CHAPTER TWO
Ultracold Hybrid Atom–Ion Systems
R. Cot^ e1 University of Connecticut, Storrs, CT, United States 1Corresponding author: e-mail address: [email protected]
Contents 1. Introduction 68 2. Atom–Ion Interactions 70 2.1 Homonuclear Case 70 2.2 Heteronuclear Case 73 3. Scattering Processes 75 3.1 Resonant Processes 75 3.2 Nonresonant Processes 83 4. Transport Properties 84 4.1 Diffusion 84 4.2 Mobility 86 4.3 Hole Mobility 88 5. Tuning Interactions: Hyperfine and Zeeman Interactions 91 5.1 Multichannel Scattering 91 5.2 Identical Nuclei 94 6. Isotopic Effects 99 6.1 Theory 100 6.2 Results for a Few Examples 102 6.3 Tuning Scattering with Magnetic Fields 103 7. Charges in a BEC 106 7.1 Ion in a BEC 106 7.2 Rydberg Electron in a BEC 111 8. Conclusions 119 Acknowledgments 119 References 119
Abstract Ultracold atomic samples doped with charged particles is a nascent field marrying two usually well-separated fields, namely trapped ions and ultracold atoms. Since the orig- inal proposals over 15 years ago, the initially slow pace has given way to rapid progress. In this chapter, we review some of the concepts relevant to this hybrid field, ranging from resonant charge transfer to the effect of isotope shifts, and the role of hyperfine
Advances in Atomic, Molecular, and Optical Physics, Volume 65 # 2016 Elsevier Inc. 67 ISSN 1049-250X All rights reserved. http://dx.doi.org/10.1016/bs.aamop.2016.04.004 68 R. Cot^ e
and Zeeman interactions in obtaining Feshbach resonances allowing control of the scattering processes taking place. The next frontier, charges in a Bose–Einstein conden- sate, is also introduced and discussed.
1. INTRODUCTION Many recent developments in Atomic, Molecular, and Optical Phys- ics have been triggered by the ability to reach ultralow temperatures, which allowed probing degenerate atomic gases (Dalfovo et al., 1999; Giorgini et al., 2008). In addition, the control of interactions in ultracold gases, eg, via Feshbach resonances (Chin et al., 2010; Kohler€ et al., 2006), made it possible to investigate a variety of many-body phenomena, such as molec- ular BEC (Greiner et al., 2003; Jochim et al., 2003; Xu et al., 2003) or BEC- BCS crossover (Altmeyer et al., 2007a,b; Du et al., 2009; Greiner et al., 2005; Kosˇtrun and Cot^ e, 2006; Partridge et al., 2005; Shin et al., 2008; Zhang and Leggett, 2009). This level of control paved the way to studies of few-body systems very sensitive to the details of the interactions, such as exotic three-body Efimov states (Kraemer et al., 2006), while mixtures (eg, 6Li– 7Li (Khaykovich et al., 2003), Li–Cs (Deiglmayr et al., 2009; Kraft et al., 2006; Mudrich et al., 2002), K–Rb (Damski et al., 2003; Ferlaino et al., 2006; Modugno et al., 2001; Simoni et al., 2003), Rb–Cs (Holmes et al., 2004;Kerman et al., 2004a,b; Sage et al., 2005), or NaCs (Haimberger et al., 2004, 2006)) have allowed studies of fermion–boson sys- tems and ultracold polar molecules. A nascent effort on ultracold samples containing ions is occurring in which resonances should be important. Although few results exist in these systems (Ciampini et al., 2002; Cot^ e, 2000a; Cot^ e and Dalgarno, 2000; Grier et al., 2009; Makarov et al., 2003;Smith et al., 2014; Zhang et al., 2009b), new studies are taking place, eg, on Yb (Cetina et al., 2012; Grier et al., 2009; Karpa et al., 2013), Rb with Yb+ (Lamb et al., 2012; Ratschbacher et al., 2012; Sayfutyarova et al., 2013;Zipkes et al., 2010a,b)orCa+ (Hall et al., 2013) or Ba+ (Krych et al., 2011; Schmid et al., 2010). As originally suggested by Makarov et al. (2003) and Smith et al. (2005), mixing atoms and ions requires hybrid traps, as shown in Fig. 1. In addition to their rele- vance to cold plasmas (Hahn, 2002; Mazevet et al., 2002; Robicheaux and Hanson, 2002), and ultracold Rydbergs (Anderson et al., 1998, 2002; Mourachko et al., 1998; Robinson et al., 2000), novel applications using the ion’s charge have been proposed, such as scanning tunneling Ultracold Hybrid Atom–Ion Systems 69
Fig. 1 Diagram of the hybrid trap apparatus. (a) A Na MOT (orange (gray in the print version)) is formed concentric with an ion cloud (gray) inside a segmented linear Paul trap (LPT) with six 589-nm MOT beams (yellow (light gray in the print version)) and a pair of anti-Helmholtz coils (exterior to the chamber). A 405-nm beam (blue (light gray in the print version)) aligned colinearly with one of the MOT beams is used for REMPI. Fluo- rescence measurements of the MOT can be made with a photomultiplier tube (PMT) or a CMOS camera. An electrically biased mesh is placed between the LPT and the Channeltron electron multiplier (CEM), which is used for ion detection. Right panels: CMOS camera image, without false coloring, of the smaller, denser, and colder type-I MOT (b) and the larger, warmer type-II MOT (c). The inner edges of the LPTs end-seg- ment electrodes can be seen in the corners of the image. Adapted from Sivarajah, I., Good- man, D.S., Wells, J.E., Narducci, F.A., Smith, W.W., 2012. Evidence of sympathetic cooling of Na+ ions by a Na magneto-optical trap in a hybrid trap. Phys. Rev. A 86, 063419. doi:10.1103/PhysRevA.86.063419 and Goodman, D.S., Wells, J.E., Kwolek, J.M., Blumel,€ R., Narducci, F.A., Smith, W.W., 2015. Measurement of the low-energy Na+-Na total collision rate in an ion-neutral hybrid trap. Phys. Rev. A 91, 012709. doi:10.1103/ PhysRevA.91.012709. microscopy of ultracold atoms (Kollath et al., 2007; Sherkunov et al., 2009), or a quantum gate using 87Rb+135Ba+ (Doerk et al., 2010). Exotic objects due to the capture of atoms around ions (Cot^ e et al., 2002), or atom-like mol- ecules with multiple atoms orbiting a heavy ion (Gao, 2010) were also predicted. In this chapter, we give an overview of ultracold samples doped with charged particles. We start by describing the atom–ion interaction, paying special attention to its long-range behavior. We follow by a survey of the principal processes taking place during the scattering of an atom and an 70 R. Cot^ e ion. We focus our attention first on atom and ion being from the same ele- ment, and then for different isotopes. We briefly mention the case of differ- ent elements, since it is outside the scope of this chapter. We describe the case of resonant and quasi-resonant charge transfer and how Feshbach res- onances can help tuning the scattering processes. We finally discuss the case of charges in a Bose–Einstein condensate (BEC), considering an ion or Rydberg electrons in a BEC.
2. ATOM–ION INTERACTIONS The interaction between an ion and another particle depends on the nature of that particle. For example, if it is another ion (atomic or molecu- lar), the leading long-range interaction behaves as 1/R, while a neutral par- ticle with a permanent electric dipole moment (eg, a polar molecule) leads to a 1/R2 long-range interaction. In this chapter, we focus our discussion on the case of an ion interacting with a nonpolar neutral object, for which the interaction scales as 1/R4 at large separation. Although a few experiments are starting to combine ultracold atoms with molecular ions, most experiments in ultracold samples involve ions and atoms. We discuss the interaction potential for two different cases: processes where both the ion and the atom are from the same element, and those where they are different.
2.1 Homonuclear Case In homonuclear systems X+X+, the charge can on either center, leading to an even or odd electronic wave function under symmetrization (inversion symmetry through the midpoint between the two nucleus, ie, a gerade (g) or ungerade (u) state. The multiplicity depends on the system considered. If X is an alkali metal element (Li, Na, K, Rb, Cs, or Fr), the electronic spin is 1 and 0 for X and X+, respectively, while it is 0 and 1 respectively for 2 2 alkaline earth metals (Be, Mg, Ca, Sr, Ba, or Ra): in both cases, the total electronic spin is S 1, leading to doublet molecular potentials. More com- ¼ 2 plex elements could lead to higher multiplicity, such as Cr and Cr+, with 5 1 3 11 electronic spin of 3 and , respectively, giving S , ,…, , and doublet, 2 ¼ 2 2 2 quartet, … , deca-doublet molecular curves. Since most atoms and ions used in ultracold experiments are alkali or alkaline earth metals (because of the existence of simple cycling transitions Ultracold Hybrid Atom–Ion Systems 71 to cool them), we discuss their case in what follows. As mentioned above, they have similar behaviors, ie, a single active electron for alkali metals, or 2 + hole for alkaline earth metals, giving two molecular potential curve Σg and 2 + 2 + 2 + Σu . The electronic ground state is X Σg for alkali metals, and X Σu for + + alkaline earth metals. Fig. 2 shows potential curves for Li2 , Na2 and + + Rb2 (left column), while the right column depicts those for Be2 , and + + Ca2 . We also include the curves for Yb2 , a rare-earth atom with an atomic structure similar to that of alkaline earth atoms. We note that the g/u order is inverted between the two class of elements. Usually, the ground state is much deeper than the upper state, though the existence of a double-well + + for Be2 and Ca2 masks the shallow long-range well that appears for all. This inner well is caused by the structure of excited electronic state, which varies “widely” with the element considered: a detailed discussion can be found in + + Banerjee et al. (2010) for Be2 and Banerjee et al. (2012) for Ca2
8 Li + 0.1 2 0 0 2 + 4 Σu –0.1 2 + Σg 0.1 15 20 25 0 –5 0 2 + Σg 2 + –0.1 + Σu –4 Be2 10 15 20 –10 8 Na + 0.1 2 0 +
a.u.) 0 a.u.) Ca2 2 +
–2 2 + Σg 4 Σu –0.1 –2 0.1 15 20 25 –5 0 2 + 2 + 0 Σg Σu –0.1 –4 Energy (10
Energy (10 Energy 10 15 20 –10 + 8 Rb2 0.1 0 2 + 0 + Σg 2 + Yb2 4 Σ –0.1 u 0.1 15 20 25 0 2 + –5 2Σ + 0 Σg u –0.1 –4 10 15 20 –10 0 5 10 15 20 25 30 0 5 10 15 20 R (a.u.) R (a.u.) Fig. 2 Potential curves for elements of group I (left column) and group II (right column). The inset shows the long-range shallow wells. Note that the g/u order is inverted for 2 + both group. Also, some of the group II systems have double wells for the Σg states, as opposed to group I elements. Though not a group II element, Yb is also shown because its ground state is also s2 like the alkaline earth atoms. 72 R. Cot^ e
The scattering properties depend on the long-range form of those inter- action potential curves, For elements with one “active” electron (or hole) discussed above, the long-range interaction contains two contributions
V R V R V R , (1) LRð Þ¼ dispð ÞÆ exchð Þ where Vdisp(R) is the dispersion term, and Vexch(R) the exchange term. To illustrate them, we consider the simplest “ion” scattering with its atomic par- ent, ie, a proton H+ interacting with H. Within the Born-Oppenheimer (BO) approximation and neglecting relativistic effects, the asymptotic expansion gives the following terms (in atomic units) (Damburg and Propin, 1968; Kaiser et al., 2013)
H + 9 15 213 7755 1773 V 2 R ⋯ (2) disp ð Þ¼ÀR4 À 2 R6 À 4 R7 À 64 R8 À 2 R9 À and
H + 1 25 131 3923 V 2 R 2Re R 1 1+ exchð Þ¼ À À 2 R À 8 R2 À 48 R3 À384 R4 145,399 5,219,189 509,102,915 37,749,539,911 ⋯ À3840 R5 À46,080 R6 À 645,120 R7 À 10,321,920 R8 À (3) where corresponds to the 1sσu electron configuration and+to the 2pσu configuration.À In general, for multielectron atoms, the dispersion term is written (in atomic units) as
X + C4 C6 V 2 R ⋯, (4) disp ð Þ¼ÀR4 À R6 À where C4 αd/2 is given by the static dipole polarizability αd of the neutral atom, and¼C α /2+c depends on the static quadrupole polarizability α 6 ¼ q 6 q and a van der Waals dispersion term c6 due to induced moments of the neu- tral atom. Higher order terms could be added, but truncation to the first two orders is usually sufficient. The exchange term can be found using the treatment of Bardsley et al. (1975) and takes the form, in atomic units,
+ X2 1 α βR B C V R AR eÀ 1+ + + … , (5) exchð Þ¼2 R R2 ! Ultracold Hybrid Atom–Ion Systems 73
2 + 2 + Table 1 Long-Range Expansion Coefficients for the Σu and Σg States of Some Homonuclear Alkali and Alkaline Earth Molecular Ions + + + + + + Parameter Li2 Na2 Rb2 Be2 Ca2 Yb2
αd 1.641 [2] 1.627 [2] 3.192 [2] 3.812 [1] 1.606 [2] 1.440 [2]
αq 1.423 [3] 1.849 [3] 6.495 [3] 3.00 [2] 3.073 [3] 2.560 [3] c6 2.635 [2] 1.20 [1] – 1.242 [2] 1.081 [3] 1.453 [2] C ( α /2) 8.205 [1] 8.135 [1] 1.596 [2] 1.906 [1] 8.032 [1] 7.20 [1] 4 ¼ d C ( α /2+c ) 9.750 [2] 9.365 [2] 3.2475 [3] 2.742 [2] 2.618 [3] 1.4253 [3] 6 ¼ q 6 A 1.53 [ 1] 1.11 [ 1] 5.322 [ 2] 1.094 [0] 3.64 [ 1] 6.673 [ 1] À À À À À E 0.1981 0.1888 0.1535 0.3426 0.2247 0.2298 α (calculated) 2.1774 2.2547 2.6096 1.4161 1.9834 1.9501 β (calculated) 0.6294 0.6145 0.5541 0.8278 0.6704 0.6779 B (calculated) 0.5191 0.4934 0.3179 0.5779 0.5655 0.5711 C (fitted) 9.52 [0] – 1.922 [1] 3.576 [1] 7.050 [1] 7.830 [1] À À À À À
All values are in atomic units. The numbers in square brackets indicates powers of ten. The values of c6 and C are from fitting and depends on the fitting range (–, not determined). where the parameters α, β and B are related by simple expressions to the ionization potential E of the neutral atoms. They are given by 1 β p2E,α 2ν 1 ,B ν2 1 ν , with ν 1=β: (6) ¼ ¼ð À Þ ¼ À 2 ¼ ffiffiffiffiffi Although A, related to the amplitude of the electronic wave function at large separation, can be calculated, it is more often obtained from numerical fit of Eq. (5) to ab initio data. Similarly, the second order expansion coefficient C + is fitted to ab initio data. We note that, consistent with analytic results of H2 (Damburg and Propin, 1968; Kaiser et al., 2013), C is negative. Table 1 lists the long-range parameters of a few homonuclear systems.
2.2 Heteronuclear Case Here again, the mixtures probed experimentally consist mostly of alkali and alkaline earth elements, such as Na+Ca+, Rb+Ca+, Rb+Ba+, etc. For those systems, there are two valence electrons giving singlet and triplet molecular potential curves. Due to the different atomic ground state energy of atoms X and Y and of their ions X+ and Y+, the curves correlated to X+Y+ and X+ +Y have different asymptotes, with singlet and triplet 74 R. Cot^ e
0.08
A 1Σ+ 0.04 Na(2S) + Ca+(2S) a 3Σ+ 0
Energy (a.u.) Na+(1S) + Ca(1S) X 1Σ+ −0.04 0 5 10 15 20 25 30 35 R (a.u.) Fig. 3 Potential curves for the lowest asymptotes of NaCa+. The ground state X1Σ+ cor- relates to Na+ and Ca in their ground states, while the second asymptote to Na and Ca+ in their ground states: it gives rise to two electronic states, a singlet A1Σ+ and a triplet a3Σ+ state. multiplicity. Fig. 3 shows the lowest potential curves for a (Na+Ca)+: the ground state is the singlet X1Σ+, while the singlet A1Σ+ and triplet a3Σ+ states merge at the second asymptote. The long-range form of those potentials includes a dispersion term 4 Vdisp(R) similar to Eq. (4), but with the leading RÀ term depending on the value of αd of the neutral partner for a given asymptote. An exchange term Vexch(R) contributes only for asymptotes with a singlet/triplet splitting. Unless spin-flip is considered (Makarov et al., 2003), this exchange term can be neglected in the scattering process. For example, for NaCa+ singlet/trip- A α βR let, by fitting V R eÀ , we find with A 1.95, α 9.5, and β 1.4 exc ¼ 2 ¼ ¼ ¼ (all in a.u.). As mentioned in the introduction, several atom–ion heteronuclear mix- tures are being investigated, and all have different electronic structures, Some will have singlet/triplet states, as exemplified in da Silva Jr. et al. (2015) for the case of Rb atoms and alkaline earth ions (Ca+, Sr+, Ba+) and Yb+, and between Li and Yb+, while other may have states with higher multiplicity, such as in Tomza (2015) for Cr atoms interacting with Ca+, Sr+, Ba+, or Yb+, for which the lowest potential curves corresponds to the X6Σ+ and a8Σ+ electronic molecular states. A thorough discussion of the many possible curves is outside the scope of this chapter, but we refer the reader to the papers referred above or in the introduction for more details. Ultracold Hybrid Atom–Ion Systems 75
3. SCATTERING PROCESSES Various outcomes emerge from the scattering of atoms and ions. Two main categories exist: resonant, and nonresonant processes corresponding to the homonuclear and heteronuclear cases, respectively. We give a general overview of these two types of processes.
