Three-body Loss In Lithium From Functional Renormalization

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  See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/23574129 Three-body loss in lithium from functionalrenormalization  Article   in  Physical Review A · December 2008 DOI: 10.1103/PhysRevA.79.053633 · Source: arXiv CITATIONS 33 READS 24 3 authors:Some of the authors of this publication are also working on these related projects: The angulon quasiparticle   View projectRichard SchmidtHarvard University 23   PUBLICATIONS   351   CITATIONS   SEE PROFILE Stefan FloerchingerCERN 34   PUBLICATIONS   517   CITATIONS   SEE PROFILE Christof WetterichUniversität Heidelberg 316   PUBLICATIONS   14,598   CITATIONS   SEE PROFILE All content following this page was uploaded by Christof Wetterich on 01 December 2016. The user has requested enhancement of the downloaded file.    a  r   X   i  v  :   0   8   1   2 .   1   1   9   1  v   1   [  c  o  n   d  -  m  a   t .  o   t   h  e  r   ]   5   D  e  c   2   0   0   8 Three-body loss in lithium from functional renormalization R. Schmidt, S. Floerchinger, and C. Wetterich Institut f¨ ur Theoretische Physik Universit¨ at Heidelberg Philosophenweg 16, D-69120 Heidelberg, Germany  We use functional integral methods for an estimate of the three-body loss in a three-component 6 Li ultracold atom gas. We advocate a simple picture where the loss proceeds by the formation of a three-atom bound state, the trion. In turn, the effective amplitude for the trion formation fromthree atoms is estimated from a simple effective boson exchange process. The energy gap of thetrion and other key quantities for the loss coefficient are computed in a functional renormalizationgroup framework. Ultracold fermion gases with three components shownew features as comparedto the well studies systems withtwo components. For degenerate fermions with SU(3)symmetry one finds in the unitarity limit of infinite scat-tering length the interesting tower of Efimov states [1].These are series of three-atom bound states with a geo-metrically decreasing gap parameter, reflecting the vio-lation of scale symmetry by a limit cycle scaling behav-ior of the renormalization flow [2, 3]. In the vicinity of the Feshbach resonance the lowest three-atom state, thetrion, is below the open channel energy level. At low tem-perature one can infer a phase structure different fromthe BCS-BEC crossover for the two-component system,namely an intermediate trion phase without superfluid-ity separating the superfluid BCS and BEC phases [3].Interesting quantum phase transitions may describe thephase transition between phases at vanishing tempera-ture.The trion bound state is also expected to persist if theSU(3) symmetry is violated by a different location andstrength for the Feshbach resonances between differentpairs of atomic components. Recent measurements of the three-body loss coefficient in a three-component sys-tem of   6 Li [4, 5] may find an interpretation in this way[6, 7]. We investigate here a simple setting, where theloss arises from the formation of an intermediate trionbound state, which subsequently decays into unspecifieddegrees of freedom – possibly the “molecule type” dimersassociated to the nearby Feshbach resonances. In turn,the trion formation from three atoms proceeds by the ex-change of an effective bosonic field, as shown in Fig. 1.We estimate the loss coefficient  K  3  as being proportional trion decay productstrioneffective bosonatoms FIG. 1: Three-body loss process involving the trion. to  p  =  3  i =1 h i g i m 2 φi 1  m 2 χ − i Γ χ 2  2 .  (1)Here  m 2 χ  and Γ χ  are the trion gap parameter and decaywidth, while  m 2 φi  describes a type of gap parameter forthe effective boson, such that its propagator can be ap-proximated by  m − 2 φi  . The Yukawa couplings  h i  couplethe fermionic atoms to the effective boson, and the trioncoupling  g i  accounts for the coupling between trion, atomand effective boson. We sum over the “flavor” indices i  = 1 , 2 , 3. We will estimate  m 2 χ ,  m 2 φi ,  h i  and  g i  from thenon-perturbative renormalization flow which arises froma simple truncation of the exact flow equation for theaverage action or flowing action [8], for reviews see [9].Recently we used the method of functional renormal-ization to describe a SU(3) invariant system of threefermion species close to a