions in the vicinity of a p-wave Fes-hbach resonance. In the single-channel model the radial wavefunction of their p-wave relative motion at distances r Re, where Re is a characteristic radius
of interaction, has the following form [15]:
ψ3D(r) = i {j1(kr) + ikf (k)h1(kr)} , (2.1)
where k is the relative momentum, j1(kr) and h1(kr) are spherical Bessel and
Hankel functions, and f (k) is the p-wave scattering amplitude, which is related to the scattering phase shift δ(k) as f (k) = 1/ [k(cot δ(k) − i)]. The p-wave S-matrix element is given by S(k) = exp 2iδ(k). It is convenient to write the wavefunction (2.1) at r → ∞ as ψ3D = (1/2ikr) {exp (−ikr) + S(k) exp (ikr)}.
In the presence of inelastic collisions the intensity of the outgoing wave is reduced in comparison to the incoming wave by a factor of |S(k)|2 < 1, which implies that the phase shift δ(k) is a complex quantity with a positive imaginary part. For low collisional energies E = ~2k2/m we can use the effective range expansion
k3cot δ(k) = −1/w1− α1k2, where w1 is the scattering volume and α1 > 0 is the
effective range. Then, the scattering amplitude becomes
f (k) = −k
2
1/w1+ α1k2 + ik3
, (2.2)
and in order to describe inelastic collisions in the vicinity of the resonance, we add an imaginary part to the inverse of the scattering volume: 1/w1 → 1/w1+ i/w10,
where w10 > 0 [92,93]. Therefore, the S-matrix element reads as S(k) = 1/w1+ α1k
2+ i(1/w0 1− k3)
1/w1+ α1k2+ i(1/w10 + k3)
. (2.3)
For the inelastic rate constant α3D(k) = vσ3Din(k), where v = 2~k/m is the relative
velocity and σ3Din(k) = 3π [1 − |S(k)|2] /k2 is the p-wave inelastic scattering cross section [15], we obtain: α3D(k) = 48π~ mw0 1 k2 [1/w1+ α1k2]2 + [1/w10 + k3] 2, (2.4)
2.2. Two-body inelastic collisions in 3D 13 where m is the atom mass and an additional factor of 2 is included since we consider collisions of identical particles. Both w1 and w10 depend on the external
magnetic field, and w1 changes from +∞ to −∞ as one crosses the Feshbach
resonance. However, the field dependence of w01 is weak. Setting w01 to be field in-dependent we are able to accurately reproduce the results of coupled-channel cal-culations of the inelastic rate constant and the data of the JILA experiment [58]. Due to the spin-dipole interaction between colliding atoms, the resonant mag-netic field (at which the p-wave scattering volume diverges) is different for orbital angular momentum projections ml= 0 and ±1 [89]. For40K atoms in the
92, −72
state the resonance for ml = 0 occurs at 198.8 G, and for |ml| = 1 at 198.3 G.
However, apart from the difference in the position of the resonance, the scattering volume w1 for ml = 0 is the same as it is for |ml| = 1. Moreover, the effective
range α1 is also practically the same for all ml’s. In order to clearly demonstrate
the effect of suppressed enhancement of two-body losses near the resonance in reduced dimensionalities, in the main text of the paper we omit the doubling of the resonance due to the spin-dipole interaction. Then, the p-wave Feshbach resonance for 40K atoms in the
92, −72 state in 3D occurs at B ≈ 198.6 G for all orbital angular momentum projections. Consequently, the rate constant also has a single peak. We discuss the effects associated with the spin-dipole induced doubling of the resonance in the Appendix 2A.
Sufficiently far from resonance, where the dominant term in the denominator of Eq. (2.4) is 1/w1, the rate constant becomes
α3D(k) ≈ 48π~ m w2 1 w01k 2. (2.5)
At T = 0 we average the rate constant over the Fermi step momentum distri-bution, and in the off-resonant regime Eq. (2.5) yields
hα3Di0 ≈ 144π 5 w2 1 w0 1 E3D F ~ , (2.6) where E3D
F = ~2kF2/2m is the Fermi energy, kF = (6π2n3D)1/3 is the Fermi
mo-mentum for a single-component 3D gas, and n3D is the 3D density.
Near the resonance on its negative side (w1 < 0) the largest contribution to
the rate constant comes from momenta close to ˜k3D = 1/pα1|w1|. In the
near-resonant regime, where |w1|(1/w10 + ˜k3D3 ) 1 and ˜k3D kF, the rate constant
exhibits a sharp peak, which is slightly shifted with respect to the position of the resonance at zero kinetic energy (1/w1 = 0). The maximum value of the rate
constant can be estimated as (see Appendix 2B.1 for the derivation)
hα3Di0 ≈ 576π2 ~ α1m 1 1 + w10k˜3 3D ˜k3D kF !3 . (2.7)
Using Eq. (2.4) we calculate numerically hα3Di0 for 40K atoms in magnetic
fields from 195 to 205 G. The results are presented in Fig. 2.1 for EF3D = 1 µK and 4 µK (corresponding to densities n3D ≈ 3.6 × 1013 cm−3 and 2.9 × 1014 cm−3,
respectively). In order to determine w10 we fit Eq. (2.4) to the results of coupled-channel numerical calculations of the relaxation rate for a gas of 40K atoms in
92, −72 state at a fixed collisional energy of 1 µK [94], using the values of w1
and α1 that have been measured in the JILA experiment [89]. Then we obtain
w10 = 0.53×10−12cm3. For the scattering volume w1 and the effective range α1 we
take the values measured in the JILA experiment [89] for |ml| = 1 and manually
shift the position of the resonance from 198.3 to 198.6 G, so that the effect of the spin-dipole interaction is compensated. The off-resonant expression (2.6) shows perfect agreement with the numerical results, and the near-resonant expression (2.7) leads to a slight overestimate. However, Eq. (2.7) correctly captures that in the vicinity of the maximum hα3Di0 ∼ (EF3D)−3/2, in contrast to the off-resonant
case, where the rate constant behaves as hα3Di0 ∼ EF3D.
