Universal relations between atomic dipolar relaxation and van der Waals interaction
Yuan-Gang Deng, Yi-Quan Zou, Gao-Ren Wang, Qi Liu, Su Yi, Meng Khoon Tey, Li You
UUniversal relations between atomic dipolar relaxation and van der Waals interaction
Yuan-Gang Deng,
1, 2, ∗ Yi-Quan Zou, ∗ Gao-Ren Wang, ∗ Qi Liu, Su Yi, † Meng Khoon Tey,
1, 5, ‡ and Li You
1, 5, § State Key Laboratory of Low Dimensional Quantum Physics,Department of Physics, Tsinghua University, Beijing 100084, China. Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing & School of Physics and Astronomy,Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China. School of Physics, Dalian University of Technology, Dalian 116024, China. CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China. Frontier Science Center for Quantum Information, Beijing, China. (Dated: January 28, 2021)Dipolar relaxation happens when one or both colliding atoms flip their spins exothermicallyinside a magnetic ( B ) field. This work reports precise measurements of dipolar relaxation in a Bose-Einstein condensate of ground state Rb atoms together with in-depth theoretical investigations.Previous perturbative treatments fail to explain our observations except at very small B -fields.By employing quantum defect theory based on analytic solutions of asymptotic van der Waalsinteraction − C /R ( R being interatomic spacing), we significantly expand the applicable range ofperturbative treatment. We find the B -dependent dipolar relaxation lineshapes are largely universal,determined by the coefficient C and the associated s -wave scattering lengths a sc of the statesbefore and after spin flips. This universality, which applies generally to other atomic species as well,implicates potential controls of dipolar relaxation and related cold chemical reactions by tuning a sc . While nominally weak, the paradigm magnetic dipole-dipole interaction (MDDI) plays an essential role in manyimportant phenomena and systems, ranging from quan-tum many-body phases [1–9], to d -wave Feshbach reso-nances [10–12] in atomic quantum gases, to diverse func-tional materials [13, 14] as well as protein folding [15]in biological systems. The very dipolar interaction alsogives rise to inelastic relaxation inside an external mag-netic ( B -) field, which together with three-body inelas-tic decay constitute the two leading mechanisms limitingcoherence times of ultracold atomic gases [16–22], hencetheir applications in quantum science and technology. Itis therefore essential to develop a general and clear un-derstanding of how dipolar relaxation is affected by in-teratomic interactions in order to offer ways to controland suppress such relaxations.At zero magnetic field ( B = 0) and away from any res-onance, cold elastic scattering cross sections from MDDIare essentially energy independent and proportional tothe fourth power of dipole moments [20, 24]. At finitefield B (cid:54) = 0, two-body dipolar relaxation arises due toexothermic spin flips. Quantitative comparisons betweenmeasured dipolar losses [17–22] and theories remain lim-ited, partly because effects due to MDDI are rather weakto measure accurately except in atoms with large mag-netic dipole moments such as Cr [20, 21] and Dy [22].Experimental observations at small B -fields are largelyconsistent with theories based on Born approximation(BA) [20, 22] which neglects the wave function depen-dence on interatomic interactions, or distorted-wave Bornapproximation (DBA) which partially accounts for phaseshift of long-range wave function [21]. Further improvedunderstanding calls for more quantitative studies con-trasting experiment and theory at larger B -fields, using accurate coupled-channel (CC) [25–27] calculations as re-liable benchmark checks.In this work, we measure B -field dependence of weakdipolar losses in a Bose-Einstein condensate (BEC) of Rb atoms with high accuracy. Different from earlierresults in other atomic species [16–22], which include asmall B rise of ∝ √ B , we observe a tail-off at large B pre-ceded by wavy structures. Upon detailed analysis of themeasurement data, a simple scaling relation between one-and two-spin-flip dipolar-loss rates is revealed, implicat-ing the existence of a universal relationship underlyingthe two processes. This is intriguing since atomic dipo-lar relaxations in heavier atoms such as Rb are knownto be strongly affected by short-range second-order spin-orbit interaction (SOI) [26] which operates on multichan-nel short-range wave functions not known to obey a sim-ple relation. Extending earlier perturbative approaches[20–22], we show that dipolar relaxation lineshapes fromincluding both long-range MDDI and short-range SOI be-have universally, largely determined by the interatomicvan der Waals (vdW) interaction − C /R plus the re-spective s -wave scattering lengths a sc of the initial andfinal spin-flipped channels. Experiment — Our experiments are carried out in aBEC of F = 1 Rb atoms confined by an optical dipoletrap (see supplemental material [28] for details). Thehigh atom number detection resolution of our setup (cal-ibrated using quantum shot noise of coherent spin state)[29, 30], together with small fluctuations of condensateatom numbers, enables small spin-flip losses (3 to 4 or-ders of magnitude smaller than those of Cr or Dy ofthe same densities) to be accurately measured. Fig-ure 1(a) shows the remaining atom numbers (as func-tions of holding time) for a pure BEC prepared in sin- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n B (G ) γ b g ( s − ) τ (s ) N m F (a) × B (G ) K ( c m s − ) × -16 ( − , − ) B (G ) K ( , ) ( c m s − ) × -16 (c) (b)(d) x 10 -2 FIG. 1. Measured dipolar loss rates of Rb atoms in F = 1.(a) The measured loss of BEC atoms prepared in m F = 1(black triangles), 0 (blue open circles), and − B = 10 .
38 G. (b) The measured loss rate (for τ < m F = 1) is essentiallyindependent of magnetic field within the range we investigate.(c) and (d) display the measured two-body dipolar relaxationrates K ( m F ,m F )2 for atoms in m F = 0 or -1, respectively. Solidlines denote CC results using the state-of-the-art Rb poten-tial [23]. Experimental data are limited to
B <
12 G byallowed current to the field generating coils. Error bars rep-resent one standard deviation over five measurements. gle m F (= − , ,
1) states at B = 10 .
38 G. The low-est energy m F = 1 state, which experiences no two-body dipolar loss, shows the longest 1 /e lifetime andserves to calibrate an essentially B -independent back-ground loss rate of γ bg (cid:39) . ± . − for all spinstates [Fig. 1(b)]. The two-body loss coefficients K (0 , and K ( − , − are extracted for atoms initially preparedin m F = 0 [Fig. 1(c)] or − Rb molecular potentials[23] with the quality of experimental data.
Perturbative model — The atomic spin-spin interactionresponsible for dipolar relaxations is (in atomic units) V ss ( R ) = α ( 1 R + κ so e b so R )[ S · S − S · ˆ R )( S · ˆ R )](1)for two atoms located at r and r [23, 32]. It consists ofMDDI ∝ /R and second-order SOI ∝ κ so exp( − b so R )with identical electron spin dependence. Here, α is thefine structure constant, R = | R | = | r − r | , ˆ R = R /R .For Rb atoms, κ so = − . a − B and b so = 0 . a − B l = 0 l = 2 E sr E two flips } β (a) β , ( c m s - ) β (B/2) β (B)CCx10 -16 (c) R/R vdW po t en t i a l β one fliptwo flips-10+1 } } E E β β m F (b) B (G) __2 __2E N f FIG. 2. Self-similarities in dipolar loss lineshapes. (a) Dipo-lar relaxation occurs by flipping atomic spin, transformingspin angular momentum into relative atomic motion. At con-densate temperatures, the incoming channel is s -wave whilethe spin-flipped outgoing channel is d -wave. The latter fea-tures a centrifugal barrier having great influence on the dipo-lar loss rates. For Rb, a d -wave shape resonance existsat E sr = 1 . E vdw (dashed line) above the d -wave thresh-old, where E vdW is the vdW energy scale [31]. (b) The al-lowed dipolar loss channels for atoms prepared in m F = − E N f released through N f spin flips isnormally much larger than trap depth, both atoms are lostafter spin flips. (c) The extracted one-flip loss rate [ β ( B ) = K ( − , − ( B ) − K (0 , ( B )] (crossed open squares) and two-fliprate data with scaled B -field [ β ( B/
