aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Spin Mixing in Spinor Fermi Gases
Ying Dong , and Han Pu , Department of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China Department of Physics and Astronomy, and Rice Quantum Institute,Rice University, Houston, Texas 77251-1892, USA Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China (Dated: June 10, 2018)We study a spinor fermionic system under the effect of spin-exchange interaction. We focus onthe interplay between the spin-exchange interaction and the effective quadratic Zeeman shift. Weexamine the static and the dynamic properties of both two- and many-body system. We find thatthe spin-exchange interaction induces coherent Rabi oscillation in the two-body system, but theoscillation is quickly damped when the system is extended to the many-body case.
PACS numbers: 03.75.Ss, 67.10.Db, 05.30.Fk
I. INTRODUCTION
Spinor quantum gas has received tremendous attentionever since the first creation of a spin-1 Bose-Einstein con-densate in an optical trap [1]. Most of these studies dealwith spinor Bose gases. The most salient feature of spinorcondensates is the presence of spin-exchange interactionwhich drives coheret spin-mixing dynamics. A naturalquestion can be asked is: can similar behavior be ob-served in a quantum degenerate fermionic system? Thecurrent work is an attempt to address this question.Quantum degenerate Fermi gases have been realized inmany laboratories [2–5]. Different with the normal elec-tronic system, many fermionic atoms have spins higherthan 1/2 in their lowest hyperfine manifold. These large-spin ultracold Fermi gases provide us a unique opportu-nity to investigate exotic many-body physics [6–8], andhave stimulated a great deal of theoretical interest [9–12].Considerable experimental progress has also been maderecently in the system of Sr ( f = 9 / , where f is totalhyperfine spin) [13] and Yb ( f = 5 / ) [14]. Both Srand
Yb have an alkaline-earth-like atomic structurewith all electron shells filled, thus their hyperfine spinscompletely come from nuclear spins. This will lead toa spin-independent atom-atom interaction since the nu-clear spins are deep within the atom. In these alkaline-earth-like fermionic system, the spin-independent inter-action can give rise to the so-called SU ( N ) symmetry,with N = 2 f + 1 [15–17]. However, for a more generalcold Fermi system including non-alkaline-earth atomswith large spins, the SU ( N ) symmetry may not be re-served. As a result, one can find a more rich phasesdiagram in its ground state [18]. In a seminal work pub-lished very recently [19], a spinor Fermi gas of K wasrealized. By taking advantage of the spin conservation,the effective spin of the system can be tuned from 1/2 to9/2.In this article, we focus on the simplest large-hyperfine-spin systems with f = 3 / with four internal components.This can be either true hyperfine spin (e.g., Cs, Beand
Hg) or effetive spin such as realized in Ref. [19].If we consider atoms with nonzero electron spins due to -1/2-3/2 3/21/2
FIG. 1: (Color Online) Schematic of a spin-mixing process. partially filled electron shells, then the interaction amongatoms will be spin-dependent. One of the key features inthis kind of systems is that there will be spin-exchangeinteractions which constantly mix different spin compo-nents. For example, two atoms with respective hyperfinespins − / and +1 / interact and become two atomswith hyperfine spins − / and +3 / , as schemcaticallydepicted in Fig. 1. A similar case of spin mixing has beenwell studied for spinor condensates [22–25].The paper is organized as follows: We present themodel Hamiltonian in Sec. II. The ground state proper-ties and the spin-mixing dynamics are discussed in Secs.III and IV, respectively. Finally, we conclude in Sec. V. II. MODEL HAMILTONIAN
