Fermionic suppression of dipolar relaxation: Observation of universal inelastic dipolar scattering
Nathaniel Q. Burdick, Kristian Baumann, Yijun Tang, Mingwu Lu, Benjamin L. Lev
FFermionic suppression of dipolar relaxation:Observation of universal inelastic dipolar scattering
Nathaniel Q. Burdick, Kristian Baumann, Yijun Tang, Mingwu Lu, and Benjamin L. Lev
Department of Applied Physics, Stanford University, Stanford CA 94305Department of Physics, Stanford University, Stanford CA 94305 andE. L. Ginzton Laboratory, Stanford University, Stanford CA 94305 (Dated: October 3, 2018)We observe the suppression of inelastic dipolar scattering in ultracold Fermi gases of the highlymagnetic atom dysprosium: the more energy that is released, the less frequently these exothermic re-actions take place, and only quantum spin statistics can explain this counterintuitive effect. Inelasticdipolar scattering in non-zero magnetic fields leads to heating or to loss of the trapped population,both detrimental to experiments intended to study quantum many-body physics with strongly dipo-lar gases. Fermi statistics, however, is predicted to lead to a kinematic suppression of these harmfulreactions. Indeed, we observe a 120-fold suppression of dipolar relaxation in fermionic versus bosonicDy, as expected from theory describing universal inelastic dipolar scattering, though never beforeexperimentally confirmed. Similarly low inelastic cross sections are observed in spin mixtures, alsowith striking correspondence to universal dipolar scattering predictions. The suppression of relax-ation opens the possibility of employing fermionic dipolar species—atoms or molecules—in studiesof quantum many-body physics involving, e.g., synthetic gauge fields and pairing.
PACS numbers: 34.50.-s, 03.65.Nk, 67.85.-d
Spin-statistics play a prominent role in determiningthe character and rate of elastic collisions among ul-tracold atoms or molecules [1–3], often leading to theenhancement or suppression of thermalization. For ex-ample, elastic collisions mediated by short-range inter-actions between spin-polarized fermions are suppressedat low velocity. The reason lies in the requirement thatthe total two-particle state—the tensor product of spinand orbital—must be antisymmetric both before and af-ter a collision [4]. Because the orbital wavefunction mustbe of odd parity for spin-polarized fermions, collisionsbetween two such atoms are inhibited by the p -wave cen-trifugal energy barrier [5]. For van der Waals interac-tions, this leads to a kinematic suppression of the elas-tic cross section as k i →
0, where the wavevector k i isproportional to the relative incoming momentum. Thefermionic suppression of thermalizing elastic collisionshas an important, well-known consequence: inefficientevaporative cooling near quantum degeneracy [6].This unfavorable scaling is modified in the case of 3Ddipolar interactions. The long-range, r − nature of thedipolar interaction leads to an elastic cross section in-dependent of k i and proportional to the fourth powerof the magnetic dipole moment µ regardless of quantumstatistics in the limit k i → T F . Here“universal” means short-range physics plays no role; scat-tering only depends on atomic parameters through µ andmass [8] and not on, e.g., the difficult-to-calculate phase-shifts of partial-waves at short range [11]. Indeed, re-cent experiments employing the highly dipolar fermionic gases KRb [3], Dy [12], and Er [13] have observed ef-ficient evaporative cooling at T F and below, providinga route to preparing quantum degenerate dipolar Fermigases without the use of sympathetic cooling [14].But while large dipoles promote useful elastic colli-sions, they also enhance inelastic dipolar collisions amongatoms in spin mixtures and in metastable Zeeman sub-states [15]. Rapid heating or population loss are a re-sult of the ensuing spin relaxation and are detrimentalto experiments exploring quantum many-body physicsor