Independent Control of Scattering Lengths in Multicomponent Quantum Gases
aa r X i v : . [ phy s i c s . a t o m - ph ] A ug Independent Control of Scattering Lengths in Multicomponent Quantum Gases
Peng Zhang , Pascal Naidon and Masahito Ueda
1, 2 ERATO, JST, Macroscopic Quantum Control Project, Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan Department of Physics, University of Tokyo, Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
We develop a method of simultaneous and independent control of different scattering lengths inultracold multicomponent atomic gases, such as K or K- Li mixture. Our method can be usedto engineer multi-component quantum phases and Efimov trimer states.
Introduction.
Recently, multicomponent gases of de-generate fermions [1, 2, 3] or boson-fermion mixtures[4] have attracted broad interest both theoretically andexperimentally. Novel quantum phases [1] and Efimovstates [3, 5] have been predicted in three-componentFermi gases. If interspecies scattering lengths can be al-tered independently, one can engineer Efimov states andthe quantum phases of a three-component Fermi gas, andcontrol an effective interaction between two-componentfermions immersed in a Bose gas [4]. The magnetic Fesh-bach resonance (MFR) technique [6] can control only onescattering length, or a few scattering lengths but not in-dependently. The optical Feshbach resonance technique[7] can be directly generalized to the independent controlof more than one scattering length [8]. However, it wouldsignificantly shorten the lifetime of the system unless theatom has a long-lived electronic excited state (e.g., the P state of Yb atom [9]).In this Letter, we combine the two ideas of MFRand rf-field-induced Feshbach resonance [10] to proposea method for independently controlling two scatteringlengths in three-component atomic gases. In our scheme,atoms are dressed via couplings between different hyper-fine states. With a proper magnetic field, the indepen-dent tuning of scattering lengths of atoms in differentdressed states can be achieved by individual control of theRabi frequencies and detunings for different couplings. Control of a single scattering length.
We first consideratoms with three ground hyperfine levels (Fig. 1a) | f i , | f i , and | g i . In this Letter, we use the Dirac bracket |i to denote the hyperfine levels of one or two atoms; | ) toindicate the spatial states of the relative motion betweentwo atoms, and |ii for the total two-atom state, whichincludes both spatial motion and hyperfine state. We as-sume that | f i is coupled to | f i through Rabi frequencyΩ and detuning ∆. Such a coupling can be realized with atwo-color stimulated Raman process (TCSRP), i.e., cou-pling the hyperfine states | f , i via a common excitedstate | e i (not shown in the figure) by two laser beams[11]. The atomic loss caused by the spontaneous decayof | e i can be suppressed when the laser frequencies arefar detuned from the resonance. We assume the Rabifrequencies to be at least one order of magnitude smallerthan the typical depth ( ∼ a few 100MHz) of optical traps(see e.g., [12]), so that the loss due to the decay of thestate | e i may be ignored. A stable coupling can also be cK(a) (Ω,∆) f f g 1 (Ω,∆) cf g 2(b)f h K (Ω,∆) f l gf W | Φ res ) f g 2r | Φ res ) f l g(c) cE c (B) K l f l gr (e) (Ω,∆) (Ω’,∆’) (d) (Ω,∆) f f d f f’ f g f d g f’ FIG. 1: (color online) (a) Schematic hyperfine levels used forthe control of a single scattering length. The states | f , i arecoupled to form two dressed states | f h,l i . (b) Bare channelswith hyperfine states | i ≡ | f i| g i and | i ≡ | f i| g i are cou-pled with parameters (Ω , ∆), and | i is coupled to the boundstate | φ res ) in the closed channel with hyperfine state | c i viainteraction W . The state | i can decay via a HFR processwith two-body loss rate K . (c) Dressed channels. a lg is res-onantly enhanced when the threshold energy of the channelwith | f l i| g i crosses the bound state | Φ res ) in | c i . (d), (e) Hy-perfine states used for the control of two scattering lengthswith our first (d) and second (e) methods. All the levels areplotted in the rotating frame of reference. In (b) and (c), r indicates the interatomic distance. realized via an rf field that induces a direct transition be-tween two hyperfine states. The Rabi frequency causedby the rf field can be 0 . − . . | i ≡ | f i| g i is stable, while the channel with | i ≡ | f i| g i can decayto another channel with hyperfine state | a i through ahyperfine relaxation (HFR) process. We further assumethat a static magnetic field is tuned to the region of MFRbetween the diatomic channel with | i and a bound state | φ res ) in a closed diatomic channel with hyperfine state | c i (Fig. 1b). The binding energy E c ( B ) of | φ res ) can becontrolled by magnetic field B . Therefore, in the rotat-ing frame of