3.1 Resonant Processes 3.1.1 Theory In the collision between an ion X+ and its neutral parent atom X, where X stands for elements of group I (Li, Na, K, Rb, Cs, or Fr) or group II (Be, Mg, Ca, Sr, Ba, or Ra), the atom and ion approach each other along two possible potential curves corresponding to the electronic molecular 2 + 2 + states Σg and Σu . Two possible outcomes can take place; elastic scattering where the charge remains on the same center
2 + X + 2 g X+X+ 8 orX 9 X+X+, (7) ! > > ! > 2 + >
2 + X + 2 g X+X+ 8 andX 9 X + +X: (8) ! > > ! > 2 + >
x g g u u 2 Itot θ f θ + f π θ + f θ f π θ ð Þ¼4jjð Þ ð À Þ ð ÞÀ ð À Þ (9) 1 x + À f g θ f g π θ + f u θ + f u π θ 2, 4 jjð ÞÀ ð À Þ ð Þ ð À Þ where x depends on the statistics of the atom/ion nuclear spin s; x (s+1)/ ¼ (2s+1) for integer s (boson) or x s/(2s+1) for half-integer s (fermion). In Eq. (9), f g and f u refer to the scattering¼ amplitude along the g and u potential curves, respectively. In practice, f g(θ) and f u(θ) both fall off rapidly with θ, and hence there is little overlap of the scattering amplitudes at θ and π θ so that cross terms g u À like f (θ) f (π θ) can be neglected in Eq. (9). We may therefore approx- imate theÁ totalÀ differential cross section by
1 I θ f g θ + f u θ 2 elð Þ¼4j ð Þ ð Þj Itot θ Iel θ + Ich θ , with 8 , (10) ð Þ’ ð Þ ð Þ > 1 g u 2 <>Ich θ f π θ f π θ ð Þ¼4j ð À ÞÀ ð À Þj > :> where Iel(θ) describes elastic scattering and Ich(θ) charge exchange scattering. We note that neglecting cross terms in Eq. (9) removes the dependence on s. 172 Fig. 4 shows results of Itot(θ) for Yb with s 0 for three collision ener- gies. It illustrates how the scattering peak at small¼ angles (θ 0) corresponds to the elastic process (as expected classically), while charge transfer domi- nates the peak at large angles (θ π) at higher energies; as the collision energy decreases, that dominance becomes less striking. The cross sections for a given scattering energy are obtained by integrat- ing over the solid angle Ω, ie, σ dΩI θ . If we adopt the standard ¼ ð Þ procedure, expand in partial waves ‘, and express f u and f g in terms of g u R the phase shifts η‘ and η‘, we obtain the cross sections for the total scattering (Mott et al., 1965; Zhang et al., 2009b)
∞ 2π g σ 2‘ +1 sin 2ηu + sin 2η , (11) tot k2 ‘ ‘ ¼ ‘ 0 ð Þð Þ X¼ 1 σ g + σu , (12) ¼ 2 el el ÀÁ π where σ g,u 4 ∞ 2‘ +1 sin 2η g,u, and charge exchange el 2 ‘ 0 ‘ k ¼ ð Þ X Ultracold Hybrid Atom–Ion Systems 77
Total Elastic Transfer
106 E = 10-1 a.u.
104
102
100 7 10 E = 10-5 a.u.
105
103
109
Differential cross section (a.u.) E = 10-10 a.u. 107
105
103 0 60 120 1800 60 120 1800 60 120 180 Angle q (degree) 172 Fig. 4 Differential cross sections Itot(θ), Iel(θ), and Ich(θ) of Yb ion-atom collisions at high (top row), medium (middle row), and low (bottom row) energy. The scattering peak at small angles corresponds to Iel(θ). The peak at large angles, clearly due to charge transfer at higher energies, diminishes at lower energies, where Ich still dominates, but less noticeably. From Zhang, P., Dalgarno, A., Co^te, R., 2009. Scattering of Yb and Yb+. Phys. Rev. A 80, 030703. doi:10.1103/PhysRevA.80.030703.
∞ π 2 u g σch 2‘ +1 sin η η : (13) k2 ‘ ‘ ¼ ‘ 0 ð Þ ð À Þ X¼ Here, k p2μE=ℏ is the wave number and μ is the reduced mass. The elas- ¼ tic cross section can be defined from those two expressions: σel σtot σch. ffiffiffiffiffiffiffiffi ¼ À The expression for σch can be obtained from the asymptotic form of the scattering wave functions on either g or u potential curves,
ikR ik R g=u e elec: elec: 1 Ψg=u k,R e Á + f θ Φ , where Φ ϕ ϕ : ð Þ¼ k ð Þ R g=u g=u ¼ p2ð A Æ BÞ (14) ffiffiffi ikR ik R e Here, eÀ Á is the (unscattered) plane wave, the outgoing scattered R g/u elec. spherical wave with scattering amplitude f , and Φg/u the even/odd 78 R. Cot^ e
electronic wave functions, where ϕA/B are the electronic wave functions with the charge centered on the nucleus A or B, respectively. If the charge is initially on a given nucleus, say A, its wave function is simply
ikR ikR 1 1 ik R g u e 1 g u e Ψ Ψg + Ψu e Á + f + f ϕ + f f ϕ , ¼ p2 2 ð Þ R A 2ð À Þ R B &' ÀÁ (15) ffiffiffi from which we identify f 1 f g + f u for the elastic process (ie, the charge el ¼ 2ð Þ remains on nucleus A), and f 1 f g f u for the exchange process (ie, the ch 2 ¼ ð À Þ 2 2 charge is transferred to nucleus B), and recover Iel fel and Ich fch from Eq. (10). ¼j j ¼j j The phase shifts η g,u k are determined from the continuum ‘ ð Þ eigenfunctions y g,u R , which are the regular solutions of the partial-wave ‘,kð Þ equation obtained from the time-independent Schrodinger€ equation
2 d 2 2μ ‘ ‘ +1 g,u + k Vg,u R ð Þ y R 0, (16) dR2 À ℏ2 ð ÞÀ R2 ‘,k ð Þ¼ where Vg,u(R) are the potential curves corresponding to the g and u elec- tronic states, respectively. One obtains η g,u k by matching the solution ‘ ð Þ to the asymptotic form π y g,u R sin kR ‘ + η g,u k : (17) ‘,k ð Þ À 2 ‘ ð Þ
3.1.2 Energy Regimes and Approximations Different scattering regimes exist depending on the collision energy E. At ultracold temperatures, only s-waves (‘ 0) contribute, and the phase shifts may be described by the effective range¼ expansion (valid at small k) (O’Malley et al., 1961)
1 π 2μ C 42μ C k 2μ kcotηg,u k + 4 k + 4 k2 ln C + k2 , 0 a 3 2 a2 3 2 a 4 2 4 ð Þ¼À g,u ℏ g,u ℏ g,u rffiffiffiffiffiℏ !Oð Þ (18)
Here, ag,u is the scattering length for the g and u states, respectively. They are obtained by fitting the cross sections at low E, which take simple forms Ultracold Hybrid Atom–Ion Systems 79
σ 4π a2 + a2 with σg,u 4πa2 , tot ¼ g u el ¼ g,u 2 9 σ π a a , (19) ch g u > ¼ À > ÀÁ 2 2 = σel σtot σch π 3ag +2agau +3au : ¼ À ¼ > > Fig. 5 shows the elastic cross sections of the g and u states;> for collisions between Na and Na+, with the low energy s-wave regime reaching the con- 2 2 stant value 4πag and 4πau , respectively. As the energy increases, considerable structure appears due to the con- tribution of higher partial waves and the occurrence of shape resonances. For large ‘, the centrifugal barrier ensures that the inner part of the potential con- tributes negligibly so that the phase shifts are determined by the long-range interactions, and hence the semiclassical approximation (Mott et al., 1965) can be used μ ∞ V R η‘ k 2 dR ð Þ : (20) ð Þ’Àℏ 1 2 R0 k2 ‘ + =R2 Z Àð 2Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
109 109 2 + Σu 108 108
107 4174 E -1/3 107
6 2 + 6 s L/2 10 Σg 10
(a.u.) s (a.u.) L el 5 ch 5 s 10 s 10
104 104
103 103 (a ln E - b)2
102 102 -16 -14 -12 -10 -8 -6 -4 -2 0 -16 -14 -12 -10 -8 -6 -4 -2 0
log10E (a.u.) log10E (a.u.) Fig. 5 Left panel: comparison of the elastic cross sections of both g and u states of Na+Na+ with the semiclassical formula of Eq. (24). Right panel: comparison of the charge exchange cross section with σ and 1σ . Adapted from Cot^ e, R., 2000. From L 2 L classical mobility to hopping conductivity: charge hopping in an ultracold gas. Phys. Rev. Lett. 85, 5316–5319. doi:10.1103/PhysRevLett.85.5316. 80 R. Cot^ e
Here, R0(E) is the outer classical turning point for a given scattering energy 4 E; if it is in the asymptotic region, so that V (R) C4/R , the phase shift is given by the simple expression ’À
πμ2C E πμ2C E ηsc k 4 4 : (21) ‘ ð Þ’ 2ℏ4 ‘ + 1 3 ’ 2ℏ4 ‘3 ð 2Þ This approximation gets better at larger ‘ for a given energy E. This is illus- trated in Fig. 6 for scattering along the g and u states: the left panel shows g=u 6 sc η k (modulo π) for E 10À a.u. and the approximate η from ‘ ð Þ ¼ ‘ Eq. (21), which is good for ‘>50, and the right panel shows the g-state 2 2 partial scattering cross section σ‘ 2‘ +1 sin η‘ at E 10À a.u., with 2 ¼ð Þ ¼ the fast oscillation of sin η‘ at smaller values of ‘ averaging to 1/2 until the last lobe at ‘ L where the semiclassical ηsc is valid. This allows for a ’ ‘ useful approximation for the cross sections
2π 4π ∞ 2π σg,u E L2 + d‘ 2‘ ηsc 2 L2 1+ ηsc 2 , (22) el ð Þ’k2 k2 ð ‘ Þ ¼ k2 ð ‘ Þ ZL ÂÃ so that σg σu , and according to Eq. (12), σ 1 σg + σu el ¼ el tot ¼ 2 ½ el el¼ π 2 L2 1+ ηsc 2 . As shown in Fig. 6, a reasonable choice for L is such that k2 ð ‘ Þ ηsc π=4 or sin 2ηsc 1=2, which gives, according to Eq. (21), L ’ÂÃL ’
2.0 1500 E = 10−6 a.u. g l 1.0 −2 σ = l h Semiclassical E = 10 a.u. l
0.0 ) g l h 2 + ( 1000 Σ 2 1.0 g Phase shift −
2 sc (2l +1)sin (ηl ) −2.0 +1) sin l sc 2 2l (ηl ) =(2 u l l 1.0 Semiclassical h s 500 0.0
2 + 1.0 Σu Phase shift −
−2.0 0 0 10 20 30 40 50 60 70 80 90 100 0 500 1000 1500 2000 Partial wave l Partial wave l g u + Fig. 6 Left panels:exactphaseshiftsη‘ (top)andη‘ (bottom)forNa+Na scattering at 6 sc E 10À a.u. as a function of ‘ (plotted between π/2 and π/2). The dashed line shows η‘ ¼ À 2 from Eq. (21). Right panel:partialelasticcrosssectionfortheg-state vs ‘ at E 10À a.u., ¼ sc sc and comparison with the semiclassicalapproximationwhenconsideringsinη‘ or η‘ . Adapted from Cot^ e, R., 2000. From classical mobility to hopping conductivity: charge hopping in an ultracold gas. Phys. Rev. Lett. 85, 5316–5319. doi:10.1103/PhysRevLett.85.5316. Ultracold Hybrid Atom–Ion Systems 81
πμ2C E 1=3 2μ2C 1=3 L 4 4 E (23) ¼ 2ℏ4 ηsc ¼ ℏ4 L and
2 1=3 2 g,u 4μC4 π 1=3 σtot E σ E π 1+ EÀ : (24) ð Þ¼ el ð Þ’ ℏ2 16 For charge exchange, at energies above the ultracold regime, the cross sec- 4 tion is dominated by the scattering from the attractive RÀ potential of one of the two potential curves (see Fig. 2). Small values of the angular momentum ‘ contribute mainly to the process, and σch varies approximated as the classical Langevin formula σ πb2 , where b is the maximum impact parameter L ¼ max max still allowing penetration of the wave function with the maximum partial wave number ‘ . The impact parameter b ‘ + 1 =k can be found from the effec- max ð 2Þ tive long-range interaction appearing in Eq. (16)
2 1 2 C ‘ + ℏ V R 4 + 2 , (25) eff ð Þ’ÀR4 2μR2 where we assume ‘ ‘ +1 ‘ + 1 2 for large enough ‘. The location of the ð Þð 2Þ centrifugal barrier R is found by setting dV /dR 0 and its height is top eff ¼ given by Veff(Rtop), namely
1 1 4 4μC 2 ‘ + ℏ4 R 4 V R 2 : (26) top 1 2 2 eff top ¼ ‘ + ℏ !) ð Þ¼4μ 4μC4 ð 2Þ
For a given energy E, as ‘ grows, ‘max is attained when the centrifugal barrier reaches E: for ‘<‘max the wave function penetrates the inner region of the potential and charge exchange takes place. The value of ‘max is found when E V (R ), giving ¼ top 1 2 4μ ‘max + pC4E, (27) 2 ¼ ℏ2 ffiffiffiffiffiffiffiffiffi so that the Langevin cross section σL becomes
π 1 2 C4 σ πb2 ‘ + 2π , (28) L ¼ max ¼ k2 max 2 ¼ E rffiffiffiffiffiffi 82 R. Cot^ e where we used ℏ2k2 2μE. The initial state of the system is distributed equally among both¼g and u states, ie, half on each curve, so that σ 1 σg + σu . As mentioned above, one of the curves dominates (much ch ¼ 2ð L LÞ deeper than the other) and the centrifugal barrier of the shallow curve grows fast with increasing ‘, preventing penetration of the wave function to short separation and leading to negligible charge exchange. Hence, only one curve contributes to charge exchange scattering, and we write,
1 C4 σch E σL π , where C4 αd=2: (29) ð Þ¼2 ¼ rffiffiffiffiffiffiE ¼
This result can also be obtained from the definition of σch. Starting from Eq. (13), we can break the sum over three different contributions: π final ‘i 2 u g σch 2 S1 + S2 + S∞ , where Si ‘ ‘init 2‘ +1 sin η‘ η‘ . The first ¼ k ½¼ ¼ i ð Þ ð À Þ sum runs from ‘ 0 to ‘*, the second from ‘*+1 to ‘ , and the third from ¼ P max ‘max +1 to ∞. Here, ‘* and ‘max are the maximum ‘ preventing penetration within the shallow and the deep potential well, respectively. For ‘>‘max, the phase shifts of both curves can be approximated by the semiclassical sc expression (20) and quickly become equal to each other by reaching η‘ given by Eq. (21), leading to a negligible contribution S 0. In S , both ∞ 1 phase shifts vary significantly and independently with ‘, leading to an average value of sin 2 ηu ηg 1, while only the phase shift associated with the shal- ð ‘ À ‘Þ2 low well follows Eq. (20) in S , so that again sin 2 ηu ηg 1. As in the case 2 ð ‘ À ‘Þ2 of the total cross section, we approximate 2‘ +1 sin 2 ηu ηg ‘ for both ð Þ ð ‘ À ‘Þ ‘max 2 u g 1 2 sums, leading to S1 + S2 ‘ 0 2‘ +1 sin η‘ η‘ ‘max, and hence ¼ ¼ ð Þ ð À Þ2 2 2 recovering Eq. (29) σch π‘Pmax/2k σL/2 as before. Fig. 7 shows the par- ch¼ g u tial charge exchange cross section σ‘ 2‘ +1 sinΔη‘ (with Δη‘ η‘ –η‘) + 4 ¼ð Þ for Na+Na at E 10À a.u., for which ‘max 103 according to Eq. (27): ch ¼ σ‘ drops rapidly at a slightly lower value ‘ 90, reflecting the fact that the finite depth of the potentials gives a slightly higher centrifugal barrier when 4 compared to the pure RÀ interaction assumed in the derivation of ‘max. At higher collision energies, the influence of the long-range attractive force tends to cancel and the cross section is determined by the exponential decay of the difference between the g and u potentials, and it varies as (see Fig. 5 right panel) Ultracold Hybrid Atom–Ion Systems 83
a 1.0 2.0
l 0.8 u
h 1.0 l h – 0.6 l ∆ g 2
h 0.0
0.4 sin = l
h −1.0 ∆ 0.2 −2.0 0.0
b 4 l 160 E = 10− a.u. h ∆ 2 120 + 1) sin l 80 (2 = l
ch 40 s
0 0 10 20 30 40 50 60 70 80 90 100 110 Partial wave l g u + 4 Fig. 7 (a) Phase shift difference Δη‘ η‘ η‘ for Na+Na at E 10À a.u. (black line: left 2 ¼ À ¼ axis) and sin Δη‘ (red (gray in the print version) line: right axis). (b) Partial charge exchange cross section vs ‘.
σ E a lnE b 2: (30) chð Þ¼ð À Þ For example, if E is given in eV and σch in a.u., a 1.33 and b 22.943 + ¼ ¼ for Na+Na (Cot^ e and Dalgarno, 2000), and a 1.88 and b 32.44 for Yb+Yb+ (Zhang et al., 2009b). ¼ ¼ At high energies beyond our calculations, oscillations about the mean value have been predicted (Tharamel et al., 1994). At much higher energy still, not considered here, the cross section is expected to decrease as a high 6 power of E as EÀ for the Brinkman–Kramers cross section (Shakeshaft and Spruch, 1979).
3.2 Nonresonant Processes As discussed in Section 2.2, the asymptotes of an ion X+ approaching an atom Y of different species (both in their respective ground state) depend on which is neutral and which is charged. If the X+ +Y is lower than X+Y+, then charge exchange corresponds to an excitation of the system requiring a photon or a high enough scattering energy. Here, since we 84 R. Cot^ e consider mainly low-energy processes, and no laser stimulating transitions, charge exchange is not possible and only elastic or spin exchange scattering can take place (if hyperfine and Zeeman interactions are considered, Feshbach resonances could also take place (Tomza, 2015)). If X+ +Y is higher than X+Y+, radiative charge transfer can take place, where by emitting a photon, the system passes from the X+ +Y to the X+Y+, A simple semiclassical treatment can be employed because many partial waves contribute even at low temperatures (Makarov et al., 2003). The cross section is given by
2μ ∞ ∞ A R σtr 2π bdb dR ð Þ , (31) 2 2 ¼ E 0 R 1 V R =E b =R rffiffiffiffiffiZ Z 0 À ð Þ À pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where b is the impact parameter, R0 the classical turning point, and V (R) the potential curve in the entrance channel. The Einstein coefficient A(R) (in atomic units, and α being the fine structure constant) is
4 A R α3ω3 R D2 R , where ℏω R V R V R , (32) ð Þ¼3 ð Þ ð Þ ð Þ¼ ð ÞÀ exitð Þ with Vexit(R) being the potential curve of the exit (lower) state, and D(R) the dipole transition moment. Using this treatment, the rate coefficient for radi- ative charge transfer was found to be basically constant and very small for + 16 3 Na+Ca : 2.3 10À cm /s. More sophisticated treatment are discussed, for example, inÂda Silva Jr. et al. (2015).
4. TRANSPORT PROPERTIES A charge, eg, an ion, immersed in a gas will diffuse through its stochas- tic scattering with surrounding atoms. The resulting diffusion coefficient is one of the transport properties of the system. Another related quantity is the charge mobility. Here, we describe both in different temperature regimes for nondegenerate gases: charges in a BEC are discussed in Section 7.