atom chip magnetometry with highly dipolar gases inmetastable spin states [16–22].Inelastic dipolar collisions among highly magneticatoms in magnetostatic traps were considered in the con-text of bosonic Cr gases at fifty to hundreds of µ K [7]and at a few hundred nK [9] and Dy gases at hundredsof mK [23] and at a few hundred µ K [24]. The authors ofRef. [7] derived an expression for inelastic dipolar scat-tering using the first-order Born approximation and ob-served rapid collisional loss in a single isotope of bosonicCr [25]. While the loss rate proved similar to that ex-pected from theory, the theory’s universality was unex-plored. The role Fermi statistics might play in suppress-ing dipolar relaxation was discussed in Ref. [9], but hasnever been experimentally investigated.By comparing dipolar relaxation rates in both ultra-cold bosonic and fermionic dysprosium, we find that spinrelaxation is enhanced among bosons while suppressedamong fermions. This supports the conclusion that quan-tum statistics play a substantial role in these collisions:The more energy that is released, the less frequentlythese exothermic reactions take place, and only quan-tum spin statistics can explain this counterintuitive ef-fect. The strikingly close correspondence of our spin re- a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t laxation data to theory predictions—with no free param-eters and despite the unclear a priori validity of the the-ory to atoms with µ ’s as large as Dy’s—represents a cleardemonstration of universal inelastic dipolar scattering.Following Refs. [7, 9], we now describe two-particledipolar scattering within the first-order Born approxima-tion, and in doing so quantify the role quantum statis-tics play in suppressing or enhancing dipolar relaxation.Dipolar scattering changes the orbital momentum of thecollision partners by ∆ l = 0 , ± m F = 0 , ± m F + ∆ m l = 0, where m l is the orbital projection.The dipolar relaxation cross section σ dr connects the-ory predictions to the experimentally measured colli-sional loss rate β dr via β dr ∝ (cid:104) ( σ + σ ) v rel (cid:105) thermal , wherea thermal average must be taken, σ ( σ ) is the single(double) spin-flip cross section, and v rel is the relativevelocity; see Supplemental Material for details [26]. Thefollowing expressions list the cross sections for the elas-tic ( σ ) and σ processes for a maximally stretched andweak-field-seeking initial two-body spin state | F, m F = + F ; F, m F = + F (cid:105) [7, 9]: σ = 16 π F (cid:18) µ ( g F µ B ) m π ¯ h (cid:19) [1 + (cid:15)h (1)] , (1) σ = 8 π F (cid:18) µ ( g F µ B ) m π ¯ h (cid:19) [1 + (cid:15)h ( k f /k i )] k f k i . (2)While the full theory is used in data analysis, we neglect σ in this initial discussion since σ /σ = F − (cid:28) | m F | states [26]. Thislimit is satisfied for bosonic Dy ( F = 8) and fermionic Dy ( F = 21 / F is the total angular momen-tum; see Fig. 1a [27].The kinematic factors in σ are a function of the ratioof output to input relative momenta: by conservationof energy k f /k i = (cid:113) m ∆ E ¯ h k i , where ∆ E = g F µ B B isthe Zeeman energy in a magnetic field B , k i = µv rel / ¯ h , µ = m/ g F is the g -factor [28].The ratio h ( x = k f /k i ) of the exchange to the directterms in the cross section monotonically increases from h (1) = − / h ( x → ∞ ) = 1 − /x ; see Refs. [7, 26].The ratio x is varied between 2–14 in this work.Quantum statistics of the colliding particles are re-flected in the value of (cid:15) : ± x (cid:29) B ,low T —the inelastic cross section (collisional loss rate)vanishes as 4 (cid:112) T /B (4 T / √ B ) for (cid:15) = −
1, while it in-creases as 2 (cid:112)
B/T (2 √ B ) for (cid:15) = +1 and (cid:112) B/T ( √ B ) +8 -8+21/2 -21/2(a) Dy, boson
Dy, fermion(b) ... +21/2, +8+19/2, +7+17/2, +6 ... -17/2-19/2-21/2 ... ... (g)-17/2-19/2-21/2 ... ... (f)(c) -17/2-19/2-21/2 ... ... (d) -17/2-19/2-21/2 ... ... (e)-17/2-19/2-21/2 ... ...