2) = K (0 , ( B/ β withboth MDDI and SOI included. (Bohr radius a B ) [23, 26]. The magnitude of SOI becomeslarger than MDDI for R < . a B , a separation muchshorter than the vdW length scale [31] (= 82 . a B for Rb atoms). At small B , dipolar relaxation selectionrules can be discussed using the hyperfine spin F = S + I ( S = 1 / I = 3 / Rb). Inside a B -field along z -axis, the z -component of the total angular momentum, m + m F + m F , remains conserved in the presence of V ss .Here, m F j and m are the eigenvalues of, respectively, F jz for atom j = 1 ,
2, and l z , the z -component of molecularorbital angular momentum. m changes during inelasticdipolar relaxation at the expense of m F j , constrained bythe selection rule ∆ l = 0 , ± s -wave ( l = 0) to d -wave ( l = 2) channel [21]through V ss ( R ) as illustrated in Fig. 2(a).Adopting distorted-wave [33] and single-channel [34]approximation, we show that the total cross section fordipolar relaxation from channel a to b can be approxi-mated by [28, 35] σ a → b ≈ E / a E / b (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ G b ∗ ( R ) (cid:101) U ss ( R ) G a ( R ) dR (cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) (cid:104) ϕ b | ( F ⊗ F ) N f | ϕ a (cid:105) (cid:12)(cid:12)(cid:12) . (2)Here, G b ( a ) l ( R ) and E b ( E a ) denote, respectively, the ra-dial wave function and the kinetic energy of the outgo-ing(incoming) wave. ( F ⊗ F ) N f is the rank-2 sphericaltensor with N f = 1 , F = 1 ground states, direct spin-fliplosses can occur only in a total of seven channels as dic-tated by the spin part in Eq. (2). For atoms preparedin m F = −
1, the incident spin state | ϕ a (cid:105) = | m F = − m F = − (cid:105) , the corresponding one- and two-flip fi-nal spin states are | ϕ ( N f =1) b (cid:105) = ( | −
1; 0 (cid:105) + | − (cid:105) ) / √ | ϕ ( N f =2) b (cid:105) = |
0; 0 (cid:105) . Each spin flip changes m F j by 1( (cid:126) ) and a maximum of 2 ( (cid:126) ) occurs for two flips. In theprocess, an energy of E N f = | N f g F µ B B | [36] is released,giving E b = E a + E N f .The nearly identical s -wave scattering lengths of F = 1 Rb: a = 101 . a B ( a = 100 . a B ) for total spin( F + F ) of “0” (“2”), results in an approximate SU(2)collision symmetry, which encourages treating scatteringapproximately by a single channel (apart from the spinand orbital angular parts) with a sc = a ≈ a for arbi-trary spin states. Based on Eq. (2), the dipolar losses forthe seven channels are thus describable by two irreduciblerates β and β , respectively for one- and two-spin flips[28]. Dipolar relaxation for atoms prepared in m F = − K ( − , − = β + β , while only two-flip process isallowed for atoms in m F = 0, i.e. K (0 , = β [Fig.2(b)]. The one-flip rate β can thus be determined using β = K ( − , − − K (0 , based on experimental data, andis found to be remarkably self-similar to β according to β ( B ) ≈ β ( B/
2) [Fig. 2(c)].To illuminate the physics behind the observed line-shapes including their self-similar dependence on B , weexpand earlier perturbative approaches to larger B -fieldrange (beyond BA [20, 22] and DBA [21]) using a single-channel semi-analytic quantum defect theory (QDT) [34],assuming that both incoming and outgoing wave func-tions are distorted by the same vdW interaction downto small R [37]. As QDT using vdW potential ignoresshort-range details of the wave functions, which subse-quently lead to inaccuracies in treating the SOI term,we will first assume (cid:101) U ss = α ( g F /g s ) (cid:112) π/ /R [36],i.e., treating only the MDDI term (see [28] for