For the classical gas (T E3D
F ) averaging α3D(k) over the Boltzmann
distri-bution of atoms, in the off-resonant regime we obtain: hα3DiT ≈ 72πw2 1 w0 1 T ~ . (2.8)
On the negative side of the resonance in the near-resonant regime, where |w1|(1/w10 + ˜k33D) 1 and ˜k3D kT, with kT = pmT /~2 being the thermal
momentum, the rate constant has a sharp peak slightly shifted with respect to the resonance at zero kinetic energy (see Appendix 2B.2for details):
hα3DiT = 96π3/2 ~ α1m 1 1 + w01˜k3 3D ˜k 3D kT !3 . (2.9)
Direct numerical calculation of hα3DiT using Eq. (2.4) shows a perfect
agree-ment with both off-resonant and near-resonant expressions, as shown in Fig. 2.2
for T = 300 nK and 1 µK.
Thus, we see that the inelastic rate constant has a drastically different tem-perature (Fermi energy) dependence in the near-resonant regime compared to the off-resonant case. For deep inelastic collisions the energy dependence of the rate constant is completely determined by the wavefunction of the initial state of colliding particles. In order to gain insight into the behavior of the inelastic rate constant, we analyze the behavior of the wavefunction of the relative motion of two atoms at distances where Re r k−1. Using Eq. (2.1) and the
expres-sions for the amplitude f (k) and the phase shift δ(k) written after this equation, we have:
ψ3D(r) ≈ i
(1/w1+ α1k2) kr/3 − k/r2
1/w1+ α1k2+ ik3
2.2. Two-body inelastic collisions in 3D 15 195 197 199 201 203 205 10−18 10−16 10−14 10−12 10−10 a) E3D F = 1 µK 1/w1= 0 B (G) hα 3 D i0 (c m 3/ s) 195 197 199 201 203 205 10−18 10−16 10−14 10−12 10−10 b) E3D F = 4 µK 1/w1= 0 B (G) hα 3 D i0 (c m 3/ s)
Figure 2.1: Three-dimensional inelastic rate constant hα3Di0 for
40K atoms in the
92, −72 state at T = 0 versus magnetic field B for EF3D = 1 µK in (a) and EF3D = 4 µK in (b). Dashed red curves correspond to the off-resonant regime described by Eq. (2.6), and the blue point marks the near-resonant peak value given by Eq. (2.7). It is shifted in the direction of higher fields by 7.8 mG in (a) and by 10 mG in (b) with respect to the magnetic field at which 1/w1 = 0.
In the off-resonant regime, the terms containing 1/w1 are the leading ones
in both the numerator and denominator of Eq. (2.10), and ψoff
3D ≈ ikr/3. This
leads to αoff
3D(k) ∼ k2, in agreement with Eq. (2.5). In the off-resonant regime,
collisions with all momenta in the distribution function contribute to the inelastic rate constant. In contrast, in the near-resonant regime on the negative side of the resonance (w1 < 0) only a small fraction of relative momenta contributes to α3D.
These are momenta in a narrow interval δk ∼ ˜k3D2 /α1 around ˜k3D. Accordingly, in
the classical gas the fraction of such momenta is F3D ∼ ˜k3D2 δk/kT3 ∼ ˜k43D/(α1kT3).
In this near-resonant regime, we have |1/w1 + α1k2| ∼ ˜k33D. Then, putting the
rest of k’s equal to ˜k3D in Eq. (2.10) and taking into account that ˜k3Dr 1
at r approaching Re, we see that the relative wavefunction in the near-resonant
regime is ψres
3D ≈ 1/(˜k3Dr)2. The ratio of the near-resonant inelastic rate constant
to the off-resonant one is R3D ∼ (ψres3D/ψ off 3D)
2F
3D, where we have to put r ∼ Re in
the expressions for the relative wavefunctions. This yields R3D ≡ hαres3DiT/α off 3D T ∼ 1/(kTRe) 5. (2.11)
This is consistent with Eqs. (2.8) and (2.9), since α1 ∼ 1/Re and w1 in the
off-resonant regime is ∼ R3
e [and we may omit unity compared to w 0
1k˜33D in the
denominator of (2.9)].
From Figs. 2.1 and 2.2, we see that there is a difference in the asymmetry of the profiles between hα3Di0 and hα3DiT. This difference can be explained as
follows. The resonance takes place for particles with relative momenta close to ˜k3D = 1/pα1|w1|, which (in 40K) grows with the magnetic field. At low
EF (at T = 0) or low T (in the Boltzmann gas), the number of particles with
195 197 199 201 203 205 10−18 10−16 10−14 10−12 10−10 a) T = 300 nK 1/w1= 0 B (G) hα 3 D iT (c m 3/ s) 195 197 199 201 203 205 10−18 10−16 10−14 10−12 10−10 b) T = 1 µK 1/w1= 0 B (G) hα 3 D iT (c m 3/ s)
Figure 2.2: Three-dimensional inelastic rate constant hα3DiT for
40K atoms in the
92, −72 state versus magnetic field B for T = 300 nK in (a) and T = 1 µK in (b). Dashed red curves correspond to the off-resonant regime described by Eq. (2.8), and the blue point marks the near-resonant peak value according to Eq. (2.9). It is shifted in the direction of higher fields by 6.7 mG in (a) and by 10 mG in (b) with respect to the magnetic field at which 1/w1 = 0.
rate constant increases [89]. In a strongly degenerate gas, at a magnetic field corresponding to ˜k3D ∼ kF, the rate constant hα3Di0 rapidly decreases, since
due to the Fermi step momentum distribution there are no particles that can experience resonant scattering at higher B fields (Fig. 2.1). In contrast, in a classical gas the high-field tail of hα3DiT decreases towards the off-resonant
values more gradually due to the Boltzmann momentum distribution of colliding particles (Fig. 2.2).
2.3
Two-body inelastic collisions in 2D
We now consider inelastic collisions in the two-dimensional case and again omit the doubling of the resonance due to the spin-dipole interaction. The resulting single-peak structure of the relaxation rate constant is realized for the magnetic field perpendicular to the plane of the translational motion. In this case the relative wavefunction at short interparticle distances corresponds to the 3D mo-tion with |ml| = 1. The spin-dipole interaction only shifts the peak of the rate
constant, and the discussion of this shift is moved to the Appendix 2A. In the Appendix2Awe also present calculations taking into account the spin-dipole dou-bling of the resonance and the emerging double-peak structure of the relaxation rate for the magnetic field parallel to the plane of the translational motion.