To begin we consider a homogeneous dilute gas offermionic atoms with hyperfine spin f = 3 / in a boxwith volume V . The second quantized Hamiltonian ofthe system is given by H = ˆ d r [ X λ ψ † λ ( r )( − ~ m ∇ + p λ ) ψ λ ( r ) (1) + X λ ,λ ,λ ,λ U λ λ λ λ ψ † λ ( r ) ψ † λ ( r ) ψ λ ( r ) ψ λ ( r )] , where ψ λ ( λ = − / , − / , / , / is the atomic fieldannihilation operator associated with atoms in the hy-perfine spin state | f = 3 / , m f = λ i . The summation in-dices in (1) run through the values − / , − / , / , / . p λ is the bare atomic energy for spin state λ . We willconsider an effective quadratic Zeeman shift such that p − / = p / and p − / = p / as the linear Zeeman shiftcan be gauged away. If we consider the s -wave scatteringonly, then the interaction between atoms can be charac-terized by the coefficients U λ λ λ λ which are obtainedfrom the two-body interaction model ˆ U = g ˆ P + g ˆ P .Here, P F is the projection operator which projects thepair into a total hyperfine spin F state, g F is the inter-action strength in the total spin F channel, which, inthe calculation, will be replaced in favor of the s -wavescattering length a F via the regularization procedure: g F → m π ~ a F − V X k m ~ k . (2) For Fermi gases, there is no s -wave interaction in the oddtotal spin ( F = 1 , 3) channels, since these channels areforbidden by Pauli’s exclusion principle.We expand the field operators with plane wave function ψ λ = P k λ k e i k · r / √ V and complete the spatial integralto the Hamiltonian into momentum space: H = X k ,λ E λ k λ † k λ k + X k , k ′ , p [ c ( α † k ′ + p β † k − p ν k µ k ′ + µ † k ′ + p ν † k − p β k α k ′ ) + c ( α † k ′ + k β † k − p β k α k ′ + µ † k ′ + p ν † k − p ν k µ k ′ )+ g V ( α † k ′ + p µ † k − p µ k α k ′ + β † k ′ + p ν † k − p ν k β k ′ + α † k ′ + p ν † k − p ν k α k ′ + µ † k ′ + p β † k − p β k µ k ′ )] , (3)Here, c = ( g + g ) / V , c = ( g − g ) / V and E λ k = ~ k / m + p λ . For notational simplicity, wehave used α , β , µ and ν to denote the annihilation op-erators for the component with spin quantum number m f = 1 / , − / , / and − / , respectively. Fromthe Hamiltonian above, we can see clearly that the termswith coefficients c describe spin-mixing processes suchas that depicted in Fig. 1. Thus spin mixing requires c = 0 or g = g . In the case g = g , spin mixingdoes not exist and the population in each spin compo-nent is individually conserved. Under such a condition,the interaction obeys SU (4) symmetry [11, 12].As our focus is on the effect of spin-mixing interaction,we will consider the case with g = g . Great simplifi-cation can be further achieved by assuming g = 0 and g = 0 , in which case the second line of Eq. (3) vanishes.This is the case we will consider in this work. We notethat the essential physics does not change qualitativelyif g = 0 . III. GROUND STATE PROPERTIES
Throughout this work, we take temperature to be zero.We will first consider the mean-field ground state prop-erty of the system. In the case of g = 0 , it is obviousfrom Hamiltonian (3) that superfluid pairing can occurbetween spin components ( , − ) and ( , − ) . If wedenote C = P k h β − k α k i and C = P k h ν − k µ k i , theHamiltonian under the mean-field approximation can be written as H = X k ,λ ( E λ k + h λ ) λ † k λ k + X k [∆ ∗ β − k α k − ∆ ∗ ν − k µ k + h.c. ] , (4)where h λ = g P k h λ † k λ k i / V and ∆ = g V ( C − C ) . Todiagonalize this Hamiltonian, we can perform the Bogoli-ubov transformation α k = u k a k + v k b †− k , β − k = u k b − k − v k a † k µ k = s k u k + t k v †− k , ν − k = s k v − k − t k u † k (5)Here, the new coefficients should meet the following con-ditions to ensure the anticommutativity of new operators. | s k | + | t k | = 1 , and | u k | + | v k | = 1 . (6)Then we can write down the Bogoliubov-de Gennes(BdG) equations as (cid:18) E α k + h α − ˜ µ ∆∆ ∗ −E β k − h β + ˜ µ (cid:19) (cid:18) u k v k (cid:19) = E (cid:18) u k v k (cid:19)(cid:18) E µ k + h µ − ˜ µ − ∆ − ∆ ∗ −E ν k − h ν + ˜ µ (cid:19) (cid:18) s k t k (cid:19) = E (cid:18) s k t k (cid:19) (7)where we have introduced the chemical potential ˜ µ intothe equations. We assume that there are N particlesin total, and the population in opposite spin states areequal, i.e., N α = N β and N µ = N ν . Then we have N = P k ( | v k | + | t k | ) and the order parameter is given by ∆ = − g V X k ( u k v ∗ k − s k t ∗ k ) . (8) n ± / n ± / n ± / n ± / p= p= p= p= p= p= p= p= p= p= p= p= k k / F k k / F p E / F Δ / E F c)( FIG. 2: (Color Online) (a), (b) The momentum distribu-tion in spin components ± / and ± / for quadratic Zeemanshift p = 0 (solid blue), p = 1 (dot-dashed orange) and p = 10 (dashed red), in units of the Fermi energy E F . In the calcula-tion we set ¯ a = − for (a1) and (a2), and ¯ a = 1 for (b1) and(b2), where the dimensionless interaction strength is definedas ¯ a = 16 a k F /π with k F being the Fermi wavenumber. (c)Order parameter ∆ as a function of p . Without loss of generality, we take the effectivequadratic Zeeman shift as p α = p β = p ± / = 0 and p µ = p ν = p ± / = p ≥ . For a given p and the s -wavescattering length a , we solve the BdG equations self-consistently, and typical results are presented in Fig. 2.The upper panels of Fig. 2 display the momentum dis-tribution in different spin components at various values of p . As p increases, the population in the spin components m f = ± / decreases, and that in m f = ± / increases.This can be easily understood from the energetic pointof view. However, somewhat surprisingly, for large p , themomentum distribution in m f = ± / components ap-proaches a step function, exemplifying a normal Fermisea. Indeed, Fig. 2(c) shows that the magnitude of theorder parameter decreases as p increases. Intuitively, onemight have thought that for very large p , the populationin the m f = ± / components becomes negligible andthe system reduces into a two-component spin-1/2 Fermigas, which should exhibits a superfluid ground state atzero temperature. Results presented in Fig. 2 apparentlycontradicts this intuitive picture.To gain some insights into this problem, let us considerthe so-called Cooper problem where two extra particleswith attractive interaction lie on the surface of a filledFermi sea noninteracting atoms. The eigenvalue equa-tions describing the two interacting particles in k -space p E / F EE / F FIG. 3: (Color Online) The bound state energy of the Cooperpair as a function of p for dimensionless interaction strength(defined in Fig. 2) ¯ a = − (solid blue) and ¯ a = 1 (dashedred), respectively. are given by Eψ αβ ( k ) = E αβ ψ αβ ( k ) + g V X k ′ ( ψ αβ ( k ′ ) − ψ µν ( k ′ )) Eψ µν ( k ) = E µν ψ µν ( k ) + g V X k ′ ( ψ µν ( k ′ ) − ψ αβ ( k ′ )) (9)where E αβ = ~ k m − E F and E µν = ~ k m − E F +2 p , and ψ αβ and ψ µν are pairing amplitudes between m f = ± / , m f = ± / , respectively. The presence of the Fermisea imposes the restriction that k and k ′ lie outside ofthe Fermi sea, i.e., k, k ′ ≥ k F . A negative value of theeigenenergy E means that the two particles form a boundpairing with binding energy | E | . The larger | E | is, themore strongly the pair binds.After some mathematical manipulation, the two eigen-value equations (9) leads to the following single equationfor E : g V X k>k F (cid:18) E − E αβ + 1 E − E µν (cid:19) , (10)which can be solved numerically. The solution of E as afunction of p is displayed in Fig. 3, from which one can seethat, as p increases, | E | tends to zero. In other words,the pairs becomes less and less bound. This result isconsistent with the many-body result illustrated in Fig. 2.We therefore reach the conclusion that the energy mis-match induced by the quadratic Zeeman shift p , togetherwith the spin-exchange interaction, tends to break thepair apart. This phenomenon is reminiscent of the ef-fect by a magnetic impurity on a spin-1/2 superconduc-tor [20, 21]. In this latter system, the magnetic impurityinduces an energy difference between the two pairing par-ticles and has the tendency of destroying the pairing. IV. SPIN-MIXING DYNAMICS