FIG. 1. (Color online) (a) Zeeman m F sublevels of the rel-evant Dy ground states. Numbers indicate the maximallystretched m F states. (b–d) Single-spin-flip dipolar relaxationof spin-polarized states into spin mixtures. Arrow points fromthe incoming to the outgoing spin population. (e) and (f)Single-spin-flip dipolar relaxation of spin mixtures into spin-polarized states. (g) Single-spin-flip dipolar relaxation of aspin mixture into a different spin mixture. for (cid:15) = 0. The relative suppression ratio in this limitbecomes σ fermions1 /σ bosons1 = β fermionsdr /β bosonsdr ∝ T /B .Ultracold gases of bosonic
Dy and fermionic
Dyare prepared by laser cooling in two magneto-optical-trapstages and by forced evaporative cooling in a 1064-nmcrossed optical dipole trap, as explained in previous pub-lications [12, 29, 30]; see also Ref. [26]. The temperaturesof the boson and fermion gases, ∼
400 nK, are chosen tobe slightly above quantum degeneracy to eliminate corre-lation effects [9]:
T /T c = 1 . × cm − ]and T /T F = 1 . × cm − ] [31]. Adia-batic rapid passage while in the optical dipole trap po-larizes the atomic cloud in its absolute internal groundstate. Co-trapping Dy with
Dy is used to enhancefermionic evaporation efficiency, after which the bosonsare removed from the trap by a resonant pushing beamwith no adverse effect on the fermions. The atoms arethen prepared in the desired Zeeman substate(s) by driv-ing rf transitions, as detailed in Ref. [26]. Stern-Gerlachmeasurements are used to verify the final state purity.The atomic cloud is trapped for varying lengths of timein order to measure population decay. Decay curves arefit to a numerically integrated rate equation that includescollision terms for both one-body loss due to backgroundgas γ and two-body loss β dr : dNdt = − γN − β dr ¯ V − N , (3) Ϭ ͘ Ϭ Ϭ ͘ ϱ ϭ ͘ Ϭ ϭ ͘ ϱ Ϯ ͘ Ϭ ƚ ƌ Ă Ɖ Ɖ ŝ Ŷ Ő ƚ ŝ ŵ Ğ ; Ɛ Ϳ Ϭ Ϯ ϰ ϲ ϴ ϭ Ϭ ϭ Ϯ ϭ ϰ Ă ƚ Ž ŵ Ŷ Ƶ ŵ ď Ğ ƌ ; п ϭ Ϭ ϯ Ϳ ϭ ϲ ϭ LJ ϭ ϲ Ϯ LJ н ϴ Ͳ ϴ н Ϯ ϭ ͬ Ϯ Ͳ Ϯ ϭ ͬ Ϯ Ͳ ϭ ϵ ͬ Ϯ Ͳ ϭ ϳ ͬ Ϯ FIG. 2. (Color online) Population decay of fermionic
Dy at B = 0 . T = 390(30) nK, and ¯ n = 7(2) × cm − for m F = +21 / − / − / Dy at B = 0 . T = 450(30) nK, and ¯ n = 3(1) × cm − for m F = +8(squares). The solid curves are fits to the data using Eq. 3.(Inset) Stern-Gerlach images of initial states. Error bars rep-resent one standard error. where ¯ V = √ π ) / σ x σ y σ z is the mean collisionalvolume for a harmonically trapped thermal cloud withGaussian widths σ i [26]. The decay rate is characterizedby the lifetime τ dr = ( β dr ¯ n ) − , where ¯ n = N / ¯ V is theinitial mean collisional density.Typical decay curves for four different spin-polarizedensembles are shown in Fig. 2. The fermions are preparedin either the m F = +21 / − /
2, or − / m F = +8 state as in Fig. 1(b). The inset of Fig. 2contains Stern-Gerlach-separated images of these statesas well as the absolute ground states m F = − / m F = −