4.1 Diffusion
The diffusion coefficient D depends on the density of the ion, ρion, and of the atoms, ρat, as well as the temperature T and the averaged diffusion cross section σ , and is given by (Dalgarno et al., 1958; Kihara, 1953) h di Ultracold Hybrid Atom–Ion Systems 85
3pπ 2kBT 1+Δ0 3pπ 2kBT 1 D , (33) ¼ 16 ρ + ρ μ σ ’ 16ρ μ σ ð ion ffiffiffi atÞsffiffiffiffiffiffiffiffiffiffiffi h di ffiffiffiat sffiffiffiffiffiffiffiffiffiffiffih di where Δ0 0.13 is a small correction that depends on the power-law of the long-range interaction (Chapman and Cowling, 1970; Kihara, 1953); for 4 RÀ potential Δ0 0(Dalgarno and Williams, 1958). We also suppose ¼ the neutral gas density to be much larger than the ion density (ρat ≫ ρion). The averaged diffusion cross section σ is given by h di 1 ∞ σ dx x2 exp x σ x , where x E=k T: (34) h di¼2 ðÀ Þ dð Þ ¼ B Z0 For atoms and ions of distinguishable species approaching each other along a single potential curve, the diffusion cross section is
∞ 4π 2 σd E ‘ +1 sin η η : (35) k2 ‘ ‘ +1 ð Þ¼ ‘ 0 ð Þ ð À Þ X¼ If the ions and neutral atoms are of the same species, the bosonic or fer- mionic nature of the neutral gas must be accounted for. For bosons/fermion B/F (B/F), the diffusion cross section σd for an ion in its parent species is given by (Dalgarno, 1958)
B=F s +1 s σ E σÆ E + σÇ E , (36) d ðÞ2s +1 ðÞ 2s +1 ðÞ where+( ) corresponds to bosons (fermions), respectively, s is the nuclear spin of theÀ atom, and
4π ∞ ∞ σ + E ‘ +1 sin 2 ηg ηu + ‘ +1 sin 2 ηu ηg , k2 ‘ ‘ +1 ‘ ‘ +1 ð Þ¼ "#‘ evenð Þ ð À Þ ‘ oddð Þ ð À Þ X X (37)
4π ∞ ∞ σ E ‘ +1 sin 2 ηu ηg + ‘ +1 sin 2 ηg ηu : À k2 ‘ ‘ +1 ‘ ‘ +1 ð Þ¼ "#‘ evenð Þ ð À Þ ‘ oddð Þ ð À Þ X X (38)
For large ‘, ηg,u ηg,u and the diffusion cross section for bosons/fermions ‘ ’ ‘ +1 becomes σ E 2σ E : (39) dð Þ’ chð Þ 86 R. Cot^ e
109 109 10−3
) 7 −9
2 10 10
0 7
a 10 ) ) 2 6 0 −12 10 10 /s 2 a 6
10 > ( cm ( d
s 5 −15
10 10 D < sd: diffusion 105 Cross sections ( 104 10−18 2s 4 ch 10 103 D 10−21
103 10−24 −16 −14 −12 −10 −8 −6 −4 −2 0 −12 −10 −8 −6 −4 −2 02
log10E (a.u.) log10T (K) Fig. 8 Left panel: diffusion cross section as a function of the collision energy for Na+Na+. Right panel: corresponding σ and D as a function of T. Adapted from Co^te, R., h di Dalgarno, A., 2000. Ultracold atom-ion collisions. Phys. Rev. A 62, 012709. doi:10.1103/ PhysRevA.62.012709.
The left panel of Fig. 8 illustrates the diffusion cross section as a function of the collision energy for Na+Na+, and compares it to twice the charge trans- fer cross section: the agreement is close. On the right panel, we show the temperature dependence of σd and D computed from Eq. (33) using h i 19 3 the standard density ρ ρ 2.69 10 cmÀ . at ¼ std ¼ Â
4.2 Mobility
The charge mobility μmob characterizes how quickly a charge can move through a medium when pulled by an electric field. It is related to the con- ductivity σcond which quantifies how much of a current density j is induced by an electric field of intensity F, j σ F, where σ n q μ , (40) ¼ cond cond ¼ carrier mob where ncarrier is the density of carriers of charge q. Here, we consider singly charged ions of charge q e and density n ρ . ¼ carrier ¼ ion The charge mobility is related to the diffusion coefficient D via the Einstein relation (Dalgarno et al., 1958; McDaniel, 1964) eD μmob , (41) ¼ kBT where T is the temperature of the bath of atoms. Fig. 9 illustrates μmob for Na+Na+ for a large range of temperatures. Ultracold Hybrid Atom–Ion Systems 87
1010 a 12 -3 r at=10 cm Ion Atom Hole 9 ) 10 1 - s 1
- Ion (quantal) V 2 108 Ion (sd =2sch) b
Mobility (cm 107 Ion (classical)
106 -10 -8 -6 -4 -2 0
log10T (K) Fig. 9 Left panel: mobility as a function of temperature for Na+Na+, comparing the quantal result to the classical approximation in Eq. (42) and the estimate from Eq. (43) with σ 2σ σ . The hole mobility μ from Eq. (51) is also shown. Right d ’ ch ¼ L hole panel: In (a), the shaded region represents the probability P(x) of the neutral atom to 1/3 be within the hopping radius ρ if the ion is located at x: here L ρÀ (n ρ in ch at at ¼ at the sketch). In (b) the absence of charge in a positive ion can be regarded as a hole with a positive charge: the electron from the nearest neighbor (d1 < d2) tunnels through to the empty level of Na+ (or the hole jumps to the neutral atom). Adapted from Co^te, R., 2000. From classical mobility to hopping conductivity: charge hopping in an ultracold gas. Phys. Rev. Lett. 85, 5316–5319. doi:10.1103/PhysRevLett.85.5316.
If the scattering is described classically, the mobility corresponding to a 4 RÀ potential of an ion in a neutral gas at a standard density 19 3 ρ 2.69 10 cmÀ is given by (Dalgarno et al., 1958) std ¼ Â
ρstd 35:9 2 1 1 μmob cm VÀ sÀ , (42) ’ ρat pμαd where αd 2C4 is in atomic units,ffiffiffiffiffiffiffi and the reduced mass is in units of proton mass.¼ The ion mobility could also be estimated via
e τ 1 μion , where τ : (43) ¼ mion ¼ ρatσdv Here, τ is the average time of ion’s free motion between collisions with the surrounding neutral atoms while it diffuses, v its velocity, and σd the diffusion cross section, which, as shown in Eq. (39), can be approximated by σd 2σch. The two approximate mobilities are compared to the quantal result’ for the Na+Na+ case in Fig. 9. It shows that both expressions are valid over a large range of temperatures, from 10nK to 100K. However, at very low temperatures, the mobility increases rapidly as the averaged diffusion 88 R. Cot^ e cross section reaches a plateau, while it ultimately decreases at high temper- atures (McDaniel, 1964).
4.3 Hole Mobility Here, we consider the case of resonant charge transfer. In Fig. 9, the ion mobility μion starts increasing as the temperature becomes ultracold. At that point, since the ions and neutral atoms are moving very slowly with respect to each other, and the resonant charge transfer cross section becomes very large, the spread of atom and ion thermal de Broglie waves may allow elec- trons to hop from atoms to ions (see Fig. 9 right panel). At such low tem- peratures, electron hopping may provide an efficient charge diffusion through the neutral gas (Cot^ e, 2000b). The system becomes effectively con- ducting: this is analogous to hopping conductivity in doped semiconducting materials, where electrons hop from impurity atoms. To clarify the picture, consider the conditions that exist in a cold 11 12 3 (T 1μK 1 mK) and dilute gas (ρat 10 10 cmÀ ). The motion of À À atoms and ions is slowed down: for example, with sodium at 1 μK, v 3.8 cm/s, and the motion of atoms and ions can be neglected for short 2 1=2 times. However, the de Broglie thermal wavelength λT 2πℏ =mkBT ¼ð 1/2 Þ becomes very large: for Na atoms and ions, λ 6.88 TÀ a (a : bohr T ’ 0 0 radius) corresponds to 364 nm at 1 μK. Although the atoms are dilute, 3 ie, ρata ≪ 1(a: elastic scattering length between neutral atoms) and not 3 in the degenerate regime, ie, ρatλT ≪1, they can be within the “hopping radius” ρch for the charge transfer (see Fig. 9 right panel), defined by σ πρ2 or ρ σ =π: (44) ch ch ch ¼ ch Electrons can hop from neutral atoms to ions:pffiffiffiffiffiffiffiffiffiffiffi the system behaves like a con- ducting gas. Many electrons compete to fill in the ionized state of Na+, and it is convenient formally to consider the hop of a positive hole. Since σch fol- lows a Langevin behavior in the range of energies corresponding to temper- atures between 300 nK and 300 K (see Fig. 5 right panel), ρch 50.34 1/4 ¼ TÀ a0 with many partial waves still contributing. We need to account for the current density of both the ion and holes, and replace the conductivity σcond in Eq. (40) by σ ρ e μ + ρ e μ ρ e μ , (45) cond ¼ hole hole ion ion ¼ ion tot where μtot μhole +μion is the total mobility, and μhole and μion the hole and ion mobilities,¼ respectively. Here, we assume the number of holes, each of Ultracold Hybrid Atom–Ion Systems 89 charge e, being equal to the number of ions (equivalently, the number of electrons participating in the conductivity is equal to the number of ions), so that ρ ρ . hole ¼ ion To compute the hole mobility μhole, we first evaluate the probability that an atom (and therefore an electron) is within the hopping sphere of radius ρch. The probability to find an atom or an ion at a given position is well described by a gaussian of width λT. If, for simplicity, we consider hopping along one direction, the atom centered at xA and the ion at xI have the fol- lowing thermal distribution
1 2 2 1 2 2 x xA =2λ x xI =2λ pat x,xA eÀð À Þ T , and pion x,xI eÀð À Þ T : ð Þ¼p2πλT ð Þ¼p2πλT (46) ffiffiffiffiffi ffiffiffiffiffi The probability that an electron is within the hopping region defined by ρch for an ion at x is (see Fig. 9a, right panel)
x + ρch P x pion x,xI dz pat z,xA , (47) ð Þ¼ ð Þ x ρ ð Þ Z À ch and the total probability for an electron (localized in an atom) to be in the hopping zone will be the integral of P(x) over the whole x-space is
∞ 2 λT 1 L ρch Ptot ρat,T dx P x exp ð À 2 Þ , (48) ð Þ ∞ ð Þ’L ρch p2π À 4λT ! ZÀ À where L is the distance between the two centers:ffiffiffiffiffi on average, L xI xA 1/3 ¼j À j ρÀ . at The hole mobility is evaluated using Eq. (41)
eDh μhole : (49) ¼ kBT
For the diffusion coefficient of the holes Dh, we assume that electrons jump randomly from an atom to the ion, but they have to be within the hopping sphere of radius ρch. Hence Dh will be proportional to Ptot. For a random process where the sites are separated by a distance L, the diffusion is L2 ν /2, where ν is the average jump frequency (Pathria and Beale, h i h i 1996). We relate this frequency to the exchange energy ΔE hν, so that 2 ¼ α βR Dh PtotL ν /2. The exchange energy takes the form ΔE AR eÀ (see¼ Eq. (5)),h i where R is the separation between the ion and’ the atom, and the parameters A, α, and β are given in Table 1 for Na Na+. Only À 90 R. Cot^ e
electrons within the hopping sphere of radius ρch can hop, and we define the average frequency as
ρch 4πA 2+α βR 3AΓ 3+α h ν dR R eÀ ð Þ, (50) h i Ω ’ ρ3 β3+α Z0 ch 3 where Ω 4πρch/3 is the volume of the hopping sphere. Since the expo- ¼ nential in the integrand decays rapidly, we can replace ρch by ∞ as the outer limit of integration to obtain the approximate analytical result in Eq. (50). + 12 3/4 1 For Na Na , we obtain ν 7.81 10 T sÀ , and together Eq. (50), weÀ have h i¼ Â
3 A 3+α e λT L 2 2 Γ L =4λT μhole ð 3+α Þ 3 eÀ : (51) ’ 2hp2π β kBT ρch
Fig. 9 (left panel) shows the behaviorffiffiffiffiffi of μhole as the temperature decreases: it is negligible until a “critical” temperature where it exhibits a sharp increase and dominates over μion. As described by Eq. (45), the conductivity of the “dopped-gas” is pro- portional to its charge density ρion and the total mobility of the system μ μ +μ . In Fig. 9, we plot the different mobilities as a function tot ¼ ion hole of T at given density ρat: the total mobility is essentially equal to the ion mobility at higher temperatures. It is relatively small, and the system behaves like an insulator, with the mobility dominated by charge transfer induced by collisions which depends essentially on the motion of the massive charged ions. As T decreases, μ const. over a large range of temperatures, until tot a “critical” temperature is reached at which point the probability of electron hopping becomes sizable. Although T is then low enough so that atoms and ions are practically frozen, their de Broglie wavelength is large enough to allow the electrons to act as if they were delocalized, and a sharp increase in charge mobility occurs. The system behaves like a “conductor,” with electrons jumping from atoms to ions (or the holes from ions to atoms) with a large frequency. Fig. 10 illustrates a possible experimental setup to measure this effect, with ultracold neutral atoms trapped in a long cigare shaped geometry. A set of low power lasers ionizes a few atoms at the “tip” of the system, after which a weak electric field is applied to accelerate the charges to a detector. By measuring the time-of-flight of the positive charges, one can find how far they traveled. As a function of temperature, one should observe a sharp tran- sition from longer time at higher temperatures to much shorter time at lower Ultracold Hybrid Atom–Ion Systems 91
a Ionizing lasers c
t ion Neutral gas
b Time Detector Hole hopping t hole Ion
Ion t hole t ion Temperature
Fig. 10 Schematic of an experiment to detect charge hopping: ions are formed at the “tip” of the cold gas (a), and a weak electric field accelerate them toward a detector (b). If T is varied, the time-of-flight should exhibit two timescales (c). From Co^te, R., 2000. From classical mobility to hopping conductivity: charge hopping in an ultracold gas. Phys. Rev. Lett. 85, 5316–5319. doi:10.1103/PhysRevLett.85.5316. temperatures. Alternatively, experiments using small atomic cloud and a Rydberg atom with the Rydberg electron orbit outside the sample could provide a mean to observe the effect of charge hopping on the lifetime of the Rydberg state. Similar concepts are described in Section 7.2 for the case of a Rydberg electron in a BEC.
5. TUNING INTERACTIONS: HYPERFINE AND ZEEMAN INTERACTIONS One of the tools accessible to experiments on atom–ion system is the use of external fields to tune interactions, such as external magnetic, electric, or radiation fields. Radiation fields could be used in different fashion, for example by using electromagnetic fields to couple to excited electronic state (as discussed in the nonresonant case), or by dressing the ground-state inter- action with a detuned laser fields. Since electric fields would accelerate ions, we will focus our discussion on tuning interactions using external magnetic fields. This approach has been very successful in neutral atomic samples to modify interactions near Feshbach resonances. It allows to tune interactions from attractive to repulsive with a magnitude ranging from zero to infinity (Chin et al., 2010).
5.1 Multichannel Scattering We consider the scattering of ions and atoms with nonzero nuclear spins, so that the spin of atom/ion j is f i + s , where i and s are their respective j ¼ j j j j 92 R. Cot^ e
nuclear and electronic spins. The interaction Hamiltonian H^ int describing this system,
H^ H^ + H^ + H^ , (52) int ¼ mol hf B consists of three components, namely H^ mol describing the molecular curves (whichdominateat short range), H^ hf accounting for thehyperfineinteraction due to the nuclear and electronic spins, and H^ B specifying the interaction of those spins with an external magnetic field. Other possible terms are omitted in our treatment, such as dipole–dipole interactions which would couple dif- ferent partial waves: the largest of such term (between electron spins) is zero since one of the colliding partner has zero electronic spin. A natural identification for the scattering channels relies on the spin state f m of each scattering partner, leading to the representation ‘m : j j ji j f1m1,f2m2 ‘m f1m1,f2m2 . To simplify the notation, we introduce a compacti labelj fori j the variousi quantum numbers, eg, we label the initial channel by ν {i,‘,m} where i {f m ,f m } and the final channel by ν i ¼ 1 1 2 2 f ¼ f,‘0 ,m0 with f f m , f m . We solve the following system of coupled f g f 10 10 20 20 g equations (in atomic units)
2 d ‘0 ‘0 +1 2μ + k2 Fνi R ν H^ γ Fνi R , 2 f ð 2 Þ νf 2 f int γ (53) dR À R ð Þ¼ℏ γ h j j i ð Þ !X where Fνi R is the radial wave function corresponding to the scattering from νf ð Þ 2 2 the initial channel νi into the final channel νf. Here, ℏ kf 2μ E Ef , where ¼ ð À νiÞ E is the total energy and Ef the energy of channel νf. The solutions F R will νf ð Þ be subject to the boundary conditions
Fνi R 0 0, νf ð ! Þ¼ 1 (54) and Fνi R δ e i kf R ‘0π=2 S e +i kf R ‘0π=2 : νf ∞ νi, νf À ð À Þ νi νf ð À Þ ð ! Þ¼pkf À ! no We can extract the S-matrixffiffiffiffi from the above expression (for channels that are closed, we use the method of Johnson (Johnson, 1985; Krems and Dalgarno, 2003)), and calculate the cross section σi f for transition between the initial state i {f m ,f m } and the final state f! f m , f m ¼ 1 1 2 2 ¼f 10 10 20 20 g 2 π 2 σi f E δνi, νf Sνi νf , (55) ! k2 ! ð Þ¼ i ‘,m,‘ ,m À X0 0 Ultracold Hybrid Atom–Ion Systems 93
2 2 where ℏ ki 2μ E Ei , with Ei being the energy of channel νi. The elastic ¼ ð À Þ π2 cross section is obtained with ν ν , ie, σel: σ 1 S 2 f i i i i 2 ‘,m νi νi ¼ ¼ ! ¼ ki jjÀ ! X with ‘ ‘0 and m m0. Similarly, the total inelastic cross section is obtained¼ by summing¼ over all other channels than the initial/entrance π2 2 channel: σinel: σ S . The corresponding i f i i f 2 f i ‘,m,‘ ,m νi νf ¼ 6¼ ! ¼ ki 6¼ 0 0 ! X X rate coefficients canP be obtained by averaging over a Maxwellian velocity distribution
8k T 1 ∞ B ε=kBT Ri f T 2 dεεeÀ σi f ε , (56) ! ð Þ¼ πμ k T 0 ! ð Þ rffiffiffiffiffiffiffiffiffiffiffið B Þ Z where ε E Ei is the kinetic energy. ¼ À In the ultracold regime, only the s-wave (‘ 0) contributes as ki 0, 2 ¼ ! el,‘ 0 π ‘ 0 2 ie, σ ¼ 1 S . Since the energy is defined from the initial/ i i 2 ν¼i νi ! ¼ ki À ! entrance channel, we drop the “i” and simply write k k . Similarly we sim- i plify the notation by S S‘ 0 since ‘ ‘ 0 so that “i” and “f” uniquely fi ν¼i νf 0 ! ¼ ¼ identify the channels νi and νf respectively. The S-matrix element defines the s-wave scattering phase shift η (k): S exp 2iη k . The scattering length i ii ¼ ½ ið Þ ai αi iβi is obtained from ηi(k)/k in the limit k 0. Expanding Sii for small¼ kÀ, we have À !