In the quasi-2D geometry obtained by a tight harmonic confinement in the axial direction (z) with frequency ω0, at in-plane (x, y) interatomic separations
2.3. Two-body inelastic collisions in 2D 17 (~/mω0)1/2, the p-wave relative motion is described by the wavefunction
ψ2D(r) = ϕ2D(ρ)eiϑ 1 (2πl2 0)1/4 exp −z 2 4l2 0 ; ρ l0, (2.12)
where ϑ is the scattering angle, and z is the interparticle separation in the axial direction. For remaining in the ultracold limit with respect to the axial motion we will assume below that l0 Re [25]. Then, the 2D p-wave radial wavefunction
ϕ2D is ϕ2D(ρ) = i J1(qρ) − i 4f2D(q)H1(qρ) , (2.13)
with q being the 2D relative momentum, and J1(qρ) and H1(qρ) the Bessel and
Hankel functions. The p-wave quasi-2D scattering amplitude f2D(q) is given by
[95,96]
f2D(q) =
4q2
1/Ap+ Bpq2− (2q2/π) ln l0q + iq2
, (2.14)
where the 2D scattering parameters are 1/Ap = (4/3
√ 2πl2
0) [l30/w1+ α1l0/2 − C1]
and Bp = (4/3
√
2π) [l0α1− C2], with numerical constants C1 ≈ 6.5553 × 10−2 and
C2 ≈ 1.4641 × 10−1. The amplitude f2D is related to the S-matrix element S2D
as f2D(q) = 2i(S2D(q) − 1). The 2D (confinement-influenced) resonance occurs at
1/Ap = 0 and is thus shifted with respect to the 3D resonance (1/w1 = 0). Like
in the 3D case, in the presence of inelastic processes we have to replace 1/w1 by
1/w1+ i/w10, which yields
S2D(q) = 1 Ap + Bp− 2 ln l0q π q2+ i 1 A0 p − q2 1 Ap + Bp − 2 ln l0q π q2+ i 1 A0 p + q2 , (2.15)
with A0p = 3√2πw10/4l0 > 0. Then, writing the wavefunction (2.13) at ρ → ∞ as
ϕ2D ≈ (1/
√
2πiqρ) {exp (−iqρ) + iS2D(q) exp (iqρ)} we see that the intensity of
the outgoing wave is reduced by a factor of |S2D(q)|2 < 1 compared to the
incom-ing wave. The 2D inelastic cross section is defined as σ2Din = (2/q) (1 − |S2D(q)|2),
and for identical particles one has an additional factor of 2. Then, for the inelastic rate constant, α2D(q) = (2~q/m)σ2Din, we obtain:
α2D(q) = 32~ mA0 p q2 1 Ap + Bp− 2 ln l0q π q2 2 + 1 A0 p + q2 2. (2.16)
Sufficiently far from the resonance, where the dominant term in the denominator of Eq. (2.16) is 1/Ap, the rate constant becomes
α2D(q) ≈ 32~ m A2 p A0 p q2 ≈ 24 √ 2π~ ml0 w12 w01q 2. (2.17)
At T = 0, averaging the off-resonant rate constant (2.17) over the Fermi step momentum distribution we obtain
hα2Di0 ≈ 12√2πw12 l0w01 EF2D ~ , (2.18) where E2D
F = ~2q2F/2m is the 2D Fermi energy, qF =
√
4πn2D is the Fermi
mo-mentum for a single-component 2D gas, and n2D is the 2D density.
Near the 2D resonance on its negative side (Ap < 0) the largest contribution
to the rate constant comes from momenta close to ˜q2D = 1/pBp|Ap|. Then, in
the regime, where (A0p)−1/2 ˜q2D qF, the rate constant has a sharp peak. The
maximum value of the 2D rate constant at T = 0 can then be estimated as hα2Di0 ≈ 128π~ A0 pBpm 1 q2 F ≈ 128π~ α1m w01 1 q2 F , (2.19)
where we took into account that A0pBp = w10(α1l0 − C2)/l0 ≈ w01α1 for typical
confinement frequencies ω0 from 50 to 150 kHz. Therefore, the tight harmonic
confinement has almost no influence on the maximum value of hα2Di0.
The results of direct numerical calculation of hα2Di0 using Eq. (2.16) for 40K
atoms are presented in Fig. 2.3 for the confining frequency ω0 = 120 kHz and
Fermi energies E2D
F = 1 µK and 4 µK (corresponding to densities n2D ≈ 1.3 × 109
cm−2 and 5.2 × 109 cm−2, respectively). The off-resonant expression (2.18) shows
perfect agreement with the numerical results, while the near-resonant expression (2.19) leads to a slight overestimate. However, Eq. (2.19) captures that in the vicinity of the maximum hα2Di0 ∼ 1/EF2D, in contrast to the off-resonant case,
where hα2Di0 ∼ E 2D F .
At T EF2D, we average Eq. (2.16) over the Boltzmann distribution of atoms. Then, the off-resonant expression for the rate constant follows from Eq. (2.17) and reads as hα2DiT ≈ 24√2πw2 1 l0w10 T ~ . (2.20)
On the negative side of the 2D resonance in the near-resonant regime, where (A0p)−1/2 ˜q2D qT, with qT =pmT /~2being the thermal momentum, the rate
constant has a sharp peak slightly shifted from the position of the 2D resonance at zero kinetic energy. The maximum value of the rate constant is given by
hα2DiT ≈ 32π~ A0 pBpm 1 q2 T ≈ 32π~ α1m w01 1 q2 T . (2.21)
Like in the zero temperature case, we see that the maximum value of hα2DiT is
practically independent of the confinement frequency.
Direct numerical calculation of hα2DiT from Eq. (2.16) shows perfect
2.4. Two-body inelastic collisions in 1D 19 for the confining frequency ω0 = 120 kHz and temperatures T = 300 nK and 1
µK.
