We have studied the decay of a Bose-Einstein condensate of metastable helium
atoms in an optical dipole trap. In the regime where two- and three-body losses
can be neglected we show that the Bose-Einstein condensate and the thermal
cloud show fundamentally different decay characteristics. The total number of
atoms decays exponentially with time constant tau; however, the thermal cloud
decays exponentially with time constant (4/3)tau and the condensate decays much
faster, and non-exponentially. We show that this behaviour, which should be
present for all BECs in thermal equilibrium with a considerable thermal
fraction, is due to a transfer of atoms from the condensate to the thermal
cloud during its decay.Comment: The intuitive explanation of the atomic transfer effect has been
  correcte

J. S. Borbely

R. van Rooij

S. Knoop

W. Vassen

Y. Kagan

English

arXiv

We have studied the decay of a Bose-Einstein condensate (BEC) of metastable helium atoms in an optical dipole trap. In the regime where two- and three-body losses can be neglected we show that the Bose-Einstein condensate and the thermal cloud show fundamentally different decay characteristics. The total number of atoms decays exponentially with time constant τ; however, the thermal cloud decays exponentially with time constant 43τ and the condensate decays much faster, and nonexponentially. We show that this behavior, which should be present for all BECs in thermal equilibrium with a considerable thermal fraction, is due to a transfer of atoms from the condensate to the thermal cloud during its decay. © 2012 American Physical Society

Knoop, S.

Borbely, J. S.

van Rooij, R.

Vassen, W.

