Characterization of the magnetic field through the three-body loss near a narrow Feshbach resonance
Yang Chen, Shuai Peng, Hongwei Gong, Xiao Zhang, Jiaming Li, Le Luo
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Characterization of the magnetic field through the three-body loss near a narrow Feshbachresonance
Yang Chen, ∗ Shuai Peng, ∗ Hongwei Gong, Xiao Zhang, Jiaming Li,
1, 2, 3, † and Le Luo
1, 2, 3, ‡ School of Physics and Astronomy, Sun Yat-sen University, Zhuhai, Guangdong, 519082, China Center of Quantum Information Technology, Shenzhen Research Instituteof Sun Yat-sen University, Nanshan Shenzhen, Guangdong, China 518087 Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing, Zhuhai 519082, China (Dated: February 24, 2021)The narrow s-wave Feshbach resonance of a Li Fermi gas shows strong three-body loss, which is proposedto be used to measure the minute change of a magnetic field around the resonance. However, the eddy currentwill cause ultracold atom experiencing a magnetic field delayed to the desired magnetic field from the currentof the magnetic coils. The elimination of the eddy current effect will play a key role in any experiments thatmotivated to measure the magnetic field to the precision of a part per million stability. Here, we apply a methodto correct the eddy current effect for precision measurement of the magnetic field. We first record the three-bodyloss influenced by the effect of induced eddy current, then use a certain model to obtain the time constant of theactual magnetic field by fitting the atom loss. This precisely determines the actual magnetic field according tothe time response of the three-body loss. After that, we implement the desired magnetic field to the atoms sothat we can analyze the three-body loss across the whole narrow Feshbach resonance. The results show that thethree-body recombination is the dominated loss mechanism near the resonance. We expect this practical methodof correcting the eddy current error of the magnetic field can be further applied to the future studies of quantumfew- and many-body physics near a narrow Feshbach resonance.
A narrow Feshbach resonance is a closed channel domi-nated resonance rather than a open channel dominated reso-nance as in the broad case, whose near-threshold scatteringand bound states only exist over a small fraction of width nearresonance [1, 2]. It usually has a nontrivial energy dependentcollisional phase shift as well as a narrow resonance width,which makes the theoretical modeling and experimental ob-servation become complex and difficulty [3–14]. But this alsointroduces many interesting and difference physics in com-pare to the widely studied broad Feshbach resonance. Forexample, the atomic-dimer relaxation ratio of the broad Fes-hbach resonance in an ultracold Li Fermi gas is suppressedas a . for two-body s-wave scattering length a > [15],which results in an stable atom and molecular mixture, whilethis inelastic process is predicted to be enhanced if the effec-tive range of the s-wave scattering phase shift r eff is largerthan a [3].The two lowest-energy hyperfine ground states mixture of Li ( |↑i and |↓i ) has a narrow s-wave Feshbach resonance at543.3 G, whose resonance width is estimated to be only 0.1 Gand r eff = − a of the interaction potential is larger thanthe interparticle separation[9]. Thus, the three-body recombi-nation is supposed to be strong. As our previous study showsthat three-body recombination of Li through the narrow Fes-hbach resonance follows a van der Waals universal and hasan asymmetric strength profile across the resonance [16]. Inthe Bardeen-Cooper-Schrieffer (BCS) side, the loss width isabout E F /µ B , where E F is the Fermi energy and µ B is theBohr magneton. Therefore, in an extreme low temperature,the three-body loss will be located in a very small magneticfield regime with strong strength. In a sense, a milligauss sta-bility and millisecond speed of magnetic field are suggested.In this letter, we propose to use the ultra narrow width fea- ture of the three-body loss near the Li narrow Feshbach res-onance to sense the magnetic field. This method is a straightforward way to characterize the experienced magnetic fieldof the ultracold atoms. By applying the sensitive magneticfield discriminator from the three-body loss in narrow Fesh-bach resonance, the magnetic field changing time constant ishighly determined in our system. Then we use the fitted re-sult to study the time evolution result of the atom loss in theBose-Einstein condensate (BEC) side and find that the maincollision mechanism is also the three-body recombination inthe BEC side when the magnetic field is close to resonance.This is also the first experimental