In order to qualitatively understand the temperature (Fermi energy) depen-dence of the inelastic rate constant, we analyze the structure of the initial state wavefunction, which for deep inelastic processes fully determines the energy de-pendence of α2D. Inelastic collisions occur at interparticle distances r . Re l0,
where the relative motion of colliding atoms has a three-dimensional character and ψ2D is different from ψ3D only by a normalization coefficient. Assuming the
inequality kl0 1, at distances r exceeding Re sufficiently far from the 3D
res-onance from Eq. (2.10) we have ψ3D(r) ∝ {r − 3w1/r2}. Then, according to
Ref. [95], the 2D wavefunction can be written as ψ2D(r) =
if2D(q)(2πl02)1/4
6πw1q
r − 3w1/r2 . (2.22)
Far from the 2D resonance (1/Ap = 0) the 2D scattering amplitude is f2Doff ≈
3√2πw1q2/l0, which leads to ψ2Doff ∼ (q/
√
l0) {r − 3w1/r2} and αoff2D ∼ q2/l0, in
agreement with Eq. (2.17). In the near-resonant regime on the negative side of the 2D resonance (Ap < 0) the main contribution to α2D is provided by relative
momenta in a narrow interval δq ∼ ˜q2D/Bp around ˜q2D. In the classical gas
the fraction of such momenta is F2D ∼ ˜q2Dδq/qT2 ∼ ˜q2D2 /(Bpq2T). In this
near-resonant regime we have |1/Ap + Bpq2| ∼ ˜q22D. Then, we may put the rest of
q’s equal to ˜q2D in Eq. (2.22) and use f2Dres(q) ≈ −4i (omitting the logarithmic
term in the denominator of f2D(q)). Thus, the 2D wavefunction becomes ψres2D ∼
(√l0/˜q2D) {r/w1− 3/r2}. The ratio of the near-resonant inelastic rate constant
to the off-resonant one is R2D ∼ (ψres2D/ψ2Doff)2F2D, where we have to put r ∼ Re
in the expressions for the relative wavefunctions and take into account that in the off-resonant regime w1 ∼ R3e, whereas in the near-resonant regime it is much
larger. This yields 1
R2D ≡ hαres2DiT /α off 2D T ∼ l0 Re 1 (qTRe)4 , (2.23)
which is consistent with Eqs. (2.20) and (2.21). As one can see from Eqs. (2.11) and (2.23), the ratio R2D/R3D∼ l0kT 1. Thus, in 2D the enhancement of the
inelastic rate constant near the resonance is suppressed compared to 3D.
2.4
Two-body inelastic collisions in 1D
We eventually turn to inelastic collisions in the one-dimensional case. Omitting the doubling of the resonance, induced by the spin-dipole interaction, we have a
1The quantity R
2Dremains the same if in the near-resonant regime (at Ap< 0) we are fairly
195 197 199 201 203 205 10−12 10−10 10−8 10−6 a) E2D F = 1 µK 1/w1= 0 1/Ap= 0 B (G) hα 2 D i0 (c m 2/ s) 195 197 199 201 203 205 10−12 10−10 10−8 10−6 b) E2D F = 4 µK 1/w1= 0 1/Ap= 0 B (G) hα 2 D i0 (c m 2/ s)
Figure 2.3: Two-dimensional inelastic rate constant hα2Di0 for
40K atoms in the
92, −72 state at T = 0 versus magnetic field B for EF2D = 1 µK in (a) and EF2D = 4 µK in (b). Dashed red curves correspond to the off-resonant regime described by Eq. (2.18), and the blue point marks the near-resonant peak value according to Eq. (2.19). It is shifted in the direction of higher fields by 1 mG in (a) and by 1.9 mG in (b) with respect to the magnetic field at which 1/Ap = 0. The confining
frequency is ω0 = 120 kHz.
single-peak structure of the relaxation rate constant. This structure is realized for the magnetic field parallel to the line of the translational motion or for the field perpendicular to this line [30,96]. The shift of the peak due to the spin-dipole interaction is discussed in the Appendix 2A. We also present there the calculations taking into account the spin-dipole doubling of the resonance and the resulting double-peak structure of the relaxation rate for the magnetic field forming the angle of 45◦ with the line of the translational motion.
In the quasi-1D geometry obtained by a tight harmonic confinement in two directions (x, y) with frequency ω0, the wavefunction of the relative motion in the
odd-wave channel (analog of p-wave in 2D and 3D) is ψ1D(r) = χ1D(z) 1 √ 2πl0 exp −ρ 2 4l2 0 , (2.24)
where z is the longitudinal interparticle separation, ρ = px2 + y2 is the
trans-verse separation, and l0 =
p
~/(mω0) is the transverse extension of the
wave-function. The longitudinal motion with the 1D relative momentum q at distances |z| l0 Re is described by the wavefunction
χ1D(z) = i sin qz + sgn(z)f1D(q)eiq|z|, (2.25)
with the odd-wave scattering amplitude f1D(q) given by [95–97]
f1D(q) =
−iq 1/lp + ξpq2 + iq
2.4. Two-body inelastic collisions in 1D 21 195 197 199 201 203 205 10−12 10−10 10−8 10−6 a) T = 300 nK 1/w1= 0 1/Ap= 0 B (G) hα 2 D iT (c m 2/ s) 195 197 199 201 203 205 10−12 10−10 10−8 10−6 b) T = 1 µK 1/w1= 0 1/Ap= 0 B (G) hα 2 D iT (c m 2/ s)
Figure 2.4: Two-dimensional inelastic rate constant hα2DiT for
40K atoms in the
92, −72 state versus magnetic field B for T = 300 nK in (a) and T = 1 µK in (b). Dashed red curves correspond to the off-resonant regime described by Eq. (2.20), and the blue point marks the near-resonant peak value according to Eq. (2.21). It is shifted in the direction of higher fields by 1.5 mG in a) and by 2.7 mG in b) with respect to the magnetic field at which 1/Ap = 0. The confining frequency is
ω0 = 120 kHz.
where lp = 3l0l30/w1+ α1l0+ 3
√
2|ζ(−1/2)|−1 and ξp = α1l20/3 are the 1D
scat-tering parameters, and ζ(−1/2) ≈ −0.208 is the Riemann zeta-function. The amplitude f1D(k) is related to the 1D odd-wave S-matrix element S1D(q) as
f1D(q) = (S1D(q) − 1)/2. Like in higher dimensions, in the presence of
inelas-tic processes we should replace 1/w1 with 1/w1+ i/w10, which gives the following
expression for the S-matrix element: S1D(q) = 1/lp+ ξpq2+ i(1/l0p− q) 1/lp+ ξpq2+ i(1/lp0 + q) , (2.27) where 1/l0p = l2 0/3w 0
1 > 0. Then, writing the wavefunction (2.25) at |z| → ∞
as χ1D = sgn(z)(1/2) {− exp (−iq|z|) + S1D(q) exp (iq|z|)}, we see that the
in-tensity of the outgoing wave is reduced by a factor of |S1D(q)|2 < 1 compared
to the incoming wave. The inelastic cross-section in 1D is defined as σin
1D =
(1 − |S1D(q)|2) /2, and for identical particles there is an additional factor of 2.