S 1 2iη k 1 2ik α iβ : (57) ii ’ À ið Þ¼ À ð i À iÞ From the unitarity of the S-matrix, we have S 2 S 2 + f j fij ¼j iij 2 el. f i Sfi 1, so that the elastic cross section σi and the total inelastic cross 6¼ P j jinel¼: section σi f iσi f from the initial channel are simply P ¼ 6¼ ! P π2 σel: σel,‘ 0 2 π α2 β2 (58) i i i¼ 2 1 Sii 4 i + i , ¼ ! ¼ k jjÀ ! ð Þ π2 π2 4π σinel: S 2 1 S 2 β , i k2 fi k2 ii k i (59) ¼ i f i j j ¼ i ½ Àj j ! X6¼ where we used Eq. (57). In the ultracold regime, the imaginary part of the scattering length gives a measure of the inelastic cross section or rate coefficient: πℏ μ with v ℏk=μ,Rinel: vσinel: 4 β ,sothatβ k σinel: Rinel:. ¼ i ¼ i ¼ μ i i ¼ 4π i ¼ 4πℏ i 94 R. Cot^ e
5.2 Identical Nuclei We consider an ion X+ scattering with its parent atom X (same isotope), both with a nonzero nuclear spin. We focus on alkali or alkaline earth ele- ments (including atoms such as Yb), ie, X+X+ systems with a single active electron/hole. For identical nuclei, the nuclear spins ij of nucleus j are equal (i i ), and one of the electronic spin s is zero while the other is simply 1 (if 1 ¼ 2 j 2 the charge is centered around nucleus 1, s 0 and s 1 for alkali atoms, 1 ¼ 2 ¼ 2 and vice versa for alkaline earth atoms). For identical nuclei, we must sym- metrize the spin state since the electron/hole can be on either nuclei, ie, we 1 use f1 f2 f1m1, f2m2 + f2m2, f1m1 . j ip2fj i j ig Various representationsffiffiffi of the spin states can be used to evaluate the dif- ferent terms of H^ . As mentioned previously, ‘m : f m ,f m is a natural int j 1 1 2 2i representation for the scattering channels. The total spin F f1 + f2 can be written as F I+S where I i + i is the total nuclear spin¼ and S s + s ¼ ¼ 1 2 ¼ 1 2 the total electronic spin. The molecular potentials depend only on R and therefore conserve the angular momentum ‘ and F, as well as their projec- tions m and M. They also conserve S and I separately (actually, they are inde- pendent of I), and therefore the representation ‘m : IM ,SM ‘m IM , j I Sij i j I SM is ideal to evaluate H^ . As we will see below, the same is true for H^ , Si mol B while either representation is well adapted to evaluate H^ hf . Matrix elements of the H^ int contributions can be transformed from one representation to another using angular momentum algebra. As mentioned in Section 3.1.1, the statistics of the total nuclear spin plays an important role in the scattering process. For identical nuclei, we need to take the bosonic or fermionic statistic into account. Here, we focus on alkali or alkaline earth elements (including atoms such as Yb), ie, systems with a single active electron/hole. We describe systems in which the nuclear spin is half-integer (eg, i 1/2 for H+ or 3/2 for 7Li+ or Na+), which must satisfy Fermi statistics (eg,¼6Li+ with i 1 would satisfy Bose statistics). Two nuclei ¼ 1 and 2 of nuclear spin i1 and i2, respectively, and an electron/hole with spin s are characterized by a spin wave function χ(i1,i2,s). If χs(i1,i2,s) and χa(i1,i2,s) are the spin wave functions which are symmetric and antisymmetric in the spins of the two nuclei 1 and 2, respectively, then, according to the Pauli principle, the wave function of the system X+ +X+ +e may have one of the following forms (Wu and Ohmura, 2014; Wu et al., 1960) Ultracold Hybrid Atom–Ion Systems 95
elec: ‘ odd elec: ‘ even Φg r;R Ψg R , Φg r;R Ψg R , χs ð Þ ð Þ χa ð Þ ð Þ (60) Â (Φelec: r;R Ψ ‘ even R , Â ( Φelec: r;R Ψ ‘ odd R , u ð Þ u ð Þ u ð Þ u ð Þ elec. where Φg/u (r;R) are the gerade and ungerade electronic wave functions, respectively, and Ψ‘ R the corresponding scattering wave functions with g=uð Þ even or odd parity for ‘ even or odd, respectively (r describes the electron coordinates and R the internuclear ones). For identical nuclei obeying Bose statistics, the ‘ even/odd (or g/u) in Eq. (60) are simply interchanged. Following these results, the total nuclear spin can used to project the scat- tering states onto the gerade or the ungerade potential curve, with an even or odd partial wave, by using the permutation symmetry of the nuclear spin 2i I states determined by ( 1) À , where i is the nuclear spin of one ion and I À + the total nuclear spin. For Na , i 3/2 so that I i1 + i2 gives ¼ 3 ¼I I 3,2,1,0, and the symmetry of χ is given by ( 1) À : we have χ for ¼ À s I 3,1 and χ for I 2,0. One can build operators P^ to project the scat- ¼ a ¼ g=u tering state onto the appropriate potential curve Vg/u(R)
H^ P^ V R + P^ V R : (61) mol ¼ g gð Þ u uð Þ
A convenient representation to express the matrix elements of H^ mol is ‘m : IM ,SM (where S 1/2 is fixed), in terms of which we have j I Si ¼
‘m : IMI ,SMS H^ mol ‘0m0 : I 0M 0,SM0 δ‘‘ δmm δII δM M δM M h j j I Si¼ 0 0 0 I I0 S S0 1 ‘ 3 I ‘ 3 I 1+ 1 1 1 À + 1 1 1+ 1 À V R  4 ½ ðÀ Þ ½ À ðÀ Þ ½ À ðÀ Þ ½ ðÀ Þ gð Þ (62) no 1 ‘ 3 I ‘ 3 I + 1+ 1 1+ 1 À + 1 1 1 1 À V R : 4 ½ ðÀ Þ ½ ðÀ Þ ½ À ðÀ Þ ½ À ðÀ Þ uð Þ no!
Note that for identical nuclei obeying Bose statistics, the Vg/u are interchanged. The interactions leading to the hyperfine structure have various origins, ranging from the coupling of the electric quadrupole moment of the nucleus with the electrons, to the nuclear and electron spin–spin interaction, which includes a dipole–dipole and a contact term. There is also coupling of those spins with the rotational angular momentum. A detailed discussion of the interactions for diatomic systems can be found in Brown and Carrington 96 R. Cot^ e
(2003). These terms can be R dependent and described by an effective Ham- + iltonian, like in the case of H2 (Babb and Dalgarno, 1991; Fu et al., 1992; Korobov et al., 2016), with H^ hf b I s + c I ^ρ ^ρ S + d s L + f I L , where ^ρ is the internuclear axis,¼ andð ÁLÞtheð totalÁ Þð orbitalÁ Þ momentum.ð Á Þ Theð Á constantsÞ b,c,d and f are obtained from integrating out the R dependence over the wave function, and therefore depend on the vibrational level of + + H2 : note that there is no quadrupole term for H2 . The spin–spin contact term is usually the leading contribution, and we omit all other terms and simply write a simplified effective Hamiltonian
1 2 ahfð Þ ahfð Þ ahf H^ hf i1 s + i2 s I s: (63) ¼ ℏ2 Á ℏ2 Á ¼ ℏ2 Á where a(1) a(2) a are the hyperfine constants (equal for identical nuclei) hf ¼ hf hf and I i1 + i2 is the total nuclear spin. The scattering channel representation ¼ ℏ2 f m ,f m is useful to evaluate this term, with i s fm j 1 1 2 2i ð Á Þj i¼ 2 f f +1 i i +1 s s +1 . For one electron/hole pairs, one of the part- ½ ð ÞÀ ð ÞÀ ð Þ + ners has a zero electronic spin, (X for alkali or X for alkaline earth), so that f i with s 0 for that partner, with the state of the pair given by f1m1,i2m2 , ¼ ¼ a j i so that H^ f m ,i m hf f f +1 i i +1 s s +1 f m ,i m . hf j 1 1 2 2i¼ 2 ½ 1ð 1 ÞÀ 1ð 1 ÞÀ ð Þj 1 1 2 2i As mentioned before, symmetrized states are used for identical nuclei, so that a H^ hf f f +1 i i +1 s s +1 is the hyperfine energy of the h hf i¼ 2 ½ ð ÞÀ ð ÞÀ ð Þ partner with a nonzero electronic spin (as expected).
The last term of H^ int in Eq. (52) is the Zeeman interaction H^ B, which is given by
H^ γ s γ 1 i γ 2 i B γ s γ I B , (64) B ¼ð e À nð Þ 1 À nð Þ 2ÞÁ ¼ð e z À n zÞÁ z 1 2 where γe, γnð Þ and γnð Þ are the gyromagnetic moment of the electron and of nuclei 1 and 2, respectively. For identical nuclei, γ 1 γ 2 γ , leading nð Þ ¼ nð Þ ¼ n to the expression to the right, where the B-field is oriented along the z-axis. The matrix elements of H^ are diagonal in the representation IM ,SM , with B j I Si ‘0m0 : I 0M 0,S0M 0 H^ BI‘m : MI ,SMS δ‘‘ δmm δII δM M δSS δM M ℏ γ Ms h I Sj i¼ 0 0 0 I I0 0 S S0 ð e À γ M B . n I Þ z We use these terms for H^ int together with the potential curves shown in Fig. 2 to compute the scattering properties of Be. We consider an isotope with hyperfine structure, specifically 9Be. In Fig. 11, we show the hyperfine Ultracold Hybrid Atom–Ion Systems 97
100 1
9 + mf = −1 Be 10Be 50
f = 1 mf = 1 m = 2 f s = 0, i = 0 0 0 (mK) E
f = 2 −50
mf = −2 −1
mi = 3/2 50 10 + 9 0.05 Be ms = 1/2 Be
f = s = 1/2 f = i = 3/2 0 0.00 (mK) E
ms = −1/2 −50 −0.05 mi = −3/2 0 200 400 600 800 0 200 400 600 800 1000 B (Gauss) B (Gauss) Fig. 11 Hyperfine and Zeeman structure for 9Be and 10Be isotopes and their ions. Top row: 9Be+ (left) with nuclear spin i 3/2 and inverted structure, and 10Be (right) with 10 + ¼ 9 s i 0. Bottom: Be (left) with i 0 and s 1/2, and Be (right) with i 3/2 and ¼ ¼ ¼ ¼ ¼ s 0. The energy scales reflect the difference in the hyperfine and Zeeman interactions ¼ for each atom/ion isotope. structure of 9Be and 9Be+ together with the Zeeman interaction as a function of the magnetic field. Using the labeling of channels illustrated in Fig. 12, the s-wave scattering properties are obtained as a function of the magnetic field for various entrance/initial channels. Fig. 13 shows the real part α of the scattering length a for a collision energy ε k T i i ¼ B corresponding to 10 μK. Significant structure exists, with several sharp features corresponding to Feshbach resonances. In particular, the density of those resonances at B-field less than 500 Gauss increases as NA gets larger 9 in channels {NANB} corresponding to larger mf values of Be. A consequence of the changes in scattering properties is the possibility of tuning the relative contribution of elastic and inelastic processes. In Fig. 14, we show an example of elastic vs inelastic rates for the {2,3} entrance chan- nel. We find a range of B-field (roughly 100 G < B < 370 G) where elastic dominates over inelastic scattering, providing favorable conditions to sym- pathetically cool the ions by collisions with ultracold neutral atoms. In other 98 R. Cot^ e
9 Be 9Be+ +3/2 −1 0 4 8 7 +1 +1/2 6 +2 3 f = 1 5 f = 3/2 2 −1/2 f = 2 4 3 1 2 +1 1 0 −1 3/2 − −2
Entrance channel {12} m = −5/2 Fig. 12 Definition of scattering channels for 9Be+ 9Be+, going from the lowest to the highest energy for the ion and the atom. The entrance channel {12} is constructed 3 3 by symmetrizing the state 1 9 2 9 + with 1 9 f , m 9 and 2 9 + j i Be j i Be j i Be j ¼ 2 f ¼À2i Be j i Be f 2,m 1 9 +. j ¼ f ¼À i Be
21 2 × 104 12 13 23 ) 0 a 4 24 1 × 10 18 27 22 14 0 26 17 −1 × 104
−2 × 104
4 41 43 2 × 10 44 ) Scattering length ( 31 32 34
0 42 a 1 × 104 33 36 0
−1 × 104 Scattering length ( −2 × 104 0 500 1000 0 500 1000 B (G) B (G)
Fig. 13 Real part αi of the scattering length ai for various entrance/initial channels i {N N } of 9Be+ 9Be+ as defined in Fig. 12, with N 1, 2, 3, and 4 for the top left, ¼ A B A ¼ top right, bottom left, and bottom right panels, respectively. The sharp features corre- spond to Feshbach resonances occurring at specific values of the magnetic field. regions, the opposite ratio would allow studies of inelastic processes (hyper- fine state change or charge exchange). However, as noted before in Fig. 4, distinguishing charge exchange from other scattering processes requires the angular distribution of the differential cross section, and hence several partial waves contributions are needed. At ultracold temperatures, where the main Ultracold Hybrid Atom–Ion Systems 99
a 10−8 /s) 3
ε/kB = 10 µK 10−10 Elastic {23} --> {32} {23} --> {41} Rate coef. (cm 10−12 10.0 b ) 0 a
3 5.0
0.0
Re(a) −5.0 Im(a) Scat. length (10
−10.0 0 200 400 600 800 1000 Magnetic field (G) Fig. 14 (a) Elastic rate and two main components of the inelastic rate for 9Be+ 9Be+ ini- 1 tially in the entrance/initial channel i {23}. The other channels with M m + m ¼ ¼ 1 2 ¼À2 ({16}, {27}, and {38}) are closed at ultralow energies. (b) Real and imaginary part, respec- tively, Re(a ) α and Im(a ) β , of the scattering length a for i {23}. i ¼ i i ¼À i i ¼ contribution arises from the ‘ 0 (isotropic) partial wave, one cannot iden- tify charge exchange scattering¼ between identical ion and parents: a possible solution to this problem is to consider different isotopes of the same element.
6. ISOTOPIC EFFECTS As discussed in Section 3.1, the elastic and charge transfer collisions cannot in principle be distinguished from each other in the scattering of a positive ion with its parent atom of the same isotopic composition. Although an empirical distinction is possible at high collision velocities, with the angu- lar differential scattering cross section exhibiting two peaks (one in the for- ward direction attributed to elastic collisions and the other in the backward direction attributed to charge transfer collisions), these distinctions vanish at ultralow energies where the isotropic s-wave (‘ 0) becomes the main contribution. ¼ However, for different isotopes X and X0 , there occurs a change in the kinetic energies of the particles in the charge transfer collisions and not in the 100 R. Cot^ e elastic collisions. At large internuclear separation, the binding energies for + + X+X0 and X +X0 differ by a small amount ΔE that depends on the masses, breaking the molecular g–u symmetry. The centers of mass and charge do not overlap and correction to the BO curves incorporating some aspects of the nuclear-electronic coupling are required. Below, we give an overview of the treatment, its application to few cases, and finally discuss the role played by hyperfine and Zeeman interactions. Here again, we consider alkali and alkaline earth elements.