reference, the Hamiltonian for the relativemotion of two atoms is given byˆ H = ˆ H (bg) + E c ( B ) | φ res )( φ res | ⊗ | c ih c | + ˆ W + ˆ W † , (1)where ˆ H (bg) = −∇ X i =1 , ,a | i ih i | + ˆ V (bg) ;ˆ V (bg) = X i =1 , ,a h V (bg) i ( r ) + E i i | i ih i | + [Ω | ih | + V a ( r ) | ih a | + h . c . ] ;ˆ W = W ( r ) | φ res )( φ res | ⊗ | ih c | , where we set ~ = 2 m ∗ = 1 with m ∗ being the reducedmass. In Eq. (1), V (bg) i ( r ) ( i = 1 , , a ) is the backgroundscattering potential in the channel | i i with r the distancebetween the two atoms; W ( r ) is the coupling between | i and | c i ; V a ( r ) is the coupling between | a i and | i ; E i isthe asymptotic energy of the channel | i i in the rotatingframe. Here we choose E = 0, which implies E = ∆and E a = ∆ − δ with δ being the energy gap between | i and | a i .Hamiltonian (1) shows that scattering channels | i and | i are coupled via the Rabi frequency Ω. Since thiscoupling is given by the single-atom TCSRP, it does notvanish in the limit r → ∞ . The scattering length istherefore not well defined for the bare channels | , i . Toovercome this problem, we diagonalize the Hamiltonianby introducing dressed states | f l i = α | f i + β | f i and | f h i = β | f i − α | f i with eigenvalues E h/l = ∆ / ± (Ω +∆ / / and coefficients α = Ω[Ω + ( E l − ∆) ] − / and β = ( E l − ∆)[Ω + ( E l − ∆) ] − / which can be controlledvia ∆ and Ω. Since the effective coupling between thedressed scattering channels | f l i| g i and | f h i| g i vanishes inthe limit r → ∞ , the scattering length a lg between | f l i and | g i is well defined.In presence of the inter-channel coupling ˆ W , both | f l i| g i and | f h i| g i are coupled with the bound state | φ res )in the closed channel | c i . When the threshold E l of thechannel | f l i| g i crosses the energy E c of | φ res ), the Fesh-bach resonance between | f l i| g i and | φ res ) strongly altersthe scattering length a lg between | f l i and | g i . There-fore, for a given magnetic field, one can control a lg bytuning E l through the coupling parameters (Ω , ∆). Bya straightforward generalization of the method in Ref.[13], we obtain a lg = a (bg) lg − π Λ ll − κe iη Λ al D − i (2 π ) χ / Λ aa (2) with the (Ω , ∆ , B )-dependent parameters D = E c + h c | ( φ res | ˆ W † G ( P )bg ˆ W | φ res ) | c i − E l ;Λ αβ = hh Φ ( α )bg [0] | ˆ W ˆ W † | Φ ( β )bg [0] ii ( α, β = l, a );Γ = e iη hh Φ ( a )bg [0] | ˆ W ˆ W † | Φ ( l )bg [0] ii ; κ = 2 q Im[ a (bg) lg ] χ. Here a (bg) lg is the background scattering length between | f l i and | g i in the absence of MFR with | φ res ), and χ ≡ √E l − E a ; | Φ ( l,a )bg [0] ii are the background zero-energy scattering states with incident particles in thechannels | f l i| g i and | a i , that is, we have | Φ ( l )bg [0] ii =(1 + G (+)bg ˆ V (bg) )(2 π ) − / | f l i| g i and | Φ ( a )bg [0] ii = (1 + G (+)bg ˆ V (bg) ) | χ ˆ e z ) | a i with the zero-energy backgroundGreen’s functions G ( ± )bg = ( i ± − ˆ H (bg) ) − and G ( P )bg =( G (+)bg + G ( − )bg ) / | χ ˆ e z ) satisfying ( ~r | χ ˆ e z ) = (2 π ) − / e iχz is the eigenstate of the relative momentum with eigen-value χ ˆ e z , and η is determined by ˆ V (bg) . Equation (2) shows that one can control the effec-tive interaction between atoms in the states | f l i and | g i ,which is determined by the real part Re[ a lg ] of a lg . Un-der a given magnetic field, one can tune Re[ a lg ] by chang-ing the coupling parameters (Ω , ∆) around the resonancepoint where D = 0.Due to the coupling term V a in the Hamiltonian (1),the HFR process also occurs from the dressed channel | f l i| g i to | a i . The two-body loss rate K l due to thisHFR is proportional to the imaginary part Im[ a lg ] of a lg : K l = − π Im[ a lg ] [14]. In Eq. (2), the change ofIm[ a lg ] due to (Ω , ∆) is described by i (cid:0) π (cid:1) δ / Λ aa and i Im[ κe iη Λ al ] . Due to these two terms, the two-bodyloss is enhanced in the resonance region. To avoid thisdifficulty, one should either use the scattering channelswithout hyperfine relaxation for both | f i| g i and | f i| g i ,or choose the proper atomic species for which the pa-rameters ( | Λ al | , | Λ aa | , κ ) are sufficiently small, so thatthe peak of Im[ a lg ] is much narrower than the resonanceof Re[ a lg ] . The scattering and resonance between atoms in thedressed ground states have been discussed in the litera-tures [10, 15]. A new point in our scheme is that we notonly couple the two hyperfine states | f , i , but also em-ploy the MFR between | f i| g i and | c i . As shown above,the enhancement of a lg usually occurs when the dressedchannel | f l i| g i is in the resonance region of | φ res ). There-fore with the help of the MFR, we can control the loca-tion of the resonance of a lg by varying both the dressingparameters and the magnetic field. Independent control of two scattering lengths.