Then, for the inelastic rate constant, α1D(q) = (2~q/m)σ1Din, we obtain:
α1D(q) = 8~ ml0 p q2 [1/lp+ ξpq2]2+1/l0p+ q 2. (2.28)
Sufficiently far from the 1D resonance (1/lp = 0), where the dominant term in
the denominator of Eq. (2.28) is 1/lp, the rate constant becomes
α1D(q) ≈ 8~ m l2 p l0 p q2 ≈ 24~ ml2 0 w12 w01q 2. (2.29)
At T = 0 the off-resonant rate constant averaged over the Fermi step momen-tum distribution reads
hα1Di0 ≈ 8w21 l2 0w01 EF1D ~ , (2.30) where E1D
F = ~2qF2/2m is the 1D Fermi energy, qF = πn1Dis the Fermi momentum
for a single-component 1D gas and n1D is the 1D density.
In the vicinity of the 1D resonance on its negative side (lp < 0) the largest
contribution to the rate constant comes from momenta ∼ ˜q1D = 1/pξp|lp|. Then,
in the near-resonant regime, where 1/l0p ˜q1D qF, the rate constant shows a
narrow peak with the value hα1Di0 ≈ 8π~ l0 pξpm 1 qF = 8π~ α1m w10 1 qF . (2.31)
As in the 2D case, the maximum value of hα1Di0 is almost independent of the
confinement frequency.
The results of direct numerical calculation of hα1Di0 using Eq. (2.28) for 40K
atoms are presented in Fig. 2.5 for the confining frequency ω0 = 120 kHz and
Fermi energies EF1D = 1 and 4 µK (corresponds to densities n1D ≈ 4.1 × 104
cm−1 and 8.2 × 104 cm−1, respectively). The off-resonant expression (2.30) and
near-resonant expression (2.31) agree with numerical results, although Eq. (2.31) leads to a small overestimate of hα1Di0.
At T E1D
F , averaging the rate constant over the Boltzmann distribution of
atoms we obtain the following off-resonant expression: hα1DiT ≈ 12w2 1 l2 0w10 T ~ . (2.32)
On the negative side of the 1D resonance in the near-resonant regime, where 1/l0p ˜q1D qT, with qT = pmT /~2 being the thermal momentum, the rate
constant displays a sharp peak, slightly shifted with respect to the position of the 1D resonance at zero kinetic energy (1/lp = 0). The maximum value of the rate
constant is hα1DiT ≈ 4√π~ l0 pξpm 1 qT = 4 √ π~ α1m w01 1 qT . (2.33)
Direct numerical calculation of hα1DiT on the basis of Eq. (2.28) shows good
agreement with both off-resonant and near-resonant expressions, as shown in Fig. 2.6 for the confining frequency ω0 = 120 kHz and temperatures T = 300 nK
and 1 µK. Note that in the vicinity of the peak value the rate constant is pro-portional to 1/√T , while in the off-resonant regime it has a linear dependence on T .
We see that in 1D, as well as in higher dimensions, the temperature (Fermi energy) dependence of the inelastic rate constant in the near-resonant regime is
2.5. Two-body inelastic rate near the resonance in 2D and 1D: Conclusions 23 very different from that in the off-resonant case. Similarly to the 2D case, at dis-tances where the relaxation occurs (Re . r l0), the 1D relative wavefunction
ψ1D has a 3D character and differs from the 3D wavefunction only by a
normal-ization coefficient. Sufficiently far from the 3D resonance, assuming that r is still larger than Re, the 1D wavefunction can be written as [95]
ψ1D(r) = − f1D(q) √ 2πl0 6πw1 r − 3w1/r2 . (2.34)
Far from the 1D resonance (1/lp = 0) the 1D scattering amplitude is f1Doff ≈
−3iw1q/l20, which yields ψ1Doff ≈ (iq/
√
2πl0) {r − 3w1/r2} and αoff1D ∼ q2/l20, in
agreement with Eq. (2.29). In the near-resonant regime on the negative side of the confinement-influenced resonance (lp < 0) the situation changes. Here the main
contribution to α1D is provided only by relative momenta in a narrow interval
δq ∼ 1/ξp around ˜q1D, and in the classical gas the fraction of such momenta is
F1D ∼ δq/qT ∼ 1/(ξpqT). In this near-resonant regime we have |1/lp+ ξpq2| ∼ ˜q1D
and f1Dres(q) ≈ −1. Then, the 1D wavefunction becomes ψres1D ∼ l0{r/w1− 3/r2}.
The ratio of the near-resonant inelastic rate constant to the off-resonant one is R1D ∼ (ψres1D/ψ1Doff)2F1D, where we have to put r ∼ Re in the expressions for
the relative wavefunctions. Taking into account that in the off-resonant regime w1 ∼ R3e and in the near-resonant regime it is much larger, we obtain:
R1D ∼ l0 Re 2 1 (qTRe)3 , (2.35)
which is consistent with Eqs. (2.32) and (2.33). From Eqs. (2.11), (2.23), and (2.35) we find that R1D/R3D ∼ (kTl0)2 ∼ (kTl0)R2D/R3D. Thus, in the 1D case
the enhancement of the inelastic rate near the resonance is even weaker than in 2D and certainly much weaker than in 3D.
2.5
Two-body inelastic rate near the resonance
in 2D and 1D: Conclusions
In this section, we analyze how the inelastic rate is enhanced on approach to the resonance in reduced dimensionalities and conclude.