6.1 Theory Here, we summarize the treatment described in Bodo et al. (2008) and Zhang et al. (2009a, 2011), which is based on the approach of Hunter et al. (1966). The slight difference in masses causes shifts of the energy asymptotes and nonadiabatic corrections that couple the curves Vg and Vu 2 + 2 + corresponding to the two lowest Σg and Σu BO states, respectively. The coupled equations to solve for those two states are (in atomic units)
2 d 2 ‘ ‘ +1 d ‘ ‘ I + k I ð Þ +2G F ð Þ R 2μCFð Þ R , (65) dR2 À R2 dR ð Þ¼ ð Þ ! where I is the 2 2 identity matrix, and k2 is the diagonal matrix with 2  2 elements kα and kβ related to the threshold energy ΔE by 2 2 1 1 1 k k 2μΔE with μÀ mÀ + mÀ (mi: mass of isotope i). In this β À α ¼ ¼ A B equation, the radial wave function vector is (anti)symmetrized, ‘ ‘ ‘ ‘ ‘ ie, Fð Þ χ ‘ +χð Þ, and Fð Þ χ ‘ χð Þ, where χ ‘ and χð Þ are the radial 1 ¼ αð Þ β 2 ¼ αð ÞÀ β αð Þ β wave functions for the BO states α and β. The elements of the first derivative @ coupling matrix G are Gαβ Gβα α β , where α and β are the ¼À ¼h j@Rj i j i j i BO eigenstates. The elements of the matrix C are given by
1 Cαα V αα + V ββ + V αβ 1 d ¼ 2ð Þ 9 with V αβ Vαβ Gαβ 1 À 2μdR > Cββ V αα + V ββ V αβ > 2 ¼ 2ð ÞÀ > mp 1 @ >, where V δ ε +ε + α β 1 d > αβ αβ α αβ 2 > ¼ 2μh j@R j i Cαβ V αα V ββ Gαβ=> ¼ 2ð À ÞÀdR 1 α L^2 + L^2 β , 1 d > À 2μR2 h jð x yÞj i Cβα V αα V ββ + Gαβ > ¼ 2ð À Þ dR > > > (66) ;> Ultracold Hybrid Atom–Ion Systems 101 where ε is the eigenenergy of the BO state α, and εmp α T^ β is the mass α αβ h j j i ℏ N polarization correction, with T^ e . Eq. (65) satisfies the i, j 1 i j ¼À2M ¼ r Ár proper scattering boundary conditions inX an atomic representation. Here,
L^x and L^y are the x and y components of the electronic orbital angular momentum operator, respectively. Fig. 15 shows the potential curves obtained by diagonalization of VαβR) in Eq. (66) in the case of Li. The charge exchange and the elastic cross sections can be expressed in terms of the scattering S-matrix (similar to Eq. (55)) as
π ‘ 2 σ α β 2‘ +1 Sð Þ and ch: k2 αβ ð ! Þ¼ α ‘ ð Þj j X (67) π ‘ 2 σ α α 2‘ +1 1 Sð Þ , el: k2 αα ð ! Þ¼ α ‘ ð Þj À j X respectively. A modified Numerov algorithm that accounts for the presence of the linear derivative can be used to compute F ‘ R (Bodo et al., 2008). ð Þð Þ The S-matrix is obtained in the asymptotic region from the scattering wave ‘ 1 Fð Þ R J R + N R K and S (1+iK)À (1 iK), where J (R) and ð Þ¼ ‘ð Þ ‘ð ÞÁ ¼ Á À ‘
7 + 6 0.0000 Li + Li 2 + Σ 7 6 + u Li + Li −0.0002
2 Σg −0.0004 2 Σu 7 6 + Li + Li 0.0006 7 + 6 − Li + Li Potential energy (a.u.) 0.0008 2 + − Σg
−0.0010 15 20 25 30 35 40
Internuclear distance (a0) Fig. 15 Nonadiabatic potential energy curves of (6Li+ 7Li)+ as a function of internuclear distances obtained by diagonalizing the matrix for Vαβ: the off-diagonal matrix elements were enlarged by 10 before the diagonalization to make their effect visible. Adapted from Zhang, P., Bodo, E., Dalgarno, A., 2009. Near resonance charge exchange in ionatom collisions of lithium isotopes. J. Phys. Chem. A 113 (52), 15085–15091. doi:10.1021/ jp905184a. PMID: 19746948. 102 R. Cot^ e
N‘(R) are function matrices of the Riccati–Bessel and Riccati–Neumann functions. We note that in the case of collisions of the same isotopes, all the off-diagonal couplings in Eq. (65) disappear, and we recover our previ- π ous expression σ ∞ 2‘ +1 sin 2 ηg ηu ) for resonant charge ch 2 ‘ 0 ‘ ‘ ¼ k ¼ ð Þ ½ À exchange. X
6.2 Results for a Few Examples Only a few systems involving isotopes of H (Bodo et al., 2008), Li (Zhang et al., 2009a), and Be (Zhang et al., 2011), have been theoretically investi- gated. All of those systems exhibit similar features. In Fig. 16, we show the elastic and charge transfer cross sections computed using Eq. (67). These panels show the elastic cross sections for D+ +H, 7Li++6Li, and 10Be++9Be, and the charge transfer cross sections. We distinguish between “quenching” for the inelastic exothermic process (D+ +H D+H+ , 7Li++6Li 7Li ! ! +6Li+, and 10Be++9Be 10Be+9Be+), and “excitation” for the reverse ! inelastic endothermic process. As the scattering energy decreases, the elastic cross section changes from 1/3 the EÀ behavior to a constant, as in the case of identical isotopes. The 1/2 quenching cross section changes from the EÀ behavior at higher energies to the expected 1/k Wigner threshold law at ultralow energy. The excitation
107 108 7Li+ + 6Li → 7Li + 6Li+ (quenching) 9Be + 10Be+ → 9Be + 10Be+ (elastic) D+ + H → D + H+ (quenching) 108 106 106 105
) 6
2 10 0 9Be + 10Be+ → 9Be+ + 10Be (quenching) a 4 10 + + D + H → D + H (elastic) 7Li+ + 6Li → 7Li+ + 6Li (elastic) 104 103 104
102 2
Cross section ( 10 102 101
0 10 + + 0 D + H → D + H (excitation) 0 6Li+ + 7Li → 7Li+ + 6Li (excitation) 10 10 9Be+ + 10Be→ 9Be + 10Be+ (excitation) 10−1 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 10−13 10−11 10−9 10−7 10−5 10−3 10−1 Collision energy (eV) Collision energy (eV) Collision energy (eV) Fig. 16 Cross sections for elastic, quenching, and excitation processes in H, Li, and Be samples. Adapted from Bodo, E., Zhang, P., Dalgarno, A., 2008. Ultra-cold ion-atom colli- sions: near resonant charge exchange. New J. Phys. 10 (3), 033024. doi:10.1088/1367-2630/ 10/3/033024 for H; Zhang, P., Bodo, E., Dalgarno, A., 2009. Near resonance charge exchange in ionatom collisions of lithium isotopes. J. Phys. Chem. A 113 (52), 15085–15091. doi:10.1021/jp905184a. PMID: 19746948 for Li; and Zhang, P., Dalgarno, A., Co^te, R., Bodo, E., 2011. Charge exchange in collisions of beryllium with its ion. Phys. Chem. Chem. Phys. 13, 19026–19035 for Be. Ultracold Hybrid Atom–Ion Systems 103
Table 2 Parameters for scattering with Different Isotopes and for Identical Isotopes 2 + 2 + Identical ag ( Σg ) au ( Σu ) 25 System ΔE (10 eV) α (a.u.) β (a.u.) isotopes (a.u.) (a.u.) D+ +H D+H+ 369.47 +15.3 +30.6 H+ +H 28.8 +725.2 ! À (42.842 K) D+ +D +485.7 95.6 À 7Li++6Li 7Li+6Li+ 7.47 +286 +145 6Li++6Li 918 1425 ! À À (0.867 K) 7Li++7Li +14.337 +1262 10Be++9Be 10Be+9Be+ 5.77 981.4 +50.3 9Be++9Be 941.8 + 569.4 ! À À (0.670 K) 10Be++10Be 229.2 +1237.0 À The equivalent of ΔE in Kelvin is given in parenthesis below its value in eV. Source: From Bodo, E., Zhang, P., Dalgarno, A., 2008. Ultra-cold ion-atom collisions: near resonant charge exchange. New J. Phys. 10 (3), 033024. doi:10.1088/1367-2630/ 10/3/033024 for H; Zhang, P., Bodo, E., Dalgarno, A., 2009. Near resonance charge exchange in ionatom collisions of lithium isotopes. J. Phys. Chem. A 113 (52), 15085–15091. doi:10.1021/jp905184a. PMID: 19746948 for Li; and Zhang, P., Dalgarno, A., Cot^ e, R., Bodo, E., 2011. Charge exchange in collisions of beryllium with its ion. Phys. Chem. Chem. Phys. 13, 19026–19035 for Be. cross section, which is related to the quenching by microscopic reversibility, is basically equal to the quenching until the energy approaches the value of ΔE, at which point it rapidly decreases to zero. At higher energies, both inelastic processes follow the a lnE b 2 as described by Eq. (30). In the ultracold limit, the cross sectionsð can beÀ expressedÞ in term of a complex scat- tering length a α iβ as described in Eqs. (58) and (59): for the case of ¼ À identical isotopes, the scattering lengths are real, as seen in the expressions (19) for the cross sections. In Table 2, we give values for ΔE and the scat- tering lengths. As seen in Fig. 16, the quenching and excitation cross sections start to depart from each other at a scattering energy roughly one order of magni- tude larger than ΔE, a value much higher than the ultracold regime. At ener- gies corresponding to mK or μK, the excitation process is closed, and only elastic scattering can take place if the colliding partners approach each other following the lower nonadiabatic potential energy curve, for example the curve corresponding to 7Li+6Li+ in Fig. 15. It might be possible to tune or control this process at ultracold temperatures using Feshbach resonances in a manner similar to ultracold neutral atomic samples.
6.3 Tuning Scattering with Magnetic Fields To bridge the large energy gap between quenching and excitation processes, which arises from the isotope shift ΔE between the two asymptotes of 104 R. Cot^ e
+ + X +X0 and X+X0 , one can use Feshbach resonances as introduced in Section 5. The particles being distinguishable, the scattering channels are not sym- metrized. In general, an arrangement label λ could be employed to identify + the scattering states, with λ 1 corresponding to X +X0, and λ 2 to + ¼ ¼ X+X0 , with the representation ‘m : λ,f1m1,f2m2 . This is useful if the spin j i 7 9 states of both isotopes have the same quantum numbers, such as Be and Be (both have the same nuclear spin i 3). In other cases, λ can be omitted, ¼ 2 since ( fmf) identifies the isotope uniquely as well as the arrangement. This is the case, for example, for 9Be and 10Be with i 3 and 0, respectively: ¼ 2 the spin f uniquely identifies 9Be ( f 3), 9Be+ ( f 1 or 2), 10Be ( f 0), ¼ 2 ¼ ¼ or 10Be+ ( f 1), so that the order in f m ,f m uniquely identifies the ¼ 2 j 1 1 2 2i arrangement 9Be++10Be or 9Be+10Be+ (see also Fig. 11).
The Hamiltonian is given by Eq. (52), with both H^ hf and H^ B given as before by Eqs. (63) and (64), respectively. The molecular component H^ mol does not account for the exchange of identical nuclei anymore as in Eq. (62) (the g u symmetry is broken). Instead, for each arrangement λ, there will be À a manifold corresponding to the f1m1,f2m2 hyperfine states. Since the scat- tering conserves the total M, eachj manifoldi contains M m +m compo- ¼ 1 2 nents in the s-wave regime (with ‘ m 0). For each manifold, the 2 2 system in Eqs. (65) and (66) which¼ couples¼ the two arrangements λÂ1 and 2, will be replaced by a 2M 2M system of coupled equations: I is¼ just the unit matrix, k2 is a block diagonal with the first block a M M diagonal  matrix with element k2 corresponding to arrangement λ 1 and the second 1 ¼ 2 @ M M diagonal matrix with element k (for λ 2). Since Gαβ α β  2 ¼ ¼h j@Rj i couples the BO states α and β , G is made of four M M diagonal matri- j i j i  ces, with elements G11, G12, G21 G12, and G22, respectively. In addition to corrections to the BO curves¼À due to the isotopic shift, C includes the effect of both H^ and H^ through V αβ: both mix the f m ,f m within hf B j 1 1 2 2i the manifold of the specific arrangement λ and change the threshold energy as a function of the magnetic field B.Fordifferentarrangements,theM M  matrix resulting from V αβ is not affected by H^ hf and H^ B. Basically, by cou- pling the states within the M M block diagonal matrix via V αα, the hyper-  fine and Zeeman interactions also affect the coupling between arrangements and can give rise to magnetically tunable Feshbach resonances. Note that a Ultracold Hybrid Atom–Ion Systems 105 treatment based on multichannel quantum-defect theory also predicts such resonances (Li et al., 2014). To illustrate these points, we consider two isotopes of Be, namely 9Be (i 3) and 10Be (i 0). Their hyperfine and Zeeman energies are shown ¼ 2 ¼ in Fig. 11 for both neutrals and ions. We compute the excitation process 9Be+ +10Be 9Be+10Be+, which is closed at ultracold temperatures. The ! BO curves and couplings U V αβ and G are shown in the left panel αβ αβ of Fig. 17. We show the real and imaginary parts (α and β) of the scattering length in Fig. 18 at an energy corresponding to 10 μK for the entrance chan- nel {8,1} corresponding to 9Be+ in its state 8 f 1,m 1 (counting j ij ¼ ¼À10 i from the lowest to the highest energy in Fig. 11) and Be in its state 1 0,0 (the only state as show in Fig. 11). The total M 1 can be real- jizedij onlyi in the exit channel {1,2} of the excitation process¼À 9Be+10Be+, ie, 9Be in 1 3 , 3 and 10Be+ in 2 1 ,1 . The only other combina- j ij2 À 2i j ij2 2i tion with M 1 corresponds to an hyperfine state change into {2,1} within the initial¼À arrangement 9Be+ +10Be with 9Be+ in 2 2, 1 and 10Be in 1 0,0 . Fig. 18 shows that the reaction rate coefficient,j ij À propor-i tional tojβiis basicallyj i zero except at the resonant magnetic field, where it can reach sizable values. This panel also illustrates the sensitivity of the results to the details of the potentials, as modified slightly in the middle panel. A variation within 10% of the parameters A and B in the exchange energy affects the position of the barrier (see Fig. 17) and hence of the coupling and position of the resonance.
0.06 0.3 ab6 Ugu × 10 0.2 0.04 0.1 U × 104 d No shift gg 0.04 (A–10%) 0.02 0.0 2 + U × 104 B Σ (A+10%) 2Σ + uu 0.02 g −0.1 4 6 8 10 12 14 16 18 0.00 0
2 + Σu −0.02 −0.02 (Hartree) V 0.3 Energy (a.u.) −0.04 X2 + −0.04 0.2 c Σ −0.06 0.1 G × 104 −0.06 gu 0.08 0.0 − 5 10 15 20 −0.1 Internuclear distance (a ) −0.08 4 6 8 10 12 14 16 18 0
2 4 6 8 10 12 14 16 18 20 Separation (a.u.)
Fig. 17 Be results. (a) BO curves, couplings Uαβ V αβ (b) and Gαβ (c) between the BO 2 + curves. (d) variation of the barrier in the Σg BO state. 106 R. Cot^ e
−42 ) 0 a ) (
a −43
−44 0 ) Re( 0 a
) ( Original a −0.01 ∆A = −9%
Im( ∆B = +6% ∆A = +10% ∆B = 0 −0.02 0 1000 2000 3000 Magnetic field (G) Fig. 18 Scattering length (top: real part; bottom: imaginary part) in the charge transfer 8,1 1,2 . The resonance is moving as the barrier in 2Σ + is varied from the original f g!f g g by ΔA and ΔB.
7. CHARGES IN A BEC The previous sections dealt with the behavior of ultracold samples containing ions in which binary collisions govern the dynamics of the sys- tem. A natural extension to the case of degenerate quantum gases, where many-body phenomena prevail, is presented here, first for the case of an ion in a BEC, and then for Rydberg electrons in a BEC. These two types of systems are actively being investigated by various experimental groups.