Thereare two methods to realize the independent control oftwo scattering lengths in a three-component system withour approach. The first method (Fig. 1c) is to use thestates | g i , | f , i , and an additional hyperfine state | d i . ∆ (MHz) a l ( a ) −10 −5 0 5 10−1000010002000 −17 −15 −13 −11 K ( c m / s ) ∆ ( ∆ ’)(MHz) R e [ a ] ( a ) K l K l’ Re[a l’g ]Re[a lg ] B=157.3G B=157.6G B=157.9G K− Li Mixture K gas (a)(b)
FIG. 2: (color online) (a): Control of two scattering lengthsin the three-component gases of a K- Li mixture. The scat-tering length a lg is plotted in units of the Bohr radius a asa function of ∆ with Ω = 40MHz and B = 157 .
3G (bluedashed curve), 157.6G (black solid curve) and 157.9G (reddash-dotted curve). (b): Control of two scattering lengthsin the three-component gases of K with B = 200G andΩ = Ω ′ = 2MHz. Re[ a lg ] (blue solid curve) and K l (greendash-dotted curve) are plotted as functions of ∆; Re[ a l ′ g ] (reddashed curve) and K l ′ (black dotted curve) are plotted asfunctions of ∆ ′ . Since the scattering lengths between | f ( f ′ ) i and | g i are not available, we take the values to be the sameas those between | f ( f ′ ) i and | g i , which are given in Refs.[12, 16, 17]. As shown above, the two states | f , i are coupled andform two dressed states | f h,l i . We further assume thescattering length a dg between | g i and | d i can be tunedvia a MFR which is close to the one between | i and | c i .Therefore, in the three-component system with fermionicatoms in the states ( | g i , | d i , | f l i ) we can control a dg bychanging the B -field. Once the magnetic field is tunedto an appropriate value, we can control the scatteringlength a lg by changing the coupling parameters (Ω , ∆)with the approach described above.In the second method, we make use of five hyperfinestates as shown in Fig. 1e. We assume the static mag-netic field is tuned to an appropriate value so that thebare channel | i is near MFR with a bound state in aclosed channel | c i , while | ′ i ≡ | f ′ i| g i is also in the re-gion of MFR with another closed channel | c ′ i . We assumethat two states | , i are coupled with Rabi frequency Ωand detuning ∆, and form two dressed states | f h,l i . Sim-ilarly the states | f ′ , i are coupled with Rabi frequencyΩ ′ and detuning ∆ ′ , and form dressed states | f ′ h,l i . Thenaccording to the above discussion, the scattering length a lg between | f l i and | g i can be resonantly controlled by(Ω, ∆), while a l ′ g between | f ′ l i and | g i can be resonantly controlled by (Ω ′ , ∆ ′ ). Therefore in the three-componentsystem with fermionic atoms in the states ( | g i , | f l i , | f ′ l i ),one can independently control a lg and a l ′ g by changingthe four parameters (Ω , ∆ , Ω ′ , ∆ ′ ).In the above two methods, two conditions are requiredto make the independent resonance controls of a lg and a dg ( a l ′ g ) practical. First, the two MFRs for the barechannels | i and | d i| g i ( | ′ i| g i ) should be close to eachother (e.g., the distance between the two resonance points . ′ should belarge enough (on the order of 10MHz). Fortunately wecan find such resonances in many systems [12, 16, 17, 18].In principle, the second method can be generalized to theindependent control of n − n -component system. To this end, one should make use of n − Possible experimental realizations.
Now we discusspossible implementations of our method. In a mixtureof K and Li [12], we can realize the three-level sys-tem with the first method in the above section. To thisend, we use the hyperfine state | / , / i of Li to be | g i and consider the state | / , − / i of K to be | d i . Wealso take the states | / , − / i and | / , − / i of K as | f i and | f i , and form the dressed states | f l i . With thehelp of the MFR in the channel | d i| g i with B = 157 . B = 0 .
15G [12] and the one of | i with B = 159 . B = 0 .