measurement.We product the ultracold Fermi gases in an optical dipoletrap [16, 17]. After the gases is cooled to . µ K , about N ( t = 0) = 32500 atoms per spin is left. The experimen-tal timing sequence of magnetic field is shown in Fig. 1. Wesweep the magnetic field to a initial value B i = 570 G, whereis tested to be stable for two component Fermi gases [9, 16].Then the magnetic field is jumped to the target value B f , stayfor a variable time t . After that it is jumped back to B i anddo the atom number N ( t ) detection. As a consequence of theinduced eddy current in the metal of our cold atom apparatus,the time response of magnetic field at the atom place is slowerthan the response of the driving current in the coils, whichintroduce error for many magnetic field dependent measure-ments if we use the driving current to scale the actual magneticfield. Especially, in our narrow Feshbach resonance experi-ments, the stabilization of the magnetic field often required tobe a part per million (ppm) level, which put forward higherrequirements for the dynamic properties of the magnetic field.Fig. 1 presents the measured driven current of the coils and es-timated magnetic field during a fast sweep. The result showsthe driving current can reach 20 ppm in 5.4 ms, but a bettercurrent resolution is absent due to the measurement limited ofthe digital multimeter. Considering the induced eddy current,the slowed down magnetic field response can be expressed bya first-order step response model [18], which is expressed as B ( t ) = ( B i − B f ) e − tτ + B f (1)Where τ is the time constant. The expected value of τ is zeroin an ideal scenario, but in our system, its value is on the orderof milliseconds. ! " $ % && ’( )* + , "-."-!"-/"-0 ) $ ,8 ’ ; < , +)=32$,<’)’$)2=( ,,,, ) .->/ $ % && ’( )* + , "-/0"-00 @@ FIG. 1. Timing sequence of driving current in the coils (blue dots)and estimated bias magnetic field at the atom place (red curve). Thedriving current is measured by a 7.5 bit digital multimeter (Keith-ley DMM7510) with 0.1 number of power line cycles and auto-zerosetup. The inner figure zooms in the current dynamic at the turningpoint. The real magnetic field is calculate from Eq. (1) with the mea-sured τ = 5 ms , B i = 570 . , and B f = 543 . . Ina typical three-body loss measurement sequence, the magnetic fieldstarts from B i , then jumps to B f and stay their for a variable time t ,after that jumps back to B i to do atomic detection. Three-body recombination at the BCS regime is measured.Its magnetic dependent properties can be described by[16] L ( B ) = 3 h K ad ( πmk B T ) / exp [ − µ B ( B − B ) k B T ] (2)where K ad is the atom-dimer relaxation rate, B is the res-onant magnetic field, h is the Planck constant and k B is theBoltzmann constant. Accounting the slow magnetic response,the three-body atom loss N ( t ) is modified to N ( t ) = 1 V Z t L ( B ( t )) dt + 1 N ( t = 0) (3)Where V is the average volume of atom gas. It is very im-portant that we find L is not sensitive to the very beginningunstable magnetic field response. So we submit the time ofeach measurements with a large enough time span, like 80ms in our experiment, then do the fitting and get L . Fig. 2presents these processes. We need point out that although wederive L , we still loss some information at the very begin-ning, which prevent our to explore some rapid phenomena, such as molecular formations[21], atomic-molecular collisionand molecular-molecular collision [5], etc. By fitting the time !" & % !’ !( !%%!)%!"%!’%!(%!% * & ( %!’%%!+%%!(%%! %%!%% ,-./ FIG. 2. Typical time evolution of /N with a three-body loss in ourexperiment. Red dots are the 80 ms time shifted data of black dots. Ifwe use a linear function to fit the black dots, the fitted /N ( t = 0) will become almost zero, which is because of the very beginningdata are taken under an unstable magnetic field due to the inducededdy current. Instead, fitting the red dots will give the right L and /N ( t = 0) . shifted /N ( t ) data, we can get the L . Fig. 3(a) shows themeasured L ( B ) at the BCS regime, which follows the ten-dency described in Eq. 2 with a fitted K ad = 4 . × − cm / s . Note that the /e width of L ( B ) is only . G,which is a much smaller than the resonance width 0.1 G. Wechoose three different B f values, and measured their N ( t ) re-spectively, as shown in Fig. 3(c). We manually fit N ( t ) withEq. (3), get the time constant of actual magnetic field response τ = 5 ms . In this way, the actual magnetic field need about54 ms to reach 1 ppm range, which also indicates this delayeffect should be considered in many rapid experiments. Wefurther use this magnetic field response prediction the N ( t ) atthe BEC side, as shown in Fig. 3(b, d). The results turn outthey are also in good agreement with the theoretical calcula-tion.It is notice that the step changing of /N around 35 ms inFig. 