7.1 Ion in a BEC We explore the behavior of a dilute atomic condensate doped with ionic impurities. Although there is an analogy to early studies involving electron bubbles and ion “snowballs” in superfluid helium (Benderskii et al., 1999; Rosenblit and Jortner, 1995), a dilute weakly interacting atomic BEC favors the formation of large mesoscopic ions in highly excited and metastable states (Cot^ e et al., 2002). We are interested in the limit T 0, and consider a homogeneous BEC with the neutral gas being the parent! atom of the doping ion. The results discussed here are also valid if the ion is from a different species or isotope with negligible charge transfer (Cot^ e et al., 2002). In the absence of impu- rities, a homogeneous BEC is described by the Hamiltonian Ultracold Hybrid Atom–Ion Systems 107
2 2 ℏ k { uB { { HBEC ckck + ckcpcqck + p q: (68) À ¼ k 2mB 2ΩV kpq X X
The coupling between the atoms of mass mB with scattering length aB is 2 † uB 4πℏ aB=mB, ΩV is the quantization volume, and ck (ck) is the creation (annihilation)¼ operator of bosonic atoms with momentum k. If most atoms occupy the ground state (k 0), c{ and c can be replaced by pN and ¼ 0 0 0 Eq. (68) expanded in decreasing order of N0, with the number of atoms { ffiffiffiffiffiffi given by N N0 + k 0ckck. By keeping terms of the order pN0 or ¼ 6¼ higher, HBEC can be diagonalized via the Bogoliubov transformation † † P † ffiffiffiffiffiffi cq uqbq +vqb q, where the Bogoliubov operator bq (bq) creates (annihilates) ¼ À a quasi-particle (or phonon) of momentum q when applied to the ground state 0 : b† 0 q . The resulting effective Hamiltonian is j i qj i¼j i
{ BEC ℏωq bqbq +1=2 , (69) H ¼ q ð Þ X 2 1=2 2 2 where ℏωq E +2uBρ Eq , with Eq ℏ q =2mB and the BEC number ¼ð q B Þ ¼ density ρB N/ΩV. The excitation energy can be rewritten in term of the sound velocity¼ s which is related to the chemical potential by μ m s2 4πℏ2ρ a =m u ρ , ie, ℏω ℏqs 1+ ℏq=2m s 2. c ¼ B ¼ B B B ¼ B B q ¼ ð B Þ The presence of an ion polarizes a nearby atomq (separatedffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by a distance R) and the two interact via a polarization potential behaving asymptotically 4 as C4/R , as discussed in Eq. (4). In this system, two types of scattering pro- cessesÀ are possible: elastic and super-elastic collisions. For both pure elastic and resonant charge transfer, the energy of the colliding partners is not chan- ged, and we refer to them as elastic collisions here: according to Wigner’s threshold laws (Wigner, 1948), the rate of all elastic processes vanishes as T 0. Super-elastic scattering, where kinetic energy is released, corre- sponds! to an inelastic process where one BEC atom is captured by the ion and kinetic energy is released by the emission of phonons (excitations). Fig. 19a depicts the process: BEC atoms are accelerated by the ionic field, and after the collision, one of the neutral atoms is captured by the polariza- tion potential, and the kinetic energy released is shared by the remaining free condensate atoms and the newly formed molecular ion. We note that con- trary to the slow-down of impurities in BECs, where momentum-energy conservation prevents phonon radiation below some critical velocity (Timmermans, 1998), the capture of atoms corresponds to free-bound 108 R. Cot^ e
2 a 4π µc = ρBaB mB
R c b a u C4 −R−4 c R c
a g a
a u a
a gcR a x x R c a
x Fig. 19 (a) Diagrams of atom capture by an ion: the spontaneous capture in level v is cap followed phonon emission (with corresponding rates). (b) Wv as a function of ξ for 2 + 2 + + various values of av corresponding to the states Σ , and Σ of Na , and av Rc g u 2 for both Na and Rb. Adapted from Cot^ e, R., Kharchenko, V., Lukin, M.D., 2002. Mesoscopic molecular ions in Bose-Einstein condensates. Phys. Rev. Lett. 89, 093001. doi:10.1103/ PhysRevLett.89.093001. transitions, and does not suffer from this restriction: phonon emission takes place at any velocity. The number of atoms Nv captures in the bound level v (see Fig. 19a) can be described by a kinetic equation dN v W cap N +1 W down + W up N , (70) dt ¼ v ð v ÞÀð v v Þ v cap down up where W v is the capture rate from the condensate, and W v and W v are the loss rates to more deeply bound states and back to the condensate, respectively. The capture (proportional to Nv +1) is Bose enhanced, while cap the depletion of the level (proportional to Nv) is not. We can obtain W v from the emission rate of phonons with momentum q and energy ℏωq (Kittel, 1987; Timmermans, 1998)
2 2π μc wemis: q S q I q nq +1 δ Δε ℏωq , (71) ð Þ¼ ℏ ρBΩV ð Þ ð Þð Þ ð À Þ where S q ℏq2=2m ω is the static structure factor (for free–free transi- ð Þ¼ B q tions; Nozieres and Pines, 1999; Timmermans, 1998), and nq the phonon occupation number. Here, I(q) accounts for transitions from a continuum state, described by the single particle wavefunction Ψ0(R) for the N BEC atoms, to a bound state Ψv(R), with energy difference Ultracold Hybrid Atom–Ion Systems 109
Δε ε0 εv ℏωq. Assuming a zero-temperature infinite homogeneous ¼ À ¼ R=a 1 1 eÀ v system, we set ε0 0, and write Ψ0 R and Ψv R , ¼ ð Þ¼pV ð Þ¼p2πav R 2 2 where the binding energy corresponding to Ψv(R) is εv ℏ =2μav , giving 2 2 ffiffiffiffi ¼À ffiffiffiffiffiffiffiffiffi Δε ℏ =2μa ℏωq (μ: system reduced mass). In an isotropic Bose gas, we ¼ v ¼ 0 find 2 3 3 * iq! !r 8πav ρB I !q N d rΨ !r eÀ Á Ψ0 !r : (72) ð Þ¼ v ð Þ ð Þ ¼ 1+q2a2 2 Z v ð Þ cap Wv is obtained by integrating over all possible phonon states, 3 3 2 ie, d qwemis: !q V = 2π , and defining ξ μ =Δε 8πρ a aBμ=mB, it ð Þ ð Þ c ¼ B v is given by R 3=2 2 3=2 2 cap μc mB ξ 1+ξ ξ mB 2 À Wv 4 ð À Þ 1+ 1+ξ ξ nq0 +1 : ¼ ℏ μ 1+ξ2 μ À ð Þ pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffi (73) We find different physical conditions for the two limiting cases of ξ: ξ ∞ ! implies q0 0, ie, a phonon-like regime, and ξ 0 implies dilute con- ! ! cap ditions with a binary collision regime. Fig. 19b illustrates W v as a function cap 2 2 2 cap 3 of ξ, with Wv ∝ρBav aB for small ξ, and Wv ∝1=av pρBaB for large ξ. The 2 binary regime (ξ ≪ 1) is proportional to ρB as expected: the capture process ffiffiffiffiffiffiffiffiffi1=2 is then reduced to three-body recombination. The ρÀB dependence for ξ ≫ 1 clearly indicates the dominant contribution of phonon assisted tran- cap sitions (Kittel, 1987). Fig. 19b also shows Wv for Na (aB 50 a.u.) and for 14 3 Rb (a 100 a.u.), assuming m /μ 1 and ρ 10 cmÀ and a R , B B B v c where Rc p2μC4=ℏ is the van der Waals length. For Na, ξ 0.066 and ¼ cap 1 neglecting nq , we find W v 600 sÀ , ie, roughly 600 atoms captured 0 ffiffiffiffiffiffiffiffiffiffiffi by the ion, with another 600 emitted as phonons (excited out of the con- densate) per second: this represents a sizable transfer mechanism that should be observable. As noted in Cot^ e et al. (2002), the capture rate into more cap cap deeply bound states is smaller, with W v 1 about ten times smaller than W v , and much smaller for v 2 and deeperÀ levels, and so that spontaneous cap- ture occurs mostly in theÀ uppermost level. up down The two other rates W v and W v appearing in Eq. (70) can be shown to be small. Since kBT ≪Δε at ultralow temperatures, there are no thermal phonons with sufficient energy to promote bound atoms to the free condensate state, and phonons emitted from the capture process 110 R. Cot^ e can be neglected since they move away from the ion. In addition, transi- tions from bound levels to the continuum (into the condensate) are not allowed because of energy conservation, and hence W up 0. Collisions v of atoms within the upper bound level (or condensate atoms with the down trapped ones) may trigger the decay to a lower level. W v ) is inversely proportional to the binding energy of the deeper level (Cot^ e et al., 2002), cap and hence of the same order as W v 1, and is at least one order of magni- cap À tude smaller than W v . The repulsive meanfield energy between the Nv atoms trapped within the uppermost bound level v will grow as Nv increases, eventually “pushing” the energy level up. Using the Gross–Pitaevskii equation (GPE), the binding energy of the uppermost level v, for Nv ≫ 1, is approx- 2/3 imately εv(Nv) (mBav/6μaBNv) εv (Cot^ e et al., 2002). However, thermal ’ max fluctuations will limit the maximum number Nv around the ion. Assum- ing their energy ( kBT) equal to εv(Nv) , thermal equilibrium is reached, so that as many atoms get kicked outj of v asj there are captured into v. At that point, W up W cap, N max is found from ε N max k T, and ξ reaches its v ¼ v v vð v Þ B equilibrium value ξ μ /k T. For example, a BEC of Na atoms with eq c B a R 2000a and T 100 nK gives N max 600 ie, roughly 600 atoms v c 0 v would be trapped around the ion at equilibrium. Fig. 19b shows the trajec- tory of the system as a function of ξ for various initial values of av, for Na and Rb BECs: as ξ grows from its initial to its final equilibrium value, the capture cap rate Wv passes through a maximum before reaching its final value, where the system has reached thermal equilibrium with the rates “in” and “out” of the uppermost level being equal. Experimental probe of the dynamics of an ion immersed in a BEC is extremely difficult, requiring trapping both the ion and the neutral atoms. As mentioned in the introduction, this can be realized in hybrid traps com- bining, for example, a Paul trap for the ion and an optical trap for the BEC. This approach has been employed by Zipkes et al. (2010a) to study scattering processes of 174Yb+ in a BEC of 87Rb. However, the micromotion of the ion in the Paul trap prevents reaching the ultracold temperatures necessary to investigate the capture of atoms by the ion and the formation of a metastable molecular ionic cluster. Similar studies have been performed by Schmid et al. (2010) with Ba+ or Rb+ in a BEC of Rb. Recently, trapping of a single Ba+ ion in an optical trap was demonstrated (Enderlein et al., 2012; Huber et al., 2014), which avoid micromotion and allows to reach much lower temperatures. The same group is planning to overlap this single ion with a Rb BEC. Fig. 20 depicts the setups and concepts. Ultracold Hybrid Atom–Ion Systems 111
x a z x(t) RF micro- 2 RF b c 1.0 motio n z 0.5 2
0 s x RF )/
t 0.0 ( micro- x −0.5 motion −1.0 BEC
0 2 4 6 8 10 12 14 16 18 20 t (10−8s) Fig. 20 (a) left: Setups combining radiofrequency (RF) (gold rods and rings) and optical (green (light gray in the print version)) traps for atoms (red (gray in the print version)) and ions (blue (dark gray in the print version)) experience cooling limitations. Right: The RF-field forces the ion to quiver at ΩRF when displaced from the RF-node. (b) The atom–ion interaction leads to a displacement from the RF-node, limiting to lowest achievable mutual kinetic energy. (c) Confining atoms (BEC) and ions in a common optical potential removes RF micro-motion and allows to reach the quantum regime. Provided by T. Sch€atz.
7.2 Rydberg Electron in a BEC A different type of interaction is provided if one excites an atom from the BEC into a Rydberg state. The system, though still neutral, consists of a pos- itive ion core essentially at rest and a fast moving negative charge (the elec- tron) scattering off the BEC atoms. This provides a different type of behavior for a charge interacting with a BEC, leading to phonon-mediated interac- tion between Rydberg atoms, as sketch in Fig. 21a. Here, we consider that only the electron collides with the BEC atoms. We first recall the result obtained by Fermi (1934) on the interaction between a quasi-free electron at x and a ground state atom at r: it can be approximated at low scattering energies by a contact interaction of the form
2 2πℏ 3 Vs x,r As kr δðÞx r , (74) ðÞ¼ me ½ðÞ ðÞÀ
1 where A k kÀ tanη k is an energy-dependent s-wave scattering sðÞ¼À sð Þ length defined from the electron–atom scattering phase shift ηs(k). As pointed out in Wang et al. (2015), higher-partial waves must be included for quantitative results. For simplicity, we limit our discussion to the s-wave case and refer the readers to the original work of Greene et al. (2000) and Hamilton et al. (2002) for the treatment of the p-wave contribution. The local wave number k(r) is
2 2 2 ℏ k r y e ð Þ R 2 + , (75) 2me ¼À n δ 4πE0r ð À ‘e Þ 112 R. Cot^ e
a
b
c x (103 a.u.) 30 40 50 2 1
aB=5k a.u. 0.8 1.5 ) 1 − 0.6
8k a.u. a.u.) 3 cm − 6 1 ) (10
) (10 10k a.u. x r 0.4 ( ( e P P ∆ 20k a.u. 0.5 0.2
0 0 30 40 50 r (103 a.u.) 2 Fig. 21 (a) Two Rydberg atoms in a BEC exchange phonons: Ψe is represented by the 2 j j surface inside each sphere. (b) Surface showing Ψe along with the interaction poten- j j 2 2 tial curve within the s-wave approximation. (c) Comparison of Pe x 4πx Ψe x ðÞ¼ j ð Þj (green filled curve) and ΔPr 4πr2δρ r (solid curves) for a 87Rb BEC at density 13 3 ðÞ¼ ð Þ 3 ρ 2 10 cmÀ for various scattering lengths a (in units of k 10 a ). From Wang, J., B ¼ Â B ¼ 0 Gacesa, M., Cot^ e, R., 2015. Rydberg electrons in a Bose-Einstein condensate.Phys.Rev.Lett. 114, 243003. doi:10.1103/PhysRevLett.114.243003. where is the Rydberg constant, e and m the charge and mass, respec- Ry e tively, of the electron with angular momentum ‘e and quantum defect
δ‘e . For low-‘e states, Eq. (74) leads to an effective interaction between Rydberg and ground state atoms Ultracold Hybrid Atom–Ion Systems 113
2 2πℏ As kr 2 VRyd r ½ðÞ Ψe r , (76) ð Þ me jjð Þ where Ψe(r) is the Rydberg electronic wave function at the position of 2 the ground state atom. Both Ψe(r) and the corresponding oscillatory j j potential V (r) are depicted in Fig. 21b for a Rb Rydberg ns (‘ 0) state. Ryd e ¼ For As < 0, VRyd(r) is attractive and leads to the formation of ultra- long-range Rydberg molecules ( Greene et al., 2000). High-‘e states with negligible δ‘e are nearly degenerate, and their electronic wave functions have complex quantum interference patterns which may support very extended bound “trilobite states.” Many groups have observed trilobite-like states (Anderson et al., 2014; Bellos et al., 2013; Bendkowsky et al., 2009; Krupp et al., 2014; Li et al., 2011). As described in the previous section, HBEC in Eq. (68) can be diagonal- ized via the Bogoliubov transformation, giving the effective Hamiltonian BEC in Eq. (69). Under this transformation, the local density operator H 1 iq r { ^ρ r ΩÀ e Á c cp becomes ðÞ¼ V p,q p + q X
N0 pN0 iq r { ^ρ r + e Á uq + vq bq + b q , (77) ðÞΩV ΩV À ffiffiffiffiffiffi q 0 X6¼ ÀÁ and the Rydberg-BEC interaction H d3r^ρ r V r is INT ¼ ð Þ Rydð Þ R N0 pN0 { HINT V0 + uq + vq bq + b q Vq, (78) ΩV ΩV À ffiffiffiffiffiffi q 0 X6¼ ÀÁ where V d3rV r eiq r is the Fourier transform of V (r). The first q ¼ Rydð Þ Á Ryd and second order corrections to the ground state energy are R E 1 d3rρ V r and ðÞ¼ B Rð Þ R E1 d3rρ V r , and ¼ B Rydð Þ Z r r0 =ξ (79) 2 mB 3 3 eÀj À j E d rd r0ρ VRyd r VRyd r0 , ¼À2πℏ2 B ðÞ r r ðÞ Z j À 0j where we integrated over q after taking the thermodynamic limit 1 3 3 ΩÀ 2π À d q: at this level of approximation, N0 can be replaced V q !ð Þ by N. P R 114 R. Cot^ e
With VRyd(r) from Eq. (76), and assuming an homogeneous BEC with (1) constant ρB, E is the mean-field energy shift given by
1 2 ae 3 2 Eð Þ 2πρ ℏ ,with ae d rAs k r Ψe r : (80) ¼ B m ¼ ½ ð Þj ð Þj e Z Here, ae is an average scattering length, which is equal to As(0) for high n 3 2 Rydberg states, since then As[k(r)] As(0) and ae As 0 d r Ψe r ’ ’ ð Þ j ð Þj ¼ A 0 . Similarly, E(2) involves the Yukawa potential V (s): sð Þ Y R
2 1 3 3 2 2 Eð Þ d rd r0 Ψe r VY r r0 Ψe r0 , 2 j ð Þj ð À Þj ð Þj Z (81) r r0 =ξL 2 eÀj À j where V r r0 Q : Y ðÞ¼ÀÀ r r j À 0j 1/2 The range ξ (16πρ a )À of V equals the BEC healing length, and the L ¼ B B Y 2 2 2 2 “effective charge” Q characterizes VY strength with Q 4πℏ ae ρBmB=me . 2 The term EðÞ, which also appears in studies of self-localization of impu- rities in a BEC (Casteels et al., 2013; Santamore and Timmermans, 2011), is the self-interaction of electrons by a Yukawa potential induced via phonon exchange at two different positions. Since the Rydberg electrons are already localized by strong Coulomb forces with ion cores, the distorted BEC den- sity may reflect the oscillatory nature of Ψe and “image” the Rydberg elec- tron. The perturbed ground state 0 is given, to first order, by j i
pN0 Vq 0 u + v q , (82) 0 q q ℏω j i¼j iÀ ΩV q 0ð Þ q j i ffiffiffiffiffiffiX6¼ and leads to the BEC density distortion
r r0 =ξL mBρB 3 eÀj À j δρ r ^ρ r ρ d r0VRyd r0 , (83) ð Þh ð ÞiÀ B ¼À ℏ2π ð Þ r r Z j À 0j which “averages” VRyd within the range ξL. For ξL not too large, the oscil- latory nature of Ψe can be imaged onto δρ(r)(Karpiuk et al., 2015), but if ξL is larger than the local wavelength of the Rydberg electron, the averaging will erase this signature. Fig. 21c compares the radial probability density 2 2 2 87 Pe x 4πx Ψe x with ΔPr 4πr δρ r for a Rb(160s) Rydberg ðÞ¼ 87 j ð Þj ðÞ¼ 13 ðÞ3 atom in a Rb BEC with ρ 2 10 cmÀ : for larger values of the scat- B ¼  tering length aB, sharper oscillations with overall smaller amplitudes are Ultracold Hybrid Atom–Ion Systems 115 produced. These amplitudes, a few percent of the average BEC density, 2 point to the possibility for in situ imaging of Ψe . We note that an upper 2 j j limit τB mBξ =ℏ for the BEC response time (Santamore and Timmermans, 2011) (a¼ few microseconds here) is much shorter than the radiative lifetime of Rydberg atoms. Recent experiments by Balewski et al. (2013) have successfully excited a single Rydberg electron in a BEC of Rb atoms, by taking advantage of the blockade mechanism (Jaksch et al., 2000; Lukin et al., 2001) which prevents excitation of more than one Rydberg atom within a blockade volume. They observed a loss of BEC atoms due to the interaction with the Rydberg elec- tron, and deformation of the BEC profile. However, fluctuations in the observed variation of the BEC density prevented the imaging of the Rydberg electron wave function (Balewski et al., 2013). The same group used a time-dependent GPE to describes the dynamics of Rydberg excita- tions in a BEC (Karpiuk et al., 2015). Fig. 22 shows results of their treatment. As mentioned above, the averaging of VRyd masks the electron self- interaction for large healing lengths ξL. However, phonon exchange still mediates interactions between Rydberg atoms. To derive this interaction, we consider two Rydberg atoms, located at R1 and R2. After applying the Bogoliubov transformation, their interaction with the BEC atoms is N 0 1 2 { 1 iq R1 2 iq R2 HINT 0 + 0 + uq + vq bq + b q qe Á + qe Á : À ΩV V V q 0ð Þð Þð V V Þ ÀÁX6¼ (84) As in Eq. (78) for a single Rydberg electron, i d3riV r eiq r, where Vq Rydð Þ Á iV (r) describes the interaction of “impurity” i with BEC atoms in coor- Ryd R dinate space. We are interested in the R dependence in the effect of the BEC phonons on the interaction of the Rydberg atoms. Within perturbation theory, the first order correction E(1) gives a constant mean-field energy shift similar to the single Rydberg atom case, and so can be omitted. The second order cor- i i i rection for spherically symmetric interactions (where q q q is V ¼ VÀ ¼ V real) and assuming ρB constant is
iqRcosθq 2 ρB 3 Aq +2Bqe Eð Þ 3 d q , (85) ¼À 2π Eq +2uBρB ð Þ Z where A 1 2 + 2 2, B 1 2 , R R R , and θ is the q ¼ð VqÞ ð VqÞ q ¼ Vq Á Vq ¼j 1 À 2j q angle between R and q. The term containing Aq does not depend on R Fig. 22 Left panel: A tightly focused laser beam (red (gray in the print version)) excites Rydberg atoms in 140S (blue (dark gray in the print version)) and 180D (orange (light gray in the print version)) states. Dashed lines indicates the respective Rydberg blockade radii. Right panel: (a) and (b) calculated orbitals for 180D and 140D Rydberg electrons convolved with a finite imaging resolution of 1 μm. (c) Center part of the BEC density distribution. (d) and (f ) simulated density change caused by a single Rydberg atom without (d) and with atom number shot noise (f ). (e) Density distribution of the BEC after 50 Rydberg atoms have been consecutively excited in the center region. From Karpiuk, T., Brewczyk, M., RzaŻewski, K., Balewski, J.B., Krupp, A.T., Gaj, A., Low,€ R., Hofferberth, S., Pfau, T., 2015. Imaging single Rydberg electrons in a Bose-Einstein condensate. New J. Phys. 17 (5), 053046. http://stacks.iop.org/1367-2630/17/i 5/a 053046. ¼ ¼ Ultracold Hybrid Atom–Ion Systems 117 and contributes a constant energy shift: it can be understood as the self-localizing energy for both Rydberg atoms calculated previously. Together with E(1), these contributions can be omitted in the study of the relative dynamics of Rydberg atoms: only the term containing Bq gives an R-dependent energy shift leading to
iqRcosθq ρB 3 2Bqe U R 3 d q , (86) ð Þ¼À 2π Eq +2uBρB ð Þ Z which can be easily generalized to interactions between any two impurities immersed in a BEC (Bijlsma et al., 2000). Only small q-values contribute for large R due to eiqRcosθq , leading to the asymptotic behavior
2 R=ξL 2 UR Q eÀ =R, where Q ρBmBBq 0=π. Within the s-wave ðÞ!À ¼ approximation, we recover the effective charge Q 2 4πℏ2a2ρ m =m2 as e B B e before. Since phonon-exchange mediates the interaction, it is not surprising to obtain the same Yukawa potential as in Eq. (81). Since the electrons are really the perturbers, Q is inversely proportional to me for Rydberg atoms, giving a much stronger interaction than for more massive impurities of mass 1 mI for which Q ∝mIÀ (Bijlsma et al., 2000). Without a BEC, two Rydberg atoms experience strong long-range interactions, leading to formation of macrodimers (Boisseau et al., 2002) and the excitation blockade (Lukin et al., 2001; Tong et al., 2004); for exam- ple, two ns Rydberg atoms separated by R experience a repulsive van der 6 11 Waals (vdW)+C6/R interaction, where C6∝n (Boisseau et al., 2002; Overstreet et al., 2009; Singer et al., 2005). However, when immersed in a BEC, the exchange of phonons between two Rydberg atoms gives rise to an attractive Yukawa potential. We illustrate these results using two 87 87 13 3 Rb(50s) Rydberg atoms in a Rb BEC with ρ 10 cmÀ , and assume B ¼ that aB is tuned to 10 a.u. (eg, via a Feshbach resonance) to ensure a healing 4 length ξL 3.66 10 a.u. is much larger than the Rydberg atoms. The numerical¼ “trilobite-like” interaction shown in Fig. 23a is constructed using the first-order perturbative model (Fermi, 1934; Omont, 1977) including s and p contributions, with As(0) 16.05a0 (Bendkowsky et al., 2010); the states in the range n 21 72 were¼À included and the resulting Hamiltonian ¼ À diagonalized to obtain the 50s eigenstate. Fig. 23b shows that Bq converges 4 2 3 to a constant Bq 0 2 10 a.u. for a small q, yielding Q 1:54 10À a. ¼   u. Fig. 23c shows how the bare interaction between two Rydberg atoms 6 8 10 given by the vdW potential C6/R +C8/R +C10/R (repulsive solid-red 118 R. Cot^ e
5 25 10 a b Bare c 20 0 0 15 U(R) a.u.) 3 ) (MHz) )(MHz) r
( 10 R (10 ( −5 q −10 Ryd U B V 5 Yukawa −10 0 −20 123456 10−5 10−4 10−3 10−2 50 100 150 200 r (103 a.u.) q (a.u.) R (103 a.u.) 87 Fig. 23 (a) Potential between Rb atoms in ground and 50s states. (b) Bq as a function of q. (c) Interaction between two Rydberg 87Rb(50s) atoms in a BEC (U(R): solid black curve) and in vacuum (bare: solid red (gray in the print version) curve). The Yukawa potential dominates the tail (dash-dotted curve) while the blue (light gray in the print version) filled curve represents the lowest bound state. From Wang, J., Gacesa, M., Co^te, R., 2015. Rydberg electrons in a Bose-Einstein condensate. Phys. Rev. Lett. 114, 243003. doi:10.1103/PhysRevLett.114.243003.