In order to demonstrate the suppressed enhancement of the inelastic rate constant near the resonance in reduced dimensionalities, we calculate the ratios of hα2DiT and hα1DiT to their off-resonant values, and compare them with the ratio
of hα3DiT to its value far from the resonance. In Fig. 2.7, we plot the ratio of the
2D rate constant to its off-resonant value, hα2DiT /α off 2D
T, versus magnetic field
B for 40K atoms in the 92, −72 state at T = 300 nK and 1 µK. The off-resonant value is taken at a fixed field value of 195 G. Figure 2.8 shows the corresponding quantity in 1D. It is evident that the rate constant in 3D experiences a much
195 197 199 201 203 205 10−8 10−6 10−4 10−2 a) E1D F = 1 µK 1/w1= 0 1/lp= 0 B (G) hα 1 D i0 (c m / s) 195 197 199 201 203 205 10−8 10−6 10−4 10−2 b) E1D F = 4 µK 1/w1= 0 1/lp= 0 B (G) hα 1 D i0 (c m / s)
Figure 2.5: One-dimensional inelastic rate constant hα1Di0 for 40K atoms in the
92, −72
state at T = 0 versus magnetic field B for E1D
F = 1 µK in (a) and
EF1D = 4 µK in (b). Dashed red curves correspond to the off-resonant regime described by Eq. (2.30), and the blue point marks the near-resonant peak value according to Eq. (2.31). The corresponding magnetic field practically coincides with the magnetic field at which 1/lp = 0. The confining frequency is ω0 = 120
kHz. 195 197 199 201 203 205 10−8 10−6 10−4 10−2 a) T = 300 nK 1/w1= 0 1/lp= 0 B (G) hα 1 D iT (c m / s) 195 197 199 201 203 205 10−8 10−6 10−4 10−2 b) T = 1 µK 1/w1= 0 1/lp= 0 B (G) hα 1 D iT (c m / s)
Figure 2.6: One-dimensional inelastic rate constant hα1DiT for
40K atoms in the
92, −72 state versus magnetic field B for T = 300 nK in (a) and T = 1 µK in (b). Dashed red curves correspond to the off-resonant regime described by Eq. (2.32), and the blue point marks the near-resonant peak value according to Eq. (2.33). The corresponding magnetic field practically coincides with the magnetic field at which 1/lp = 0. The confining frequency is ω0 = 120 kHz.
2.5. Two-body inelastic rate near the resonance in 2D and 1D: Conclusions 25 195 197 199 201 203 205 1 102 104 106 a) T = 300 nK B (G) hα2DiT/αoff2D T hα3DiT/αoff 3D T 195 197 199 201 203 205 1 102 104 106 b) T = 1 µK B (G) hα2DiT/αoff2D T hα3DiT/αoff 3D T
Figure 2.7: Inelastic rate constants in 2D (red) and in 3D (black) divided by their off-resonant values at a fixed filed of 195 G for 40K atoms in the 92, −72
state versus magnetic field B for T = 300 nK in (a) and T = 1 µK in (b). The confining frequency is ω0 = 120 kHz.
stronger enhancement near the resonance than the rate constants in 2D and 1D. In other words, this means that the enhancement of the two-body inelastic rate near the resonance is suppressed in reduced dimensionalities. The effect is especially pronounced in 1D, which is consistent with our discussion in the previous section. This effect is mostly related to a weaker enhancement of the relative wave-function on approach to the resonance in 2D and 1D than in 3D. Indeed, using expressions for ψ3D and ψ2D in the near- and off-resonant regimes [written
af-ter Eqs. (2.10) and (2.22)] we see that the ratio ψ2Dres/ψ2Doff2/ ψres3D/ψ3Doff2 ∼ (˜k3Dl0)2 1 slightly away from the 3D resonance. Similarly, in 1D we have
ψres 1D/ψoff1D 2 / ψres 3D/ψ3Doff 2
∼ (˜k3Dl0)4, which is even smaller than in the 2D case.
Our results may draw promising paths to obtain novel many-body states in 2D and 1D, such as low-density p-wave (odd-wave) superfluids of spinless fermions. It is quite likely that they can be extended to the case of three-body recombination2, which will be the topic of our future research.
2In 1D one has an extra suppression of three-body recombination of identical fermions in
the off-resonant regime by a factor of (E1D
F /E∗) at T = 0 and (T /E∗) in the classical gas,
where E∗∼ 1 mK is a characteristic molecular energy. For typical densities/temperatures this
195 197 199 201 203 205 1 102 104 106 a) T = 300 nK B (G) hα1DiT/αoff1D T hα3DiT/αoff 3D T 195 197 199 201 203 205 1 102 104 106 b) T = 1 µK B (G) hα1DiT/αoff1D T hα3DiT/αoff 3D T
Figure 2.8: Inelastic rate constants in 1D (red) and in 3D (black) divided by their off-resonant values at a fixed filed of 195 G for 40K atoms in the 92, −72
state versus magnetic field B for T = 300 nK in (a) and T = 1 µK in (b). The confining frequency is ω0 = 120 kHz.
Appendices to Chapter 2
2A
Spin-relaxation taking into account the
dou-bling of the p-wave Feshbach resonance
Throughout the Chapter we assumed that the 3D p-wave Feshbach resonance occurs at the same magnetic field for all orbital angular momentum projections ml. However, in reality due to the spin-dipole interaction between colliding atoms
the binding energy of the two-body bound state, the coupling to which leads to the resonance in the scattering amplitude, depends on |ml|. As a consequence,
the resonant magnetic field (at which the scattering volume diverges) is different for ml = 0 and ±1. Then the 3D rate constant exhibits a doublet structure:
there are two distinct peaks, corresponding to ml = 0 and |ml| = 1. In reduced
dimensionalities, one can also get a double-peak structure of the relaxation rate constant, although the situation is more peculiar as the orientation of the external magnetic field plays a crucial role [30,96,97]. In this appendix we analyze these effects in more detail. We first derive an expression for the ml-dependent inelastic
rate constant in 3D and show that it has the expected doublet structure. We then discuss how our results for the rate constants in 2D and 1D are affected by the ml-dependence of the p-wave Feshbach resonance.