NARCIS 

Nonexponential one-body loss in a Bose-Einstein condensate

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Physical Review A

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VU Research PortalNonexponential one-body loss in a Bose-Einstein condensateKnoop, S.; Borbely, J. S.; van Rooij, R.; Vassen, W.published inPhysical Review A. Atomic, Molecular and Optical Physics2012DOI (link to publisher)10.1103/PhysRevA.85.025602document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Knoop, S., Borbely, J. S., van Rooij, R., & Vassen, W. (2012). Nonexponential one-body loss in a Bose-Einsteincondensate. Physical Review A. Atomic, Molecular and Optical Physics, 85(2), 025602. [025602].https://doi.org/10.1103/PhysRevA.85.025602General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.E-mail address:vuresearchportal.ub@vu.nlDownload date: 05. Nov. 2022PHYSICAL REVIEW A 85, 025602 (2012)Nonexponential one-body loss in a Bose-Einstein condensateS. Knoop, J. S. Borbely, R. van Rooij, and W. VassenLaserLaB Vrije Universiteit, De Boelelaan 1081, NL-1081 HV Amsterdam, The Netherlands(Received 20 December 2011; published 14 February 2012)We have studied the decay of a Bose-Einstein condensate (BEC) of metastable helium atoms in an opticaldipole trap. In the regime where two- and three-body losses can be neglected we show that the Bose-Einsteincondensate and the thermal cloud show fundamentally different decay characteristics. The total number of atomsdecays exponentially with time constant τ ; however, the thermal cloud decays exponentially with time constant43 τ and the condensate decays much faster, and nonexponentially. We show that this behavior, which should bepresent for all BECs in thermal equilibrium with a considerable thermal fraction, is due to a transfer of atomsfrom the condensate to the thermal cloud during its decay.DOI: 10.1103/PhysRevA.85.025602 PACS number(s): 03.75.Hh, 34.50.Cx, 67.85.−dI. INTRODUCTIONAtomic gases can be cooled and trapped to ultracold tem-peratures and densities where Bose-Einstein condensation andFermi degeneracy can be reached. These trapped gases decayin several ways due to one-body, two-body, and three-bodycollisions. Two- and three-body losses are density dependentand have been extensively studied [1–6] and applied in workon atom-atom correlations [2] and on universal few-bodyphysics [7–9].One-body loss is generally considered trivial as it usuallyresults from collisions between trapped atoms and room-temperature atoms from the background gas in the ultrahighvacuum chamber that contains the trapped atoms. A back-ground pressure of ∼10−11 mbar typically leads to a lifetimeof atoms in the trap of ∼100 s. Other causes of one-bodyloss are scattering by off-resonant light in a dipole trap andon-resonance excitation by stray laser cooling light. All theseeffects cause exponential decay of the trapped cloud with atime constant τ , typically in the range of 1–100 s. Experimentsin ultracold atomic physics measure this time constant bymonitoring the number of trapped atoms as a function of time,a procedure commonly performed using absorption imagingof the cloud on a CCD camera. Due to the high densities ina condensate, two- and three-body losses are most importantin the first instances of the decay. In studies of decay, it isgenerally assumed that the decay becomes exponential forlong enough times, where two- and three-body losses havebecome negligible due to the low density that is then reached.Zin et al. [10], however, showed theoretically that thedecay of a Bose-Einstein condensate (BEC) is expected tobe nonexponential when it is in thermal equilibrium witha substantial thermal cloud. The origin of this effect stemsfrom the transfer of atoms from the condensate to thethermal cloud during the decay. This atomic transfer occursonly when thermalization is fast compared to the changeof thermodynamic variables during the decay of the trappedcloud. When two-body and three-body decay can be neglectedthe model predicts that for an overall exponential decaytime τ , the condensate decays faster and nonexponentially,while the thermal cloud decays exponentially with a largertime constant, 43τ . To our knowledge, this has not beendemonstrated experimentally.Enhancement in the decay of a BEC in the presence of athermal cloud was observed by Tychkov et al. [11]. In thatexperiment the decay of a large (>106 atoms) condensate ofhelium atoms in the metastable 2 3S1 state (4He∗, lifetime8000 s) was monitored with and without a thermal cloudcontaining approximately the same number of atoms. Thatstudy, performed