20 26 33 curve with C6 1.074 10 , C8 7.189 10 , and C10 7.162 10 , in a.u. for 87Rb(50¼ s)(Singer et al.,¼À 2005)) is affected by immersing¼ them in a BEC: the U(R) potential (solid black curve) is attractive at large separa- tions, following the Yukawa potential Q 2e R=ξ=R (dash-dotted curve), À À before becoming repulsive at shorter range where the “bare” repulsive vdW interaction dominates the phonon-exchange contribution. In addition, because of the large mass of Rb, the resulting well of depth 17.77 MHz and equilibrium separation 60,000 a.u. (much larger than Àthe 5000 a.u. extension of the “trilobite-like” potentials) can support bound levels; the three lowest are at about 17.64 MHz, 17.56 MHz and 17.47 MHz. The ground state wave functionÀ with a spatialÀ width about 2000À a.u. is also shown in Fig. 23c. These results show how phonon-exchange modifies an otherwise repul- sive interaction into a potential well capable of binding two Rydberg atoms. Interesting phenomena could be investigated using Rydberg electrons in BECs. For example, because these wells are easily controlled by tuning aB, it would be possible to generate “synthetic” Coulomb potentials between heavy neutral atoms as aB 0. Studies of Rydberg atoms immersed in BECs open promising avenues! of research, such as bi- and multipolaron physics (Casteels et al., 2013) or “co-self-localization” (Santamore and Timmermans, 2011), possible Rydberg crystallization (Roberts and Rica, 2009), or phase diagram of Yukawa bosons (Osychenko et al., 2012). They also offer the opportunity to investigate systems where electron–phonon coupling plays a crucial role under conditions different from the standard BCS physics. Ultracold Hybrid Atom–Ion Systems 119
8. CONCLUSIONS The field of ultracold samples doped with charged particles, though still in its infancy, is making rapid progress. Whenever very different tech- niques need to be merged, such as in the case of Paul traps for ions and optical traps for atoms, unsuspected difficulties are encountered. Since the early proposals to cool ions with neutral atoms (Cot^ e and Dalgarno, 2000) and the suggestion to build hybrid traps (Makarov et al., 2003), many experi- mental and theoretical groups have joined the effort to investigate these sys- tems. New techniques and ideas are now coming online, such as optically trapping an ion and overlapping it with a BEC, which promise to help reaching the energy and temperature regimes where the dynamics is not dic- tated by binary collisions only. In these new regimes, more complex dynam- ics involving many-body phenomena may rule the behavior of those systems. The possibility of using Rydberg electrons in BECs, a very different path to obtain charges in an ultracold sample, also opens the way to study electron–phonon coupling with the degree of control attainable in AMO systems. It is impossible to predict where ultracold charged samples will make its next contribution, but it is clearly a nascent field with enormous potential for new and exciting results.
ACKNOWLEDGMENTS The author would like to thank S. Banerjee, M. Gacesa, I. Simbotin, J. Stanojevic, as well as W.W. Smith, V. Kharchenko, and J. Montgomery. Support for the National Science Foundation and the Army Research Office is gratefully acknowledged.
REFERENCES Altmeyer, A., Riedl, S., Kohstall, C., Wright, M.J., Geursen, R., Bartenstein, M., Chin, C., Denschlag, J.H., Grimm, R., 2007. Precision measurements of collective oscillations in the BEC-BCS crossover. Phys. Rev. Lett. 98, 040401. http://dx.doi.org/10.1103/ PhysRevLett.98.040401. Altmeyer, A., Riedl, S., Wright, M.J., Kohstall, C., Denschlag, J.H., Grimm, R., 2007. Dynamics of a strongly interacting Fermi gas: the radial quadrupole mode. Phys. Rev. A 76, 033610. http://dx.doi.org/10.1103/PhysRevA.76.033610. Anderson, W.R., Veale, J.R., Gallagher, T.F., 1998. Resonant dipole-dipole energy transfer in a nearly frozen Rydberg gas. Phys. Rev. Lett. 80, 249–252. http://dx.doi.org/ 10.1103/PhysRevLett.80.249. Anderson, W.R., Robinson, M.P., Martin, J.D.D., Gallagher, T.F., 2002. Dephasing of res- onant energy transfer in a cold Rydberg gas. Phys. Rev. A 65, 063404. http://dx.doi. org/10.1103/PhysRevA.65.063404. 120 R. Cot^ e
Anderson, D.A., Miller, S.A., Raithel, G., 2014. Photoassociation of long-range nD Rydberg molecules. Phys. Rev. Lett. 112, 163201. http://dx.doi.org/10.1103/ PhysRevLett.112.163201. Babb, J.F., Dalgarno, A., 1991. Electron-nuclear coupling in the hyperfine structure of the hydrogen molecular ion. Phys. Rev. Lett. 66, 880–882. http://dx.doi.org/10.1103/ PhysRevLett.66.880. Balewski, J.B., Krupp, A.T., Gaj, A., Peter, D., Buchler,€ H.P., Low,€ R., Hofferberth, S., Pfau, T., 2013. Coupling a single electron to a Bose-Einstein condensate. Nature (London) 502, 664–667. Banerjee, S., Byrd, J.N., Cot^ e, R., Michels, H.H., Montgomery Jr., J.A., 2010. Ab initio 2 + 2 + + potential curves for X Σu the and B Σg states of Be2 : existence of a double minimum. Chem. Phys. Lett. 0009-2614496, 208–211. http://dx.doi.org/10.1016/j.cplett. 2010.07.039. Banerjee, S., Montgomery, J.A., Byrd, J.N., Michels, H.H., Cot^ e, R., 2012. Ab initio poten- 2 + 2 2 + + tial curves for the X Σu ,A πu and B Σg states of Ca2 . Chem. Phys. Lett. 542, 138–142. http://dx.doi.org/10.1016/j.cplett.2012.06.011. Bardsley, J.N., Holstein, T., Junker, B.R., Sinha, S., 1975. Calculations of ion-atom inter- actions relating to resonant charge-transfer collisions. Phys. Rev. A 11, 1911–1920. http://dx.doi.org/10.1103/PhysRevA.11.1911. Bellos, M.A., Carollo, R., Banerjee, J., Eyler, E.E., Gould, P.L., Stwalley, W.C., 2013. Exci- tation of weakly bound molecules to trilobitelike Rydberg states. Phys. Rev. Lett. 111, 053001. http://dx.doi.org/10.1103/PhysRevLett.111.053001. Benderskii, A.V., Zadoyan, R., Schwentner, N., Apkarian, V.A., 1999. Photodynamics in superfluid helium: femtosecond laser-induced ionization, charge recombination, and preparation of molecular Rydberg states. J. Chem. Phys. 110 (3), 1542–1557. http:// dx.doi.org/10.1063/1.477796. Bendkowsky, V., Butscher, B., Nipper, J., Shaffer, J.P., Low,€ R., Pfau, T., 2009. Observation of ultralong-range Rydberg molecules. Nature (London) 458, 1005–1008. Bendkowsky, V., Butscher, B., Nipper, J., Balewski, J.B., Shaffer, J.P., Low,€ R., Pfau, T., Li, W., Stanojevic, J., Pohl, T., Rost, J.M., 2010. Rydberg trimers and excited dimers bound by internal quantum reflection. Phys. Rev. Lett. 105, 163201. http://dx.doi.org/ 10.1103/PhysRevLett.105.163201. Bijlsma, M.J., Heringa, B.A., Stoof, H.T.C., 2000. Phonon exchange in dilute Fermi-Bose mixtures: tailoring the Fermi-Fermi interaction. Phys. Rev. A 61, 053601. http://dx.doi. org/10.1103/PhysRevA.61.053601. Bodo, E., Zhang, P., Dalgarno, A., 2008. Ultra-cold ion-atom collisions: near resonant charge exchange. New J. Phys. 10 (3). http://dx.doi.org/10.1088/1367-2630/10/ 3/033024. 033024. Boisseau, C., Simbotin, I., Cot^ e, R., 2002. Macrodimers: ultralong range Rydberg mole- cules. Phys. Rev. Lett. 88, 133004. Brown, J.M., Carrington, A., 2003. Rotational Spectroscopy of Diatomic Molecules. Cambridge Molecular Science, Cambridge University Press, Cambridge, UK. ISBN 9780521530781. Casteels, W., Tempere, J., Devreese, J.T., 2013. Bipolarons and multipolarons consisting of impurity atoms in a Bose-Einstein condensate. Phys. Rev. A 88, 013613. http://dx.doi. org/10.1103/PhysRevA.88.013613. Cetina, M., Grier, A.T., Vuletic´, V., 2012. Micromotion-induced limit to atom-ion sympa- thetic cooling in Paul traps. Phys. Rev. Lett. 109 (25). http://dx.doi.org/10.1103/Phy- sRevLett.109.253201. 253201. Chapman, S., Cowling, T.G., 1970. The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Ultracold Hybrid Atom–Ion Systems 121
Gases. Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK. ISBN 9780521408448. Chin, C., Grimm, R., Julienne, P., Tiesinga, E., 2010. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286. http://dx.doi.org/10.1103/RevModPhys. 82.1225. Ciampini, D., Anderlini, M., Muller,€ J.H., Fuso, F., Morsch, O., Thomsen, J.W., Arimondo, E., 2002. Photoionization of ultracold and Bose-Einstein-condensed Rb atoms. Phys. Rev. A 66, 043409. http://dx.doi.org/10.1103/PhysRevA.66.043409. Cot^ e, R., 2000. From classical mobility to hopping conductivity: charge hopping in an ultracold gas. Phys. Rev. Lett. 85, 5316–5319. http://dx.doi.org/10.1103/ PhysRevLett.85.5316. Cot^ e, R., 2000. From classical mobility to hopping conductivity: charge hopping in an ultracold gas. Phys. Rev. Lett. 85, 5316–5319. http://dx.doi.org/10.1103/ PhysRevLett.85.5316. Cot^ e, R., Dalgarno, A., 2000. Ultracold atom-ion collisions. Phys. Rev. A 62, 012709. http://dx.doi.org/10.1103/PhysRevA.62.012709. Cot^ e, R., Kharchenko, V., Lukin, M.D., 2002. Mesoscopic molecular ions in Bose-Einstein condensates. Phys. Rev. Lett. 89, 093001. http://dx.doi.org/10.1103/PhysRevLett. 89.093001. da Silva Jr., H., Raoult, M., Aymar, M., Dulieu, O., 2015. Formation of molecular ions by radiative association of cold trapped atoms and ions. New J. Phys. 17 (4), 045015. http:// stacks.iop.org/1367-2630/17/i 4/a 045015. Dalfovo, F., Giorgini, S., Pitaevskii,¼ L.P.,¼ Stringari, S., 1999. Theory of Bose-Einstein con- densation in trapped gases. Rev. Mod. Phys. 71, 463–512. http://dx.doi.org/10.1103/ RevModPhys.71.463. Dalgarno, A., 1958. The mobilities of ions in their parent gases. Phil. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 0080-4614250 (982), 426–439. http://dx.doi.org/10.1098/ rsta.1958.0003. http://rsta.royalsocietypublishing.org/content/250/982/426. Dalgarno, A., Williams, A., 1958. The second approximation to the mobilities of ions in gases. Proc. Phys. Soc. 72 (2), 274. http://stacks.iop.org/0370-1328/72/i 2/a 416. Dalgarno, A., McDowell, M.R.C., Williams, A., 1958. The mobilities of ions in¼ unlike¼ gases. Phil. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 0080-4614250 (982), 411–425. http://dx.doi.org/10.1098/rsta.1958.0002. http://rsta.royalsocietypublishing.org/ content/ 250/982/411. Damburg, R.J., Propin, R.K., 1968. On asymptotic expansions of electronic terms of the + molecular ion H2 . J. Phys. B Atom. Mol. Phys. 1 (4), 681. http://stacks.iop.org/ 0022-3700/1/i 4/a 319. Damski, B., Santos,¼ L.,¼ Tiemann, E., Lewenstein, M., Kotochigova, S., Julienne, P., Zoller, P., 2003. Creation of a dipolar superfluid in optical lattices. Phys. Rev. Lett. 90, 110401. http://dx.doi.org/10.1103/PhysRevLett.90.110401. Deiglmayr, J., Pellegrini, P., Grochola, A., Repp, M., Cot^ e, R., Dulieu, O., Wester, R., Weidemuller,€ M., 2009. Influence of a feshbach resonance on the photoassociation of lics. New J. Phys. 11 (5), 055034. http://stacks.iop.org/1367-2630/11/i 5/a 055034. Doerk, H., Idziaszek, Z., Calarco, T., 2010. Atom-ion quantum gate.¼ Phys.¼ Rev. A 81, 012708. http://dx.doi.org/10.1103/PhysRevA.81.012708. Du, X., Zhang, Y., Thomas, J.E., 2009. Inelastic collisions of a Fermi Gas in the BEC-BCS crossover. Phys. Rev. Lett. 102, 250402. http://dx.doi.org/10.1103/PhysRevLett. 102.250402. Enderlein, M., Huber, T., Schneider, C., Schaetz, T., 2012. Single ions trapped in a one- dimensional optical lattice. Phys. Rev. Lett. 109, 233004. http://dx.doi.org/10.1103/ PhysRevLett.109.233004. 122 R. Cot^ e
Ferlaino, F., D’Errico, C., Roati, G., Zaccanti, M., Inguscio, M., Modugno, G., Simoni, A., 2006. Feshbach spectroscopy of a K-Rb atomic mixture. Phys. Rev. A 73, 040702. http://dx.doi.org/10.1103/PhysRevA.73.040702. Fermi, E., 1934. Sopra lo Spostamento per Pressione delle Righe Elevate delle SerieSpettrali. Nuovo Cimento 1827-612111, 157–166. Fu, Z.W., Hessels, E.A., Lundeen, S.R., 1992. Determination of the hyperfine structure of + H2 (ν 0, R 1) by microwave spectroscopy of high- Ln 27 Rydberg states of H2. Phys.¼ Rev. A¼ 46, R5313–R5316. http://dx.doi.org/10.1103/PhysRevA.46.R5313¼ . Gao, B., 2010. Universal properties in ultracold ion-atom interactions. Phys. Rev. Lett. 104, 213201. http://dx.doi.org/10.1103/PhysRevLett.104.213201. Giorgini, S., Pitaevskii, L.P., Stringari, S., 2008. Theory of ultracold atomic Fermi gases. Rev. Mod. Phys. 80, 1215–1274. http://dx.doi.org/10.1103/RevModPhys.80.1215. Greene, C.H., Dickinson, A.S., Sadeghpour, H.R., 2000. Creation of polar and nonpolar ultra-long-range Rydberg molecules. Phys. Rev. Lett. 85, 2458. http://dx.doi.org/ 10.1103/PhysRevLett.85.2458. Greiner, M., Regal, C.A., Jin, D.S., 2003. Emergence of a molecular Bose-Einstein conden- sate from a Fermi gas. Nature 426, 537–540. http://dx.doi.org/10.1038/nature02199. Greiner, M., Regal, C.A., Stewart, J.T., Jin, D.S., 2005. Probing pair-correlated Fermionic atoms through correlations in atom shot noise. Phys. Rev. Lett. 94, 110401. http://dx. doi.org/10.1103/PhysRevLett.94.110401. Grier, A.T., Cetina, M., Orucˇevic´, F., Vuletic´, V., 2009. Observation of cold collisions between trapped ions and trapped atoms. Phys. Rev. Lett. 102, 223201. http://dx. doi.org/10.1103/PhysRevLett.102.223201. Hahn, Y., 2002. Threshold lowering effects on an expanding cold plasma. Phys. Lett. A 293, 266–271. http://dx.doi.org/10.1016/S0375-9601(01)00854-4. Haimberger, C., Kleinert, J., Bhattacharya, M., Bigelow, N.P., 2004. Formation and detec- tion of ultracold ground-state polar molecules. Phys. Rev. A 70, 021402. http://dx.doi. org/10.1103/PhysRevA.70.021402. Haimberger, C., Kleinert, J., Dulieu, O., Bigelow, N.P., 2006. Processes in the formation of ultracold NaCs. J. Phys. B: At. Mol. Opt. Phys. 39 (19), S957. http://stacks.iop.org/ 0953-4075/39/i 19/a S10. Hall, F.H.J., Eberle,¼ P., Hegi,¼ G., Raoult, M., Aymar, M., Dulieu, O., Willitsch, S., 2013. Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser- cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap. Mol. Phys. 111, 2020–2032. http://dx.doi.org/10.1080/00268976.2013.780107. Hamilton, E.L., Greene, C.H., Sadeghpour, H.R., 2002. Shape-resonance-induced long- range molecular Rydberg states. J. Phys. B: At. Mol. Opt. Phys. 35, L199. http://dx. doi.org/10.1088/0953-4075/35/10/102. Holmes, M.E., Tscherneck, M., Quinto-Su, P.A., Bigelow, N.P., 2004. Isotopic difference in the heteronuclear loss rate in a two-species surface trap. Phys. Rev. A 69, 063408. http://dx.doi.org/10.1103/PhysRevA.69.063408. Huber, T., Lambrecht, A., Schmidt, J., Karpa, L., Schaetz, T., 2014. A far-off-resonance optical trap for a Ba+ ion. Nat. Commun. 5, 1. http://dx.doi.org/10.1038/ ncomms6587. Hunter, G., Gray, B.F., Pritchard, H.O., 1966. Born-oppenheimer separation for three-particle systems. I. Theory. J. Chem. Phys. 45 (10), 3806–3816. http://dx.doi.org/10.1063/ 1.1727403. http://scitation.aip.org/content/aip/journal/jcp/45/10/10.1063/1.1727403. Jaksch, D., Cirac, J.I., Zoller, P., Rolston, S.L., Cot^ e, R., Lukin, M.D., 2000. Fast quantum gates for neutral atoms. Phys. Rev. Lett. 85, 2208–2211. Jochim, S., Bartenstein, M., Altmeyer, A., Hendl, G., Riedl, S., Chin, C., Hecker Denschlag, J., Grimm, R., 2003. Bose-Einstein condensation of molecules. Science 302, 2101–2104. http://dx.doi.org/10.1126/science.1093280. Ultracold Hybrid Atom–Ion Systems 123