In 3D, if the scattering volume and the effective range depend on the value of ml, then the p-wave scattering phase shift also becomes ml-dependent. In the
low energy limit we have k3cot δml = −1/w1,ml − α1,ml, and the p-wave part of
the total scattering amplitude can be written as [15] f (k, ˆk0) = 4π X
ml=0,±1
fml(k)Y1,m∗ l(ˆk)Y1,ml(ˆk 0
), (2.36)
where ˆk and ˆk0 are unit vectors in the directions of incident and outgoing relative momenta, Y1,ml is the spherical harmonic, and fml(k) = (Sml(k) − 1)/2ik is
the p-wave partial scattering amplitude, with Sml(k) = exp {2iδml} being the
p-wave S-matrix element. In order to describe inelastic collisions we make the replacement 1/w1,ml → 1/w1,ml+ 1/w
0
1, where w01 > 0 is the same for all ml’s,
since it can be assumed to be field-independent. From Eq. (2.36) we see that the p-wave part of the scattering amplitude depends on both the incoming and outgoing momentum directions and not only on the angle between k and k0 (as it would have been in the case where the scattering phase shift is independent of ml). Then, integrating over the scattering angles Ωkˆ0, for the inelastic scattering
cross section we have:
σin(k) = 4π k2 X ml=0,±1 1 − |Sml(k)| 2 Y1,ml(ˆk) 2 . (2.37)
For identical particles the above expression should be multiplied by an extra fac-tor of 2. Taking into account that |Y1,0(ˆk)|2 = (3/4π) cos θˆk and |Y1,±1(ˆk)|2 =
(3/8π) sin2θkˆ, where θˆk is the angle between the unit vector ˆk and the
quan-tization axis, we average expression (2.37) over the incident angles Ωkˆ. Then,
the inelastic cross section can be written as ¯σin(k) = P
ml=0,±1σ¯ in ml(k), where ¯ σin ml(k) = (2π/k 2)[1 − |S ml(k)|
2]. Accordingly, for the inelastic rate constant
in 3D, α3D(k) = (2~k/m)¯σin(k), we obtain: α3D(k) = 16π~ mw10
X
ml=0,±1 k2 1 w1,ml + α1,mlk 2 2 + 1 w10 + k 3 2. (2.38)One immediately sees that if w1,ml and α1,ml are the same for all ml’s, the above
expression reduces to Eq. (2.4) of the main text, which has only one peak. However, as we already mentioned before, if one takes the spin-dipole doubling of the resonance into account, then this peak splits in two smaller peaks. The one which corresponds to |ml| = 1 is by a factor of 3/2 smaller, whereas the second
peak, corresponding to ml = 0, is smaller by a factor of 3. For 40K atoms in the
9
2, − 7
2 state the 3D resonance for ml = 0 occurs at 198.8 G, and for |ml| = 1 at
198.3 G [89]. We present this in Fig. 2.9, which displays the thermally averaged inelastic rate constant hα3DiT in a 3D classical gas at 1 µK.
In reduced dimensionalities, the two-body inelastic relaxation occurs at inter-particle distances that are much smaller than the extension of the relative wave-function in the tightly confined direction(s) [25]. Therefore, the relative motion acquires a 3D character and the related wavefunction represents a superposition of ml = 0 and ±1 contributions. This means that in principle the rate constant
in 2D and in 1D can also have the double-peak structure. However, the number of peaks and their positions depend on the relative orientation of the external magnetic field [30,96]. In the following, we first consider the quasi-2D case with the magnetic field perpendicular to the plane of the translational motion. In quasi-1D we assume that the field is perpendicular to the line of the translational motion. In both cases, one has a single-peak structure of the relaxation rate, since
2A. Spin-relaxation taking into account the doubling of the resonance 31
195
197
199
201
203
205
10
−1810
−1610
−1410
−1210
−10T = 1 µK
m
l= 0
|m
l| = 1
B (G)
hα
3Di
T(cm
3/s)
Figure 2.9: Three-dimensional inelastic rate constant hα3DiT for40K atoms in the
92, −72 state versus magnetic field B for T = 1 µK. For comparison, the black line shows the 3D inelastic rate constant from Fig. 2.2 (b) of the main text. the relative wavefunction of two atoms at short separations corresponds only to the 3D motion with |ml| = 1. The expressions for quasi-2D and quasi-1D
scat-tering amplitudes for an arbitrary orientation of the magnetic field were derived in Ref. [96]. In the case of magnetic field perpendicular to the plane (line) of the translational motion in 2D (1D), these expressions are reduced to Eqs. (2.14) and (2.26) for f2D and f1D, correspondingly, where one should use the values of w1
and α1 for |ml| = 1. Thus, the spin-dipole interaction simply shifts the position
of the peak of the inelastic rate by approximately 0.3 G, as can be seen from Fig. 2.10. For the 1D case with the magnetic field perpendicular to the line of the translational motion, particle losses in a spin-polarized gas of40K atoms were measured in Ref. [30]. The peak position observed in the experiment coincides with that in Fig. 2.10 (b).
To illustrate the effect of the spin-dipole doubling of the resonance in reduced dimensionalities, we now turn to the case where the magnetic field is parallel to the plane of the translational motion in 2D. Using the result of Ref. [96] for the 2D scattering amplitude depending on the orientation of the B-field, the inelastic scattering cross section can be written as
σ2Din(q) = 4 q nh 1 − S02D 2i cos2φˆq+ h 1 − S12D 2i sin2φˆq o , (2.39) where q is the incident relative momentum, φqˆ is the angle between q and the
external magnetic field, and S2D
|ml| = exp
n 2iδ2D
|ml|
o
195 197 199 201 203 205 10−12 10−10 10−8 10−6 a) |ml| = 1 B (G) hα 2D iT (cm 2/s) 195 197 199 201 203 205 10−8 10−6 10−4 10−2 b) |ml| = 1 B (G) hα 1D iT (cm/s)
Figure 2.10: Two-dimensional inelastic rate constant hα2DiT in (a) and
one-dimensional inelastic rate constant hα1DiT in (b) for 40K atoms in the
92, −72
state versus magnetic field B for T = 1 µK and confining frequency ω0 = 120
kHz. The magnetic field is perpendicular to the translational motion in both (a) and (b). For comparison, black lines show the corresponding inelastic rate constants from Figs. 2.4 (b) and 2.6 (b) of the main text. One can see that the spin-dipole interaction shifts the peaks by approximately 0.3 G.
elements, with δ|m2D
l| being the 2D p-wave |ml|-dependent scattering phase shifts.