45G [12], one can tune a dg by changing the mag-netic field, and then tune a lg by changing the couplingparameters (Ω, ∆). We note that, since both of the twobare channels | i and | i are stable, there is no hyperfinerelaxation in the collisions between atoms in the states | g i and | f l i . The unequal masses of K and Li can leadto many new phenomena in such a three-component het-eronuclear system ( K atoms in | d i , | f l i and Li atomsin | g i ), e.g., the appearance of an Efimov state formedby two heavy atoms and one light atom when both a dg and a lg are large enough [5].With a multi-channel calculation based on the realis-tic K- Li interaction potential, one can determine thescattering length a lg as a function of (Ω, ∆). For sim-plicity, we use a square-well model [19] for the K- Liinteraction: V (bg) i ( r ) = − v (bg) i θ (¯ a − r ), W ( r ) = wθ (¯ a − r )and V a ( r ) = 0 . where θ ( x ) is the unit step functionand ¯ a is defined as ¯ a = 4 π Γ(1 / − R vdW with Γ( x ) be-ing the Gamma function and R vdW the van der Waalslength [19]. v (bg) i , w , v a and δ are determined by theexperimental scattering parameters. In the absence ofthe coupling between | f i and | g i , our model gives thebackground scattering lengths in the channels | f i| g i and | f i| g i , and the correct resonance point and width for theMFR between | f i| g i and | c i . Although the square-wellmodel cannot provide a quantitatively accurate result for a lg , it can give a qualitative and intuitive illustration ofour method. The results of our calculation are shownin Fig. 2a. It is clear that, for every given value of themagnetic field (or the scattering length a dg ), a lg can beresonantly controlled via the laser induced coupling (Ω,∆).It is also possible to implement our method in ultracold K atoms with the second method described above. Wecan utilize five hyperfine states to construct the five-levelstructure as schematically illustrated in Fig. 1e. Theground hyperfine state | / , − / i is used for | g i , whilefour other states | / , − / i , | / , − / i , | / , − / i and | / , − / i , respectively are taken as | f i , | f i , | f ′ i and | f ′ i . To utilize the MFR in the channel | i with B = 202 . B = 8G [16] and the one in the chan-nel | ′ i with ( B = 224 . B = 10G) [17], we choose B = 200G. As shown above, dressed state | f l ( f l ′ ) i can beformed by the couplings between | f ( f ′ ) i and | f ( f ′ ) i ,and one can independently tune ( a lg , a l ′ g ) by choosingdifferent dressing parameters in the three-component gasof atoms in the states ( | g i , | f l i , | f l ′ i ).In this scheme, the upper hyperfine states | f ( f ′ ) i caninduce the HFR processes of both bare channels | i , | ′ i ≡ | f ′ i and dressed channels | f l ( f l ′ ) i| g i . The val-ues of the loss rate of the bare channels | ′ ) i| g i of Kare not available. However, the loss rate of the process | / , / i| / , / i → | / , / i| / , / i has been mea-sured to be as low as 10 − cm / s [20]. We take this valueto be the loss rate of | ′ ) i in our calculation with thesquare-well model, where V a (1 ′ a ′ ) ( r ) is assumed to be v a (1 ′ a ′ ) θ (¯ a − r ). Both intra-channel scattering lengths(Re[ a lg ] , Re[ a l ′ g ]) and the two-body loss rate ( K l , K ′ l ) of | f l ( f ′ l ) i| g i are shown in Figs. 2b and 2c. We find that,around the resonance point where the loss rates ( K l , K ′ l )peak, there are broad regions with large absolute val-ues of (Re[ a lg ] , Re[ a l ′ g ]) and small ( K l , K ′ l ). Althoughwe do not perform the calculation with realistic interac-tions potential between K atoms, our result based onthe square-well model also shows the feasibility of ourscheme in a gas of K. Conclusion and discussion.
In this Letter we proposea method for the independent control of (at least) twoscattering lengths in three-component gases by prepar-ing atoms in dressed states and via independent tuning ofthe couplings among the hyperfine states. Under suitableconditions, our method can be generalized to the controlof n − n >
3) scattering lengths in an n -componentsystem. This would be a powerful technique for engineer-ing different types of homonuclear or heternuclear Efimovstates and for control of quantum phases. We have shownthat our scheme can be implemented in cold gases of Kor a K- Li mixture. It is also possible to apply ourmethod to bosonic systems, such as the K- Rb mix- ture, where two close Feshbach resonances at B =464Gand 467.8G [18] are available.We thank M. A. Cazalilla, S. Inouye, S. Jochim, Y.Kawaguchi, T. Mukaiyama, T. Shi, C. P. Sun, Y. Taka-hashi and R. Zhao for helpful discussions. We also grate-fully acknowledge thank E. Tiesinga for providing theinteraction potentials of Li atoms. [1] T. L. Ho and S. Yip, Phys. Rev. Lett.
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