Johnson, B.R., 1985. Comment on a recent criticism of the formula used to calculate the s matrix in the multichannel log-derivative method. Phys. Rev. A 32, 1241–1242. http:// dx.doi.org/10.1103/PhysRevA.32.1241. Kaiser, A., Muller,€ T.O., Friedrich, H., 2013. Quantization rule for highly excited vibrational states of H. Mol. Phys. 111 (7), 878–887. http://dx.doi.org/ 10.1080/00268976.2012.751462. Karpa, L., Bylinskii, A., Gangloff, D., Cetina, M., Vuletic´, V., 2013. Suppression of ion trans- port due to long-lived subwavelength localization by an optical lattice. Phys. Rev. Lett. 111 (16). http://dx.doi.org/10.1103/PhysRevLett.111.163002. 163002. Karpiuk, T., Brewczyk, M., RzaZ˙ ewski, K., Balewski, J.B., Krupp, A.T., Gaj, A., Low,€ R., Hofferberth, S., Pfau, T., 2015. Imaging single Rydberg electrons in a Bose-Einstein condensate. New J. Phys. 17 (5), 053046. http://stacks.iop.org/1367-2630/17/i 5/ a 053046. ¼ Kerman,¼ A.J., Sage, J.M., Sainis, S., Bergeman, T., DeMille, D., 2004. Production and state- selective detection of ultracold RbCs molecules. Phys. Rev. Lett. 92, 153001. http://dx. doi.org/10.1103/PhysRevLett.92.153001. Kerman, A.J., Sage, J.M., Sainis, S., Bergeman, T., DeMille, D., 2004. Production of ultracold, polar RbCs* molecules via photoassociation. Phys. Rev. Lett. 92, 033004. http://dx.doi.org/10.1103/PhysRevLett.92.033004. Khaykovich, L., Schreck, F., Cubizolles, J., Bourdel, T., Corwin, K.L., Ferrari, G., Salomon, C., 2003. A Bose-Einstein condensate immersed in a Fermi sea: observation of ultra-cold mixture of Bose and Fermi gases. Phys. B 0921-4526329, 13–16. http://dx. doi.org/10.1016/S0921-4526(02)01873-2. http://www.sciencedirect.com/science/ article/pii/S0921452602018732. Kihara, T., 1953. The mathematical theory of electrical discharges in gases. B. Velocity- distribution of positive ions in a static field. Rev. Mod. Phys. 25, 844–852. http://dx. doi.org/10.1103/RevModPhys.25.844. Kittel, C., 1987. Quantum Theory of Solids. Wiley, New York. ISBN 9780471624127. Kohler,€ T., Go´ral, K., Julienne, P.S., 2006. Production of cold molecules via magnetically tunable feshbach resonances. Rev. Mod. Phys. 78, 1311–1361. http://dx.doi.org/ 10.1103/RevModPhys.78.1311. Kosˇtrun, M., Cot^ e, R., 2006. Two-color spectroscopy of fermions in mean-field BCS-BEC crossover theory. Phys. Rev. A 73, 041607. http://dx.doi.org/10.1103/ PhysRevA.73.041607. Kollath, C., Kohl,€ M., Giamarchi, T., 2007. Scanning tunneling microscopy for ultracold atoms. Phys. Rev. A 76, 063602. http://dx.doi.org/10.1103/PhysRevA.76.063602. Korobov, V.I., Koelemeij, J.C.J., Hilico, L., Karr, J.P., 2016. Theoretical hyperfine structure of the molecular hydrogen ion at the 1 ppm level. Phys. Rev. Lett. 116, 053003. http:// dx.doi.org/10.1103/PhysRevLett.116.053003. Kraemer, T., Mark, M., Waldburger, P., Danzl, J.G., Chin, C., Engeser, B., Lange, A.D., Pilch, K., Jaakkola, A., N€agerl, H.C., Grimm, R., 2006. Evidence for Efimov quantum states in an ultracold gas of caesium atoms. Nature 440, 315–318. http://dx.doi.org/ 10.1038/nature04626. Kraft, S.D., Staanum, P., Lange, J., Vogel, L., Wester, R., Weidemuller,€ M., 2006. Forma- tion of ultracold lics molecules. J. Phys. B: At. Mol. Opt. Phys. 39 (19), S993. http:// stacks.iop.org/0953-4075/39/i 19/a S13. Krems, R.V., Dalgarno, A., 2003. Disalignment¼ ¼ transitions in cold collisions of 3P atoms with structureless targets in a magnetic field. Phys. Rev. A 68, 013406. http://dx.doi.org/ 10.1103/PhysRevA.68.013406. Krupp, A.T., Gaj, A., Balewski, J.B., Ilzhofer,€ P., Hofferberth, S., Low,€ R., Pfau, T., Kurz, M., Schmelcher, P., 2014. Alignment of D-state Rydberg. Mol. Phys. Rev. Lett. 112, 143008. http://dx.doi.org/10.1103/PhysRevLett.112.143008. 124 R. Cot^ e
Krych, M., Skomorowski, W., Pawłowski, F., Moszynski, R., Idziaszek, Z., 2011. Sympathetic cooling of the Ba+ ion by collisions with ultracold Rb atoms: theoretical prospects. Phys. Rev. A 83 (3). http://dx.doi.org/10.1103/PhysRevA.83.032723. 032723. Lamb, H.D.L., McCann, J.F., McLaughlin, B.M., Goold, J., Wells, N., Lane, I., 2012. Struc- ture and interactions of ultracold Yb ions and Rb atoms. Phys. Rev. A 86 (2). http://dx. doi.org/10.1103/PhysRevA.86.022716. 022716. Li, M., You, L., Gao, B., 2014. Multichannel quantum-defect theory for ion-atom interac- tions. Phys. Rev. A 89, 052704. http://dx.doi.org/10.1103/PhysRevA.89.052704. Li, W., Pohl, T., Rost, J.M., Rittenhouse, S.T., Sadeghpour, H.R., Nipper, J., Butscher, B., Balewski, J.B., Bendkowsky, V., Low,€ R., Pfau, T., 2011. A homonuclear molecule with a permanent electric dipole moment. Science 334, 1110. Lukin, M.D., Fleischhauer, M., Cot^ e, R., Duan, L.M., Jaksch, D., Cirac, J.I., Zoller, P., 2001. Dipole blockade and quantum information processing in mesoscopic atomic ensembles. Phys. Rev. Lett. 87, 037901. Makarov, O.P., Cot^ e, R., Michels, H., Smith, W.W., 2003. Radiative charge-transfer life- time of the excited state of (NaCa)+. Phys. Rev. A 67, 042705. http://dx.doi.org/ 10.1103/PhysRevA.67.042705. Mazevet, S., Collins, L.A., Kress, J.D., 2002. Evolution of ultracold neutral plasmas. Phys. Rev. Lett. 88, 055001. http://dx.doi.org/10.1103/PhysRevLett.88.055001. McDaniel, E.W., 1964. Collision Phenomena in Ionized Gases. John Wiley & Sons, New York. Modugno, G., Ferrari, G., Roati, G., Brecha, R.J., Simoni, A., Inguscio, M., 2001. Bose- Einstein condensation of potassium atoms by sympathetic cooling. Science 294, 1320–1322. http://dx.doi.org/10.1126/science.1066687. Mott, N.F., Massey, H.S.W., Massey, H.S.W., 1965. The Theory of Atomic Collisions. The International Series of Monographs on Physics, Clarendon Press, London, UK. Mourachko, I., Comparat, D., de Tomasi, F., Fioretti, A., Nosbaum, P., Akulin, V.M., Pillet, P., 1998. Many-body effects in a frozen Rydberg gas. Phys. Rev. Lett. 80, 253–256. http://dx.doi.org/10.1103/PhysRevLett.80.253. Mudrich, M., Kraft, S., Singer, K., Grimm, R., Mosk, A., Weidemuller,€ M., 2002. Sympa- thetic cooling with two atomic species in an optical trap. Phys. Rev. Lett. 88, 253001. http://dx.doi.org/10.1103/PhysRevLett.88.253001. Nozieres, P., Pines, D., 1999. Theory of Quantum Liquids. Advanced Books Classics, Westview Press, Boulder, CO. ISBN 9780813346533. O’Malley, T.F., Spruch, L., Rosenberg, L., 1961. Modification of effective-range theory in the presence of a long-range (r[sup-4]) potential. J. Math. Phys. 2 (4), 491–498. http:// dx.doi.org/10.1063/1.1703735. http://link.aip.org/link/?JMP/2/491/1. Omont, A., 1977. On the theory of collisions of atoms in Rydberg states with neutral particles. J. Phys. (Paris) 38, 1343. http://dx.doi.org/10.1051/jphys:0197700380110134300. Osychenko, O.N., Astrakharchik, G.E., Mazzanti, F., Boronat, J., 2012. Zero-temperature phase diagram of Yukawa bosons. Phys. Rev. A 85, 063604. http://dx.doi.org/10.1103/ PhysRevA.85.063604. Overstreet, K.R., Schwettmann, A., Tallant, J., Booth, D., Shaffer, J.P., 2009. Observation of electric-field-induced Cs Rydberg atom macrodimers. Nat. Phys. 5, 581–585. http:// dx.doi.org/10.1038/nphys1307. Partridge, G.B., Strecker, K.E., Kamar, R.I., Jack, M.W., Hulet, R.G., 2005. Molecular probe of pairing in the BEC-BCS crossover. Phys. Rev. Lett. 95, 020404. http://dx. doi.org/10.1103/PhysRevLett.95.020404. Pathria, R.K., Beale, P.D., 1996. Statistical Mechanics. Elsevier Science, New York. ISBN 9780080541716. Ultracold Hybrid Atom–Ion Systems 125
Ratschbacher, L., Zipkes, C., Sias, C., Kohl,€ M., 2012. Controlling chemical reactions of a single particle. Nat. Phys. 8, 649–652. http://dx.doi.org/10.1038/nphys2373. Roberts, D.C., Rica, S., 2009. Impurity crystal in a Bose-Einstein condensate. Phys. Rev. Lett. 102, 025301. http://dx.doi.org/10.1103/PhysRevLett.102.025301. Robicheaux, F., Hanson, J.D., 2002. Simulation of the expansion of an ultracold neutral plasma. Phys. Rev. Lett. 88, 055002. http://dx.doi.org/10.1103/PhysRevLett. 88.055002. Robinson, M.P., Tolra, B.L., Noel, M.W., Gallagher, T.F., Pillet, P., 2000. Spontaneous evolution of Rydberg atoms into an ultracold plasma. Phys. Rev. Lett. 85, 4466–4469. http://dx.doi.org/10.1103/PhysRevLett.85.4466. Rosenblit, M., Jortner, J., 1995. Dynamics of the formation of an electron bubble in liquid helium. Phys. Rev. Lett. 75, 4079–4082. http://dx.doi.org/10.1103/PhysRevLett. 75.4079. Sage, J.M., Sainis, S., Bergeman, T., DeMille, D., 2005. Optical production of ultracold polar molecules. Phys. Rev. Lett. 94, 203001. http://dx.doi.org/10.1103/PhysRevLett. 94.203001. Santamore, D.H., Timmermans, E., 2011. Multi-impurity poltroons in a dilute Bose- Einstein condensate. New J. Phys. 13, 103029. Sayfutyarova, E.R., Buchachenko, A.A., Yakovleva, S.A., Belyaev, A.K., 2013. Charge transfer in cold Yb++Rb collisions. Phys. Rev. A 87 (5). http://dx.doi.org/10.1103/ PhysRevA.87.052717. 052717. Schmid, S., H€arter, A., Denschlag, J.H., 2010. Dynamics of a cold trapped ion in a Bose- Einstein condensate. Phys. Rev. Lett. 105, 133202. http://dx.doi.org/10.1103/ PhysRevLett.105.133202. Shakeshaft, R., Spruch, L., 1979. Mechanisms for charge transfer (or for the capture of any light particle) at asymptotically high impact velocities. Rev. Mod. Phys. 51, 369–405. http://dx.doi.org/10.1103/RevModPhys.51.369. Sherkunov, Y., Muzykantskii, B., d’Ambrumenil, N., Simons, B.D., 2009. Probing ultracold Fermi atoms with a single ion. Phys. Rev. A 79, 023604. http://dx.doi.org/10.1103/ PhysRevA.79.023604. Shin, Y.i., Schirotzek, A., Schunck, C.H., Ketterle, W., 2008. Realization of a strongly interacting Bose-Fermi mixture from a two-component Fermi gas. Phys. Rev. Lett. 101, 070404. http://dx.doi.org/10.1103/PhysRevLett.101.070404. Simoni, A., Ferlaino, F., Roati, G., Modugno, G., Inguscio, M., 2003. Magnetic control of the interaction in ultracold K-Rb mixtures. Phys. Rev. Lett. 90, 163202. http://dx.doi. org/10.1103/PhysRevLett.90.163202. Singer, K., Stanojevic, J., Weidemuller,€ M., Cot^ e, R., 2005. Long-range interactions between alkali Rydberg atom pairs correlated to the ns-ns, np-np and nd-nd asymptotes. J. Phys. B: At. Mol. Opt. Phys. 38. S295–S3-7. Smith, W.W., Makarov, O.P., Lin, J., 2005. Cold ion neutral collisions in a hybrid trap. J. Mod. Opt. 52, 2253–2260. http://dx.doi.org/10.1080/09500340500275850. Smith, W.W., Goodman, D.S., Sivarajah, I., Wells, J.E., Banerjee, S., Cot^ e, R., Michels, H.H., Mongtomery Jr., J.A., Narducci, F.A., 2014. Experiments with an ion-neutral hybrid trap: cold charge-exchange collisions. Appl. Phys. B 0946- 2171114 (1-2), 75–80. http://dx.doi.org/10.1007/s00340-013-5672-2. Tharamel, J., Kharchenko, V.A., Dalgarno, A., 1994. Resonant charge transfer in collisions between positive ions. Phys. Rev. A 50, 496–501. http://dx.doi.org/10.1103/ PhysRevA.50.496. Timmermans, E., 1998. Phase separation of Bose-Einstein condensates. Phys. Rev. Lett. 81, 5718–5721. http://dx.doi.org/10.1103/PhysRevLett.81.5718. 126 R. Cot^ e
Tomza, M., 2015. Ultracold magnetically tunable interactions without radiative-charge- transfer losses between Ca+,Sr+,Ba+, and Yb+ ions and Cr atoms. Phys. Rev. A 92, 062701. http://dx.doi.org/10.1103/PhysRevA.92.062701. Tong, D., Farooqi, S.M., Stanojevic, J., Krishnan, S., Zhang, Y.P., Cot^ e, R., Eyler, E.E., Gould, P.L., 2004. Local blockade of Rydberg excitation in an ultracold gas. Phys. Rev. Lett. 93, 063001. http://dx.doi.org/10.1103/PhysRevLett.93.063001. Wang, J., Gacesa, M., Cot^ e, R., 2015. Rydberg electrons in a Bose-Einstein condensate. Phys. Rev. Lett. 114, 243003. http://dx.doi.org/10.1103/PhysRevLett.114.243003. Wigner, E.P., 1948. On the behavior of cross sections near thresholds. Phys. Rev. 73, 1002–1009. http://dx.doi.org/10.1103/PhysRev.73.1002. Wu, T., Ohmura, T., 2014. Quantum Theory of Scattering. Dover Books on Physics, Dover Publications, New York. ISBN 9780486320694. Wu, T.Y., Rosenberg, R.L., Sandstrom, H., 1960. μ-Hydrogen molecular ion and collisions between μ-hydrogen atom and proton, deuteron and H atom. Nucl. Phys. 0029- 558216 (3), 432–459. http://dx.doi.org/10.1016/S0029-5582(60)81006-1. http:// www.sciencedirect.com/science/article/pii/S0029558260810061. Xu, K., Mukaiyama, T., Abo-Shaeer, J.R., Chin, J.K., Miller, D.E., Ketterle, W., 2003. For- mation of quantum-degenerate sodium molecules. Phys. Rev. Lett. 91, 210402. http:// dx.doi.org/10.1103/PhysRevLett.91.210402. Zhang, S., Leggett, A.J., 2009. Universal properties of the ultracold Fermi gas. Phys. Rev. A 79, 023601. http://dx.doi.org/10.1103/PhysRevA.79.023601. Zhang, P., Bodo, E., Dalgarno, A., 2009. Near resonance charge exchange in ionatom col- lisions of lithium isotopes. J. Phys. Chem. A 113 (52), 15085–15091. http://dx.doi.org/ 10.1021/jp905184a. PMID: 19746948. Zhang, P., Dalgarno, A., Cot^ e, R., 2009. Scattering of Yb and Yb+. Phys. Rev. A 80, 030703. http://dx.doi.org/10.1103/PhysRevA.80.030703. Zhang, P., Dalgarno, A., Cot^ e, R., Bodo, E., 2011. Charge exchange in collisions of beryl- lium with its ion. Phys. Chem. Chem. Phys. 13, 19026–19035. Zipkes, C., Palzer, S., Sias, C., Kohl,€ M., 2010. A trapped single ion inside a Bose-Einstein condensate. Nature 464, 388–391. http://dx.doi.org/10.1038/nature08865. Zipkes, C., Palzer, S., Ratschbacher, L., Sias, C., Kohl,€ M., 2010. Cold heteronuclear atom- ion collisions. Phys. Rev. Lett. 105, 133201. http://dx.doi.org/10.1103/ PhysRevLett.105.133201.