relationbetween (cid:101) U ss and V ss ( R )). The loss rate per unit parti- cle density is calculated according to β a → b = (cid:104) σ a → b v a (cid:105) a by averaging over the distribution of the incoming veloc-ity v a = (2 E a /µ ) / of the gas ensemble in the initialspin state. The appropriate G q = a,bl ( R ) is normalized tosin( k q R − lπ/ δ ql ) at large R with δ ql being the phaseshift for the l -th partial wave.The differences between BA [20, 22], DBA [21], andour approach can be highlighted using G a,bl ( R ). BA com-pletely ignores the effects of interatomic potential on thescattering wave functions, giving G a ( R ) = k a R j ( k a R )and G b ( R ) = k b R j ( k b R ) [38] which correspond, respec-tively, to the s - and d -partial wave components of the as-sociated (incoming or outgoing) plane waves. The DBAmodel adopted in Ref. [21] inserts a phase shift δ a intothe long-range wave function of the incoming s -wave, butignores any distortion to the outgoing d -wave, leading to G a = ( k a R )(1 + e i δ a ) [ j ( k a R ) + k a a sc y ( k a R )] / − tan δ a = k a a a sc . Our approach accounts for theeffects of the vdW potential on both the incoming andoutgoing wave functions, by using the semi-analytic so-lutions of − C /R for all R and replacing the effect ofshort-range potential by a single (quantum defect) pa-rameter uniquely related to a sc . This is referred to asvdW universality for cold atom collision [37], whereby G b ( R ) and G a ( R ) are determined by just three param-eters, namely, C , E a,b , and a a,b sc . With our approach,such a universality clearly permeates to dipolar relax-ation given by Eq. (2) as well. The overall B -dependentlineshapes for one or two spin-flips thus take the samefunctional form depending only on C and a a,b sc . Universal lineshape — The computed β based on theaforementioned perturbative models and CC calculationsare compared in Fig. 3(a) (all with SOI ignored). For thedemonstrated B -field range, BA and DBA fail largelyto agree with the CC results, whereas the QDT modelworks uniformly well. For BA, the prediction β ∝ √ B [28] agrees with CC only for B < .
04 G. The DBAmodel [21] predicts vanishing dipolar loss approximatelyat B ∨ = 64 (cid:126) / (9 π | g F | µ B µa a ) for positive a a sc [28] andexpands the range of agreement with CC to about 0.2 G,but deviates seriously at higher B -fields. Our model suc-cessfully extends vdW universality to dipolar relaxationby explaining the observed self-similar patterns as well asthe wavy structures. Working out the constant prefac-tors associated with the spin part in Eq. (2) and takinginto account the factor of 2 in linear Zeeman shifts E N f for one and two flips, we find analytically the observedself-similar B -scaling relation β ( B ) ≈ β ( B/ d -wave quasi-bound statefor vdW potential near threshold when a sc is slightlygreater than 0.956 R vdW . Our case of Rb ground stateatoms corresponds to a b sc ≈ a a sc = a sc ≈ . R vdW ,and the d -wave shape resonance is shifted up slightly to ab sc ( R v d W ) E b (E vdW ) β ab ( a r b . un i t s ) B ADB AQDTCC x 10 -16 β ( c m s - ) (a)(b) B (G) FIG. 3. Universal lineshape. (a) Calculated β from MDDI(without SOI) using perturbative BA, DBA, QDT, and fullCC. Green dots show a fit of the CC results using Fano line-shape with an asymmetry parameter q = 1 . . < B <
10G [28]. (b) Universal lineshapes for dipolar loss rate β a → b asa function of E b and a b sc of the spin flipped state, both in vdWscales. a a sc /R vdW = 0 . β a → b is essentially independent of theincident energy E a [28]. ∼ . E vdW [31]. Such d -wave shape resonances havebeen observed in K, K, and
Yb [11, 39, 40], in F = 2 Rb [12, 41, 42], and in F = 1 Rb at a veryhigh B ( ∼