in a magnetic trap with atoms in the m=+1magnetic substate, revealed that the condensate in the presenceof a thermal cloud decayed much faster with the cloud thanwithout. However, in that study the decay was studied inthe presence of large two- and three-body losses possiblyobscuring an effect of atomic transfer; the enhanced decay ofthe condensate could also be understood from two- and three-body inelastic collisions between condensate and thermalatoms. Furthermore, two- and three-body loss rate constantswere not known accurately and both processes were expectedto contribute about equally to the decay [6,11]. Therefore therate constants had to be determined from a fit.Recently, we have accurately determined these two- andthree-body loss rate constants over a range of magnetic fieldvalues [12]. We transferred 4He∗atoms from a magnetic trapinto an optical dipole trap and measured, both for atoms inthe m=+1 and m=−1 state, the two- and three-body lossrate constants as a function of an applied magnetic field andwith a (quasi-)pure BEC. In this paper we extend the previousexperiment to partially condensed clouds with a large thermalfraction in order to investigate atomic transfer. To reduce two-and three-body losses we study long time scales and work atsmall magnetic field using m=−1 atoms that, in contrast tom=+1 atoms, only show three-body losses.II. THEORYA. Theory of trap lossThe time evolution of the density of a trapped atomic gascan be described asṅ = −n/τ − κ2L2n2 − κ3L3n3. (1)The first term on the right takes into account one-body loss,which causes exponential decay with a time constant τ . L2and L3 are the rate coefficients for two- and three-body loss,respectively, and are defined such that they explicitly include025602-11050-2947/2012/85(2)/025602(4) ©2012 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 85, 025602 (2012)the loss of two and three atoms per loss event. The constantsin front of L2 and L3 are κ2 = 1/2! and κ3 = 1/3! for a BEC,while κ2 = κ3 = 1 for a thermal gas [13].The rate coefficients are obtained by measuring the numberof trapped atoms after a variable hold time. The analysis ofthis data requires integration of Eq. (1) over space. Because aBEC and a thermal cloud have different density distributions,extracting two- and three-body loss rate coefficients frompartially condensed samples becomes very difficult, andexperiments usually focus on either a pure BEC or a thermalsample.B. Atomic transfer modelIn Sec. II of their paper, Zin et al. [10] have derived a simplemodel for the decay of the thermal and BEC components ofa partially condensed cloud in thermal equilibrium below thetemperature threshold for Bose-Einstein condensation. Theessential assumption of the model is that thermal equilibriumholds during one-body decay of a condensate. Although two-and three-body losses may be incorporated in the model, thiscomplicates the discussion and therefore we here ensure thoseto be negligible compared to the one-body losses by using lowenough densities. If an atom is removed from the thermal cloudby, for instance, a collision with a background atom, then thereis a place free in the otherwise saturated thermal distribution,which can be filled by a BEC atom, thus maintaining the sizeof the thermal cloud at the expense of the BEC. This transferof atoms from the condensate to the thermal cloud enhancesthe decay of the condensate and increases the lifetime of thethermal cloud.For the whole cloud, containing N atoms, decay isexponential with time constant τ . The model then predictsthat the thermal cloud will also decay exponentially, however,with a larger time constant, while the condensate is expectedto decay nonexponentially [10]:Ṅ = − 1τN, (2)ṄT = − 34τNT , (3)ṄC = − 1τ(NC + 14NT). (4)Here NC and NT are the number of condensed atoms andnumber of thermal atoms, respectively, and N = NC + NT .The equations show that the thermal cloud is expected todecay exponentially with a time constant τ ′ = 43τ , independentof NC , which is assumed to be nonzero. For the BEC anonexponential decay is expected if an appreciable amountof thermal atoms is present. The model is valid up to the pointthat there are no BEC atoms left or thermal equilibrium cannotbe assumed anymore. Starting with a partially condensed cloudfinally leads to the complete depletion of the BEC; for the purethermal cloud, decay then proceeds with time constant τ ′ = τ .Equations (3) and (4) are valid when the energy of a condensateatom (which is apart from a small mean-field contribution,equal to the ground-state energy ε0 of the harmonic