8. Decay of these states, which cannot undergodipolar relaxation at this B -field and temperature, arenot presented due to their much slower decay, limitedonly by 1 /γ = 21(1) s. Table I lists the experimentaldecay rates β dr for the m F = − / − / χ analysis [26].We expect from the form of Eq. 12 that the bosonic life-time τ dr should decrease as the magnetic field increases,while the fermionic lifetime should increase. Both trendsare observed, as shown in Fig. 3(a) and (b). Whilebosonic Dy decays rapidly, the fermionic gases at 1 Glive for approximately 1 s at this density.While the relative suppression is evident in the formof Eq. 12, we may gain a more intuitive understanding ofthis relative suppression from an analysis of symmetriza-tion and selection rules. Let us first consider the spinrelaxation channel depicted in Fig. 1(b) in which spin-polarized fermions or bosons decay from the maximally β − / − / dr β − / − / dr β − / − / dr β − / − / dr exp. 10(2) 4.1(7) 60(30) 3(1)th. 6.3(3) 4.1(1) 37(1) 4.2(5)TABLE I. Collisional loss rates in units of [ × − cm s − ]. stretched state m F = + F . This case corresponds to thedata in Figs. 3(a) and (b), respectively, and to the setsof triangle and square data in Fig. 2. The collisionalreaction among fermions may be written:Fermions: | F, m F ; F, m F (cid:105) ⊗ | p, m l (cid:105) → (4) | F, m F − F, m F (cid:105) S ⊗ | p, m l + 1 (cid:105) , where S denotes the symmetric superposition. Whilethis inelastic collision is allowed by dipolar-interactionselection rules and by symmetrization, the reaction iskinematically suppressed once the temperature falls be-low the Dy p -wave threshold barrier ∼ µ K [5, 32][33].The (cid:15) = − σ is a manifestation of this kine-matic suppression due to Fermi statistics. In contrast,there is no p -wave threshold barrier in the bosonic case,Bosons: | F, m F ; F, m F (cid:105) ⊗ | s, (cid:105) → (5) | F, m F − F, m F (cid:105) S ⊗ | d, (cid:105) , since symmetrization allows an incoming s -wave channel:no centrifugal barrier must be surmounted. We see thatquantum statistics dictates that bosons possess a relativeenhancement, (cid:15) = +1, in the inelastic cross section σ .Feshbach resonances can mask the universal nature ofEq. 12 by increasing losses due to three-body inelasticcollisions. Dysprosium has a high density of Feshbachresonances, even at low field [30], and atom loss spectrafor the different m F states were measured prior to in-vestigating the magnetic field dependence of dipolar re-laxation. Magnetic fields were selected to avoid increasedloss due to sharp Feshbach resonance features in the dataof Figs. 2–4. Feshbach spectra for the m F = +8 bosonsand m F = +21 / β dr ’s of the data in Figs. 3(a)and (b). Data are in remarkable agreement with the the-ory curves, though the discrepancy of the fermion β dr ’s atfields below ∼ β dr ’s are dominated by uncertainties in the tem-peratures and trap frequencies, see Ref. [26].We next investigate whether the fermionic suppressionof dipolar relaxation is present in collisions involving spinmixtures. As predicted by theory, we observe suppressionin the decay of the m F = − , − mixture, but nosuppression in the decay of the m F = − , − mixture;see Fig. 4.These drastically different decay rates are due to thedifferent quantum statistics governing the dominant re-laxation processes. The only interspecies decay channel Ϭ ͘ Ϭ Ϭ ͘ ϭ Ě ƌ ; Ɛ Ϳ ϭ ϲ Ϯ LJ ͕ ď Ž Ɛ Ž Ŷ ͕ ŵ & с н &