common Feshbach resonance[3, 10]. In this context we explored the manifestationof the Efimov effect and formulated some predictions onthe quantum phase diagram in such systems. In contrastto this theoretical model, the system consisting of three-component  6 Li atoms, which is of current experimentalinterest [4, 5], does not possess this SU(3) symmetry.The main difference is that the resonances do not occurat the same magnetic field, and thus, for a given mag-netic field  B , the scattering lengths of different pairs of atoms, (1 , 2), (2 , 3), and (3 , 1) differ from each other.In this letter we generalize the model presented in [3]to cope with this more general situation. Our truncationof the (euclidean) average action then readsΓ k  =   x  ψ ∗ i ( ∂  τ   − ∆ − µ ) ψ i + φ ∗ i  A φi ( ∂  τ   − ∆ / 2) + m 2 φi  φ i + χ ∗  ∂  τ   − ∆ / 3+ m 2 χ  χ + h i ǫ ijk ( φ ∗ i ψ j ψ k − φ i ψ ∗ j ψ ∗ k )+ g i ( φ ∗ i ψ ∗ i χ − φ i ψ i χ ∗ )  ,  (2)where we choose natural units     = 2 M   = 1, with the  2atom mass  M  . We sum over the indices  i ,  j ,  k  whereverthey appear. Here  ψ i  denotes the fermionic atoms,  φ i  abosonic auxiliary field which mediates the four-fermioninteraction and  χ  is a fermionic field representing thebound state of three atoms. Formally, this trion fieldis introduced as the field mediating the interaction be-tween atoms  ψ  and bosons  φ . We show this schemat-ically in Fig. 2. In the limit  m 2 χ  → ∞ ,  m 2 φi  → ∞ , h 2 i /m 2 φi  → | λ i | ,  g 2 i /m 2 χ  → | λ (3) |  the action describespointlike two-body interactions with strength  λ i , as wellas a three-body interaction with strength  λ (3) . We willconcentrate on a microscopic interaction of this pointliketype. We consider the “vacuum limit” where tempera-ture and atom density go to zero. Then the chemicalpotential  µ  in Eq. (2) satisfies  µ ≤ 0. A negative chemi-cal potential  µ  has the meaning of an energy gap for thefermions when some other particle (boson or trion) hasa lower energy. The dominant difference to the SU(3)symmetric model arises from the different propagators of the bosonic fields  φ 1   = ψ 2 ψ 3 ,  φ 2   = ψ 3 ψ 1 , and  φ 3   = ψ 1 ψ 2 . Inaddition, we allow in general for different Yukawa cou-plings  h i  corresponding to different widths of the threeresonances. Also the Yukawa-like coupling  g i  that cou-ples the different combinations of fermions  ψ i  and bosons φ i  to the trion field  χ   = ψ 1 ψ 2 ψ 3  is permitted to vary withthe species involved. Although the SU(3) symmetry isexplicitly broken, the system exhibits three global U(1)symmetries corresponding to the three conserved num-bers of species of atoms. FIG. 2: Interaction between atoms  ψ  and effective bosons  φ as mediated by the trion field  χ . The renormalization flow of the various couplings fromthe microscopic (UV),  k  = Λ, to the physical, macro-scopic (IR) scale,  k  = 0, is obtained by inserting the“truncation” (2) into the exact flow equation [8], for de-tails we refer again to [3]. The flow equations for thetwo-body sector, i. e. for the boson propagator param-eterized by  A φi  and  m 2 φ i , are very similar to the SU(3)symmetric case ( t  = ln( k/ Λ)) ∂  t A φi  =  h 2 i k 5 6 π 2 ( k 2 − µ ) 2 ,∂  t m 2 φi  =  h 2 i k 5 6 π 2 ( k 2 − µ ) 3 .  (3)Since the Yukawa couplings  h i  are not renormalized, ∂  t h i  = 0 ,  (4)we can immediately integrate the equations (3). The so-lution can be found in [3]. The microscopicvalues m 2 φi (Λ)(bare couplings) have to be choosen such that the phys-ical scattering lengths (at  k  = 0) between two fermions(renormalized couplings) are reproduced correctly. Theyare given by the exchange of the boson field  φ . For ex-ample, the scattering length between the fermions 1 and2 obeys a 12  = −  h 23 8 πm 2 φ 3 ,  (5)where all “flowing parameters” are evaluated at themacroscopic scale  k  = 0 and for  µ  = 0. We use this de-scription for the scattering between fermions  ψ  in termsof a composite boson field  φ  also away from the res-onance. We emphasize that the field  φ  is not relatedto the closed channel Feshbach molecules of the nearbyresonance. It rather describes an additional “effectiveboson” which may be seen as an auxiliary or Hubbard-Stratonovich field, allowing for a simple but effective de-scription. For the numeric calculations in this note wewill use large values of   h 2 i  on the initial scale Λ. Thiscorresponds to pointlike atom-atom interactions in themicroscopic regime.Quite similar to the scattering between fermions  ψ  interms of the bosonic composite state  φ  we use a descrip-tion of the scattering between fermions  ψ  and bosons  φ  interms of the trion field  χ . As an example, a process wherethe fermion  ψ 1  and the boson  φ 1  scatter to a fermion  ψ 2 and a boson  φ 2 , is given by a tree level diagram as in Fig.2. For vanishing center-of-mass momentum the effectiveatom-boson coupling reads λ (3)1 , 2  = − g 1 g 2 m 2 χ .  (6)The flow equations for the three-body sector within ourapproximation are given by the flow of the “mass term”for the trion field ∂  t m 2 χ  = 3  i =1 2 g 2 i k 5 π 2 A φi (3 k 2 − 2 µ + 2 m 2 φi /A φi ) 2  (7)and the Yukawa-like coupling  g i  with flow equation ∂  t g 1  =  − g 2 h 2 h 1 k 5  6 k 2 − 5 µ +  2 m 2 φ 2 A φ 2  3 π 2 A φ 2 ( k 2 − µ ) 2  3 k 2 − 2 µ +  2 m 2 φ 2 A φ 2  2 − g 3 h 3 h 1 k 5  6 k 2 − 5 µ +  2 m 2 φ 3 A φ 3  3 π 2 A φ 3 ( k 2 − µ ) 2  3 k 2 − 2 µ +  2 m 2 φ 3 A φ 3  2 .  (8)The flow equations for g 2  and g 3  can be obtained from Eq.(8) by permuting the indices 1, 2, 3. For simplicity, weneglected in the flow equations (7) and (8) a contribution  3that arises from box-diagrams contributing to the atom-boson interaction. As described in [3] this term can beincorporated into our formalism using scale-dependentfields. Also terms of the form  ψ ∗ i ψ i φ ∗ j φ j  with  i  =  j , thatare in principle allowed by the symmetries are neglectedby our approximation in Eq. (2). We expect that theirquantitative influence is sub dominant as it is the casefor the SU(3) symmetric case [10].We apply our formalism to  6 Li by choosing the initialvalues of   m 2 φi  at the scale Λ such that the experimentallymeasured scattering lengths (see Fig. 3) are reproduced.For  A φi (Λ) = 1, the value of   h i  parameterizes the mo-mentum dependence of the interaction between atoms onthe microscopic scale. Close to the Feshbach resonance itis also connected to the width of the resonance  h 2 i  ∼ ∆ B .We choose here equal and large values for all three species h 1  =  h 2  =  h 3  =  h . This correspond to pointlike interac-tions at the microscopic scale Λ. Since the precise valueof   h  is not known, we use the dependence of our results on h  as an estimate of their uncertainty. The initial valuesof the couplings  m 2 χ  and  g i  are parameters in addition tothe scattering lengths which have to be fixed from experi-mental observation. For equal interaction between atoms ψ  and bosons  φ  in the UV, the parameter to be fixed is λ (3) = −  g 2 (Λ) m 2 χ (Λ) (9)with  g  =  g 1  =  g 2  =  g 3 . Pointlike interactions at themicroscopic scale may be realized by  m 2 χ (Λ) →∞ .  1000  800  600  400  2000200400     a      [     a        0      ] 0 100 200 300 400 500 600  14  12  10  8  6  4  20      E      [    M    H   z     ] Magnetic Field  B [G] FIG. 3: (Color online)  Upper panel:  Scattering length  a 12 (solid),  a 23  (dashed) and  a 31  (dotted) as a function of themagnetic field  B  for  6 Li. These curves were calculated byP. S. Julienne [11] and taken from Ref. [4]. Lower panel:  Binding energy per atom  E   of the three-bodybound state  χ  b  = ψ 1 ψ 2 ψ 3 . The solid line corresponds to theinitial value  h 2 = 100 a − 10  , while the shaded region gives theresult in the range  h 2 = 20 a − 10  (upper border) to  h 2 = 300 a − 10 (lower border). We solve the flow equations (3), (4), (7) and (8) numer-ically. For some range of   λ (3) and  µ ≤ 0 we find  m 2 χ  = 0at  k  = 0 for large enough values of the scattering lengths a 12 ,  a 23  and  a 31 . This indicates the presence of a boundstate of three atoms  χ   = ψ 1 ψ 2 ψ 3 . The binding energy peratom  E   of this bound state is given by the chemical po-tential | µ | with  µ  fixed such that  m 2 χ  = 0 [3]. To comparewith the recently performed experimental investigationsof   6 Li [4, 5], we adapt the initial value  λ (3) such that theappearance of this bound state corresponds to a magneticfield  B  = 125G, the point where strong three-body losseshave been observed. Using the same initial value of   λ (3) also for other values of the magnetic field, all microcsopicparameters are now fixed. We can now proceed to thepredictions of our model.First we find that the bound state of three atoms existsin the magnetic field region from  B  = 125G to  B  =498G. The binding energy per atom  E   is plotted in thelower panel of Fig. 3. We choose here  h 2 = 100 a − 10  ,as appropriate for  6 Li in the (1,2)-channel close to theresonance, while the shaded region corresponds to  h 2 ∈ (20 a − 10  , 300 a − 10  ).As a second prediction, we present an estimate of thethree-body loss coefficient  K  3  that has been measuredin the experiments by Jochim  et al.  [4] and O’Hara  et al.  [5]. For this purpose it is important to note thatthe fermionic bound state particle  χ  might decay intostates with lower energies. These may be some deeplybound molecules not included in our calculation here.We first assume that such a loss process does not dependstrongly on the magnetic field  B  and therefore work witha constant decay width Γ χ  for the bound state  χ . Thedecay width Γ χ  appears as an imaginary part of the trionpropagator when continued to real time G − 1 χ  =  ω −    p 2 3  − m 2 χ  + i Γ χ 2  .  (10)Instead of working with negative  µ  chosen such that m 2 χ  = 0, as done for the computation of the binding en-ergy, we now perform an energy shift such that the zeroenergy level corresponds to the open channel and there-fore  µ  = 0. In the region from  B  = 125G to  B  = 498Gthe energy gap of the trion is then negative  m 2 χ  <  0.The three-body loss coefficient  K  3  for arbitrary Γ χ  isobtained as follows. The amplitude to form a trion outof three fermions with vanishing momentum and energyis given by   3 i =1 h i g i /m 2 φi . The amplitude for the transi-tion from an initial state of three atoms to a final state of the trion decay products (cf. Fig 1) further involves thetrion propagator that we evaluate in the limit of smallmomentum    p 2 = (  i   p i ) 2 →  0, and small on-shell atomenergies  ω i  =    p 2 i ,  ω  =   i ω i  →  0. A thermal distribu-tion of the initial momenta will induce some corrections.Finally, the loss coefficient involves the unknown verticesand phase space factors of the trion decay – for this rea-son our computation contains an unknown multiplicativefactor  c K  . In terms of   p  given by Eq. (1) we obtain thethree-body loss coefficient K  3  =  c K   p.  (11)  4Our result as well as the experimental data points [4]are shown in Fig. 4. The agreement between the form of the two curves is already quite remarkable. FIG. 4: (Color online) Loss coefficient  K  3  in dependence onthe magnetic field  B  as measured in [4] (dots). The solidline is a two-parameter fit of our model to the experimentalcurve. We use here a decay width Γ χ  that is independent of the magnetic field  B . We have used three parameters, the location of theresonance at  B 0  = 125G, the overall amplitude  c K   andthe decay width Γ χ . They are essentially fixed by thepeak at  B 0  = 125G. The extension of the loss rate awayfrom the peak involves then no further parameter.Our simple prediction involves a rather narrow secondpeak around  B 1  ≈  500G, where the trion energy be-comes again degenerate with the open channel, cf. Fig.3. The width of this peak is fixed so far by the assumptionthat the decay width Γ χ  is independent of the magneticfield. This may be questionable in view of the close-by Feshbach resonance and the fact that the trion mayactually decay into the associated molecule-like boundstates which have lower energy. We have tested sev-eral reasonable approximations, which indeed lead to abroadening or even disappearance of the second peak,without much effect on the intermediate range of fields150G  < B <  400G.In conclusion, a rather simple trion exchange picturedescribes rather well the observed enhancement of thethree-body loss coefficient in a range of magnetic fieldsbetween 100G and 520G. A similar trion dominatedthree-body loss is possible for large  B  ( B   850G), wherealso a trion bound state with energy below the open chan-nel exists. However, the dimer bound states are nowabove the open channel level, such that the trion decaymay be strongly altered. The role of trion bound states inthe resonance region is an interesting subject by its own,that can be explored by our functional renormalizationgroup methods with an extended truncation. Acknowledgement   We thank S. Jochim and the mem-bers of his group for interesting discussions and sendingus their experimental data. We also thank S. 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