trappotential) can be neglected compared to the average energyof a thermal atom (ET /NT ) [10]:ε0NT /ET  1. (5)In order to observe atomic transfer experimentally, inelasticcollisions within the cloud should not play a role, but stilla high enough elastic collision rate is necessary to reachfast thermalization. The thermalization rate is given by γth =γcoll/2.7 [14], with collision rate γcoll = navv̄σel. Here, nav isthe average density, v̄ = √16kBT /πm is the mean relativethermal velocity, and σel = 8πa2 the elastic cross section atultralow temperatures, where a=142.0(0.1)a0 for 4He∗ [15].The large scattering length ensures fast thermalization downto a density of nav = 1011 cm−3, which is reached after 40 s inour dipole trap (see Sec. III).III. EXPERIMENTWe have studied one-body loss in a BEC of 4He∗atoms witha considerable thermal fraction. The experimental setup andmeasurement procedure have been described earlier [12,16];here we summarize only the most essential parts. A BEC ofabout 106 atoms is prepared in a crossed optical dipole trapat a wavelength of 1557 nm, and all atoms are transferredfrom the m=+1 magnetic substate to the m=−1 magneticsubstate by an RF sweep. We measure the remaining number oftrapped atoms after a variable hold time by turning off the trap,causing the atoms to fall and be detected by a microchannelplate (MCP) detector, which is located 17 cm below the trapcenter and gives rise to a time-of-flight of approximately186 ms. From a bimodal fit to the MCP signal, the BEC andthermal fraction are extracted as well as the temperature andthe chemical potential of the trapped gas [16].In Fig. 1 we show typical time evolutions of the partiallycondensed cloud, where the total thermal and BEC atomnumber are logarithmically plotted, normalized to the totalnumber of atoms at t = 0. Figure 1(a) shows the decay for asample with an initial BEC fraction of 80% [with a thermalfraction at a temperature of 0.22(3) μK], while in Fig. 1(b) theBEC fraction is 50% [at a temperature of 0.36(5) μK]. Theexperiments are performed in a dipole trap with a geometricalmean of the trapping frequency of 2π × 194(1) Hz and2π × 245(1) Hz, respectively. Equation (5) is easily fulfilledthroughout the decay since ε0NT /ET < 0.03. We also observethat the temperature changes very little during the decay(see the insets in Fig. 1). At short hold times, the loss isdominated by three-body recombination with a rate constantof L3 = 6.5(0.4)stat(0.6)sys × 10−27 cm6s−1 [12]. Two-bodyloss by Penning ionization is strongly suppressed for thespin-stretched states m=+1 and m=−1 [17], while for m=−1the spin-dipole interaction also does not contribute to two-bodylosses because the atom is in the lowest spin state [12]. Figure 1clearly shows that, for hold times longer than 10 s, the lossof the total atom number becomes exponential, indicating theregime of one-body loss. In both measurements the initialdensity of the BEC is 3 × 1013 cm−3. The thermal cloudinitially has a density nav ≈ 2 × 1012 cm−3, which after 40 shas decreased to nav ≈ 1 × 1011 cm−3. The thermalization ratedecreases in that time from γth ≈ 50 s−1 to γth ≈ 3 s−1, stillmuch larger than the one-body decay rate of approximately0.05 s−1, ensuring sufficiently rapid thermal equilibration.025602-2BRIEF REPORTS PHYSICAL REVIEW A 85, 025602 (2012)0 10 20 30 400.010.11normalizedatomnumber10 20 30 4000.10.20.30.4TΜKtotalBECthermal0 10 20 30 400.0010.010.11hold time snormalizedatomnumber10 20 30 4000.20.40.6TΜKtotalBECthermal(a)(b)FIG. 1. (Color online) Lifetime measurements of an ultracold4He∗gas in the m=−1 state at a magnetic field of 10 G, showing theresults of bimodal fits of the MCP signal of the remaining 4He∗atoms,where the insets show the fit temperature. The trap frequencies andthe average temperature are (a) 2π × 194(1) Hz and 0.22(3) μK and(b) 2π × 245(1) Hz and 0.36(5) μK, respectively. The solid lines andthe shaded area are obtained from fits to the atomic transfer model,as described in the text.We observe a faster decay of the BEC fraction than thethermal fraction, eventually leading to a full depletion ofthe condensate, as expected from the atomic transfer model.Inspection of the decay of the thermal cloud for times longerthan 10 s clearly shows that the thermal cloud decays witha larger time constant than the whole cloud. Fitting anexponential decay function to the data at longer hold timesyields τ = 18.0(8) s, τ ′ = 30.8(2.5) s, and τ ′/τ = 1.7(2) forthe data of Fig. 1(a), and τ = 24.4(1.8) s, τ ′ = 36.2(3.1) s, andτ ′/τ = 1.5(2) for Fig. 1(b), where we give 1σ uncertainties.We