632 G) [43]. In dipolar relaxation, interfer-ence between this shape resonance with the continuumin the d -wave outgoing channel gives rise to the cele-brated Fano lineshape [44] (as illustrated by the greendots in Fig. 3(a)). In fact, the observed wavy structuresin Fig. 1(d) consist of two Fano profiles, respectivelyassociated with one- and two-flip d -wave channels. Be-cause this d -wave shape resonance lies close to the topof the centrifugal barrier, the observed resonance line-shapes are rather broad despite of being in a high partialwave. More generally, for atomic species without SU(2)collision symmetry, we check that the vdW universalitystill holds. Specifically, for a finite range of E b , there ex-ists a unique relation between the dipolar relaxation rate β a → b between the scaled spin-flipped energy E b /E vdW and the scaled s -wave scattering lengths a a,b sc /R vdW , in-dependent of small E a . Figure 3(b) shows some exam-ples of such universal lineshapes with a a sc /R vdW = 0 . d -wave shape resonancenear a b sc /R vdW ≈ ( b ) M D D I + S O I M D D I S O I x 1 0 - 1 6 x 1 0 - 1 6 b ( cm3s-1 ) b ( cm3s-1 ) x 1 0 - 1 6 ( a ) b ( B / 2 ) b ( B ) B ( G )
FIG. 4. Contributions of SOI and MDDI to dipolar loss. (a)The loss rate β from CC as a function of B . The curve la-beled “MDDI” (“SOI”) includes only MDDI (or second-orderSOI) contribution, while “MDDI+SOI” denotes the inclusionof both. The scaling relationship β ( B ) ≈ β ( B/
2) shown ininset survives even when SOI is included, suggesting that thedipolar loss due the short-range SOI is related to the long-range potential through a simple rule. (B) The CC resultswith only SOI agrees completely (up to a constant amplitude)with Eq. (3), which depends only on B , a b sc , and C . The overall features of dipolar loss we observe are thusqualitatively explained by the effects of vdW potentialusing QDT with SOI ignored. We next compare in detailin Fig. 4 the CC predictions using full V ss (“MDDI+SOI”,solid line), MDDI (dashed-dotted line), or SOI (dashedline). For B > . R where the wave functions are of multi-channel nature and differ for different outgoing channels.It turns out that the lineshape of SOI-induced dipolarloss is describable by an analytic function [28, 45] β SOI ∝ (cid:2) ( Z cfs − K c Z cgs ) + ( Z cfc − K c Z cgc ) (cid:3) − , (3)related solely to the vdW universality through C , a b sc and E b . The validity of Eq. (3) is evident from its excel-lent agreement with the CC results including only SOI,up to a constant factor [Fig. 4(b)]. This understand-ing partially explains the scaling relationship for the SOIcase, up to an unknown amplitude.In summary, we show that atomic dipolar relaxationrates can be universally described by the long-range vdWpotential plus the s -wave scattering lengths for the in-coming and (spin flipped) outgoing channels within afinite spin-flipped energy. The distorted wave approxi-mation we adopt based on QDT wave functions includ-ing the vdW potential greatly expands the applicabilityof perturbative prediction for dipolar loss, and is foundto work well also for other alkali-metal atoms. In thefuture, it would be interesting to apply this model toother atomic species such as Dy, Cr, Er whose dipolar re-laxations are orders of magnitude stronger, but detailedknowledge of their interatomic potentials are unavailableto support accurate CC predictions.The authors thank Peng Zhang and Jinlun Lifor helpful discussions. This work is supported bythe National Key R&D Program of China (GrantNo. 2018YFA0306504, No. 2018YFA0306503 and2018YFA0307500) and the NSFC (Grant No. 91636213,No. 11654001, No. 91736311 and 11874433). ∗ These authors contributed equally to this work. † [email protected] ‡ mengkhoon [email protected] § [email protected][1] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, andT. Pfau, Reports on Progress in Physics , 126401(2009).[2] M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller,Chemical Reviews , 5012 (2012).[3] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz,Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Science , 201 (2016).[4] S. Yi, T. 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