Adopting our notations from the main text, in the low energy limit we can write q2cot δ2D
|ml| = −1/Ap,|ml|+Bp,|ml|− (2/π) ln l0q q
2. The quantities A
p,|ml| and
Bp,|ml| are given by the same expressions as Ap and Bp written after Eq. (2.14) of
the main text, except that one has to use the |ml|-dependent scattering volume
w1,|ml| and effective range α1,|ml|. Then, averaging expression (2.39) over the
angles φˆq and replacing 1/Ap,|ml| with 1/Ap,|ml|+ i/A 0
p, for the 2D inelastic rate
constant we obtain: α2D(q) = 16~q 2 mA0 p
X
|ml|=0,1 ( "1
A
0 p + q2 #2 + "1
A
p,|ml| + Bp,|ml|−2 ln l
0q
π
! q2 #2 −1 . (2.40)Similarly to the 3D case, the above expression reproduces Eq. (2.16) of the main text if there is no |ml|-dependence of the scattering parameters. In Fig.2.11 (a)
we plot the thermally averaged inelastic rate constant hα2DiT for the quasi-2D
classical gas at T = 1 µK as a function of magnetic field B. One can clearly see the emerging double-peak structure of the inelastic rate constant for magnetic field oriented parallel to the plane of translational motion. Both peaks are by a factor of 2 smaller than the single peak of the rate constant in the |ml|-independent case.
Positions of the peaks coincide with those found in the experiment measuring particle losses [30].
2A. Spin-relaxation taking into account the doubling of the resonance 33 In order to have the doubling of the resonance in the quasi-1D geometry, the magnetic field has to be neither parallel, nor perpendicular to the line of the translational motion [96]. Let us consider the situation where the B-field forms an angle β with the quasi-1D tube. Then, the 1D scattering amplitude can be written as [96] f1D(q) = −iq [1/L + iq]
−1 with 1/L = F0[F1+ G] cos 2β + F 1[F0+ G] sin2β F0sin2β + F1cos2β + G , (2.41)
where we have the functions F|ml| = 1/lp,|ml|+ ξp,|ml|q
2 and G = D
1/l0+ D2l0q2,
with numerical constants D1 ≈ −0.4648 and D2 ≈ 0.8316. Here, q is the 1D
relative momentum, and the 1D scattering parameters lp,|ml| and ξp,|ml| are given
by the same expressions as lp and ξp in the main text. The only difference is
that the scattering volume w1,|ml| and effective range α1,|ml| now depend on |ml|.
Then, replacing 1/lp,|ml|with 1/lp,|ml|+ i/l0p and repeating the steps from Sec. 2.4,
for the 1D inelastic rate constant we obtain: α1D(q) = 8~ m Im {1/L} q2 (Re {1/L})2+ (Im {1/L} + q)2, (2.42) with Re 1 L =1 K ( F1+ G sin2β sin2β Φ2+ [2F1(F1+ G) + 1/l02p + G2− F2 1 cos 2β Φ + F 1(F1 + G) 2 + 1/lp02 ) , (2.43) Im 1 L = sin 2β Φ2+ 2 (F 1+ G) sin2β Φ + (F1+ G) 2 + 1/l02p l0 pK , (2.44) where we introduced the quantities Φ = F0− F1 and
K = Φ2 sin4β + 2 (F
1+ G) Φ sin2β + (F1+ G)2+ 1/lp02. (2.45)
One can easily verify that if there is no |ml| dependence of w1 and ξp, then Φ = 0.
Thus, we have Re {1/L} = 1/lp+ξpq2and Im {1/L} = 1/lp0 and recover expression
(2.28) of the main text. Taking β = 45◦, we plot the thermally averaged inelastic rate constant hα1DiT for the quasi-1D classical gas at T = 1 µK as a function
of magnetic field B in Fig. 2.11 (b). We again see that the rate constant has a characteristic doublet structure. However, unlike in 3D and 2D, both peaks of hα1DiT are now slightly higher than the single peak of the rate constant in the
ml-independent case. The origin of this enhancement becomes more clear if we
195 197 199 201 203 205 10−12 10−10 10−8 10−6 a) ml= 0 |ml| = 1 B (G) hα 2D iT (cm 2/s) 195 197 199 201 203 205 10−8 10−6 10−4 10−2 b) ml= 0 |ml| = 1 B (G) hα 1D iT (cm/s)
Figure 2.11: Two-dimensional inelastic rate constant hα2DiT in (a) and
one-dimensional inelastic rate constant hα1DiT in (b) for 40K atoms in the
92, −72
state versus magnetic field B for T = 1 µK and confining frequency ω0 = 120
kHz. The magnetic field is parallel to the plane of the translational motion in (a) and forms an angle of 45◦ with the line of the translational motion in (b). For comparison, black lines show the corresponding inelastic rate constants from Figs. 2.4 (b) and2.6 (b) of the main text.
1/l02p and G in Eqs. (2.43) and (2.44). Then, the rate constant α1D(q) can be
written as α1D= 8~q 2 ml0 p cos2β F2 0 + cos2β +F0 F1 sin2β 2 q2 + sin 2β F2 1 + sin2β + F1 F0 cos2β 2 q2 . (2.46) The term 1/lp02is negligibly small, and by neglecting G we slightly shift positions of the peaks, since this term essentially renormalizes F|ml|[see Eq. (2.41)]. However,
the behavior of the rate constant becomes much more transparent. Indeed, in Eq. (2.46) the first term corresponds to the peak for ml= 0 and the second term to the
peak for |ml| = 1. Then, close to the resonance for ml = 0 on its negative side we
have F0 ≈ 0, and the second term in Eq. (2.46) vanishes. The first term behaves
as the rate constant given by Eq. (2.28) in the main text (where we can omit the term 1/l0p in the denominator), with an extra factor of 1/ cos2β. Therefore,
for β = 45◦ the peak value corresponding to ml= 0 becomes approximately by a
factor of 2 larger than the single peak in the ml-independent case. The resonance