attribute the relatively small lifetime to off-resonantscattering of the dipole trap light or resonant excitation bystray light, which could also explain the difference in theobtained lifetimes (our background pressure would limit thelifetime to ∼100 s). We conclude that the thermal fractiondecays significantly slower than the BEC fraction and that themeasured lifetime ratio is in reasonable agreement with thetheoretical prediction of τ ′/τ = 4/3.To compare the measurements of the BEC fraction withthe atomic transfer model we first fit an exponential decayN (t) = N0e−t/τ to the total atom number for long hold timesto obtain the overall lifetime τ as well as N0. Here N0 isthe apparent atom number at t=0 in the absence of three-body loss. In the second step, we fit an exponential decayNT (t) = N0(1 − fc)e−3t/4τ to the thermal fraction, with onlythe apparent BEC fraction fc as a fit parameter. Finally, withthe obtained parameters N0, τ , and fc, we numerically solveEq. (4) to obtain NC(t). The fit results are shown in Fig. 1 assolid lines. The 1σ errors in the fit parameters are reflected inthe band around the theoretical curve for NC . We observe thatthe atomic transfer model describes the time evolution of thethermal and BEC fraction very well.IV. CONCLUSIONSOur data provide direct verification of the atomic transfermodel. The relatively short lifetime τ ≈ 20 s, together witha large scattering length, provide optimal conditions to seethis effect. Furthermore, at the relatively small numbers oftrapped atoms needed to demonstrate atomic transfer, MCPdetection allows better fitting of the BEC fraction comparedto absorption imaging. Atomic transfer is most directly visiblein the one-body loss regime, but is important in the two- andthree-body regime as well, as was theoretically discussed inRef. [10]. As a final remark we note that one should takecare using a BEC for loss measurements. The thermal fractionmay be small in the initial stage of BEC decay, but candramatically increase after long hold times affecting the decayof the condensate, which will finally become nonexponential.ACKNOWLEDGMENTSWe thank Jacques Bouma for technical support. Thiswork was financially supported by the Dutch Foundation forFundamental Research on Matter. S.K. acknowledges financialsupport from the Netherlands Organization for ScientificResearch via a VIDI grant.[1] J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod.Phys. 71, 1 (1999).[2] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland,E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 79, 337(1997).[3] J. Söding, D. Guéry-Odelin, P. Desbiolles, G. Ferrari, andJ. Dalibard, Phys. Rev. Lett. 80, 1869 (1998).[4] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye,H.-J. Miesner, J. Stenger, and W. Ketterle, Phys. Rev. Lett. 80,2027 (1998).025602-3BRIEF REPORTS PHYSICAL REVIEW A 85, 025602 (2012)[5] J. Söding, D. Guéry-Odelin, P. Desbiolles, F. Chevy,H. Anamori, and J. Dalibard, Appl. Phys. B 69, 257 (1999).[6] W. Vassen, C. Cohen-Tannoudji, M. Leduc, D. Boiron,C. Westbrook, A. Truscott, K. Baldwin, G. Birkl, P. Cancio,and M. Trippenbach, Rev. Mod. Phys. (to be published), e-printarXiv:1110.1361.[7] B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. Lett. 83,1751 (1999).[8] T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm,Phys. Rev. Lett. 91, 123201 (2003).[9] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin,B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl,and R. Grimm, Nature (London) 440, 315 (2006).[10] P. Ziń, A. Dragan, S. Charzyński, N. Herschbach, W. Hogervorst,and W. Vassen, J. Phys. B 36, L149 (2003)[11] A. S. Tychkov, T. Jeltes, J. M. McNamara, P. J. J. Tol,N. Herschbach, W. Hogervorst, and W. Vassen, Phys. Rev. A73, 031603(R) (2006).[12] J. S. Borbely, R.van Rooij, S. Knoop, and W. Vassen, Phys. Rev.A 85, 022706 (2012).[13] Y. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, JETP Lett.42, 209 (1985).[14] H. Wu and C. J. Foot, J. Phys. B 29, L321 (1996).[15] S. Moal, M. Portier, J. Kim, J. Dugué, U. D. Rapol, M. Leduc,and C. Cohen-Tannoudji, Phys. Rev. Lett. 96, 023203 (2006).[16] R. van Rooij, J. S. Borbely, J. Simonet, M. D. Hoogerland, K. S.E. Eikema, R. A. Rozendaal, and W. Vassen, Science 333, 196(2011).[17] G. V. Shlyapnikov, J. T. M. Walraven, U. M. Rahmanov, andM. W. Reynolds, Phys. Rev. Lett. 73, 3247 (1994).025602-4

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Non-exponential one-body loss in a Bose-Einstein condensate

Non-exponential one-body loss in a Bose-Einstein condensate

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