3(d) is the integration of three-body loss from the BCSside to part of the BEC side, rather than the evidence of themolecular formation. And the different final values of /N in the deep BEC regime are because of the different overlap-ping strength between the L ( B ) and the exponential decayof magnetic field under different B f . In a sense, the pre-cision determination of the actual magnetic field is very im-portant. Furthermore, our studies show the three-body loss isthe domination lossy mechanics in the BEC side for a narrowFeshbach resonance, which is also consistent with the theoryprediction [3]. It is should be pointed out that although ourresults shows the main feature across the narrow s-wave Fes-hbach resonance is three-body recombination, the profile of L ( B ) in the BEC regime is unknown and without experi-mental measurement before. Here, we use a Gaussian profile !" ’%( !&%!"%!&&!"&!&’&!" ) * + , - . / "2*!*&"2*! ("2*! ."2*! 2 %!&&!(&!.&!2&! &!&’&! :; < = + %& * & !" &’ (!)(!"%!)%!""!)"!" * & ( (""%)"%"")"" +,-. !" &%’ (!)(!"%!)%!""!)"!"&"!) * + , - . / )2 ! ()2 ! ")2 !(’)2 !(.)2 !(2)2 !(( %!""!’"!."!2"!("!"&"!( :; < = + %" " !" &%’ (!)(!"%!)%!""!)"!"&"!) * + , - . / )2 ! ()2 ! ")2 !(’)2 !(.)2 !(2)2 !(( %!""!’"!."!2"!("!"&"!( :; < = + %" " a) b)c) d) !" &’ (!)(!"%!)%!""!)"!" * & ( (""%)"%"")"" +,-. ! " $%" & ’ % ! (% ! "" ! (" ! " ) & ""%("%""("" * + , - . , / (12 ! %( / (12 ! / (12 ! / (12 ! / FIG. 3. (a) measured L ( B ) at the BCS side. Blue curve is the fitted result with Eq. (2). (c) time-dependent /N at three different B f values,which are marked at (a). (b) measured L ( B ) at the BEC side. Blue curve is the fitted result with Gaussian profile. (d) time-dependent /N at four different B f values, which are marked at (b). Both the width of L ( B ) are much smaller than the resonance width (Black curve is thecalculated scattering length). Solid lines in (b, d) are the fitting curves with Eq. 3 and the fitted time constant of the magnetic field curve. to fit the L ( B ) as shown in Fig. 3(b). The reason is experi-ment data of L ( B ) has a very narrow unitary regimes at theresonance point, which may have some connections with thenarrow p-wave Feshbach resonance [19]. The fitted Gaussianwidth is about 3 mG, which is comparable with the width atthe BCS side.In summary, we use the narrow Feshbach resonance of LiFermi gases to realize a precision measurement of the mag-netic field near the resonance. Our method directly measuresthe magnetic field at the location of the atom cloud, eliminat-ing the effects of induced eddy current and residual magnetic.According to the result, the magnetic field response shows alarge delay in comparing with the driven current. This is akey factor to be considered in many experiments related to thenarrow Feshbach resonance. We find that three-body recombi-nation dominates the atom loss in both BEC and BCS regimeof the resonance. In the degenerate temperature, the energybroadening of the resonance is small. The technique of elimi-nating the eddy current provides a way to precisely determinethe magnetic field of the narrow Feshbach resonance, and willgive other chances for the future studies.
ACKNOWLEDGEMENTS
This work is supported by Key-Area Research and De-velopment Program of GuangDong Province under GrantNo. 2019B030330001. J. Li received supports from Na-tional Natural Science Foundation of China (NSFC) underGrant No. 11804406, Fundamental Research Funds for SunYat-sen University 18lgpy78, Science and Technology Pro-gram of Guangzhou 2019-030105-3001-0035. L. Luo re-ceived supports from NSFC under Grant No. 11774436,Guangdong Province Youth Talent Program under Grant No.2017GC010656, Sun Yat-sen University Core TechnologyDevelopment Fund. ∗ These authors contributed equally to this work. † [email protected] ‡ [email protected][1] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbachresonances in ultracold gases, Rev. Mod. Phys. , 1225-1286 (2010).[2] C. L. Blackley, P. S. Julienne, and J. M. Hutson, Effective-rangeapproximations for resonant scattering of cold atoms, Phys.Rev. A , 042701 (2014).[3] Y. Wang, J. P. D’Incao, and B. D. Esry, Ultracold three-bodycollisions near narrow Feshbach resonance, Phys. Rev. A ,042710 (2011).[4] D. S. Petrov, Three-Boson Problem near a Narrow FeshbachResonance, Phys. Rev. Lett. , 143201 (2004).[5] T. T. Wang, M.-S. Heo, T. M. Rvachov, D. A. Cotta, and W.Ketterle, Deviation from Universality in Collisions of Ultracold Li Molecules,
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