So far we have focused on the ground state of the sys-tem. Now, let us turn to the spin-mixing dynamics. Be-fore dealing with the many-body situation, it may behelpful to investigate the Cooper problem first. We takethe initial state to be the ground state of the Coopersystem under an effective quadratic Zeeman field with p = 10 E F . At t = 0 , we suddenly turn this field off sothat p = 0 and the system starts to evolve. The dynam-ics of system is governed by Eq. (9) after replacing E onthe left hand side with i ~ ∂/∂t . The equations can besimplified if we redefine two quantities as follows: ψ ± ( k , t ) = 1 √ ψ αβ ( k , t ) ± ψ µν ( k , t )] , and the dynamical equations for p = 0 can be rewrittenas i ~ ∂∂t ψ + ( k , t ) = ~ k m ψ + ( k , t ) , (11) i ~ ∂∂t ψ − ( k , t ) = ~ k m ψ − ( k , t ) + g V X k ′ ψ − ( k ′ , t ) . (12)Hence the two equations for ψ ± are decoupled and theinteraction term only appears in the equation for ψ − .The above equations can be easily solved and resultsare presented in Fig. 4. Initially, due to the presenceof the large quadratic Zeeman shift, almost all the pop-ulations are in spin states m f = ± / . Spin mixingdynamics is initiated by quenching this Zeeman shift at t = 0 . We compare the cases for two different values ofthe interaction strength: a weak interaction with ¯ a = − and a strong interaction with ¯ a = 1 . In both scenarios,damping is observed in spin mixing, and stronger inter-action gives rise to a much faster damping. This can beintuitively understood as follows: The initial state canbe regarded as a superposition of different eigenstates ofthe quenched Hamiltonian. The stronger the interaction,the larger the number of the eigenstates contained in theinitial state. For t > , different eigenstates oscillate atdifferent frequencies which results in the damping of thepopulation dynamics. The more eigenstates are involved,the faster the damping. Therefore, such damping is a re-sult of the intrinsic multi-mode nature of the Fermi gas.For a system of spinor condensate near zero tempera-ture, as all the atoms occupy the same lowest-energy or-bitals, nearly undamped spin-mixing oscillations can beobserved [23–25].We now turn our attention to the dynamics in themany-body setting. As in the Cooper problem, we pre-pare the system in the ground state with p = 10 E F andat t = 0 quench the quadratic Zeeman field to zero. Theensuing mean-field dynamics can be simulated by solvingthe time-dependent BdG equations, obtained by replac-ing the eigenenergies at the right hand side of Eqs. (7)with i ~ ∂/∂t [26]. At each time step, the order parameterwill be updated as ∆( t ) = − g V X k ( u k ( t ) v ∗ k ( t ) − s k ( t ) t ∗ k ( t )) . Representative results are shown in Fig. 5. The dy-namics exhibits qualitatively similar properties as in theCooper problem: damping is observed in both the dyan-mics of the population and that of the order parameter, N ± N ± N ± N ± t FIG. 4: (Color Online) Populations in spin states m f = ± / and m f = ± / as functions of time (in units of ~ /E F ) attwo different interaction strengths. N ± N ± N ± N ± t t Δ popu l a ti on FIG. 5: (Color Online) Time dependence of population (up-per panels) and order parameter ∆ (lower panels), in unitsof E F , for two different interaction strengths. The horizon-tal lines in the lower panels represent the value of ∆ in theground state with p = 0 . Time is in units of ~ /E F . and the stronger the interaction, the faster the damping.This can be understood using a similar intuitive argu-ment we presented for the Cooper problem. We note thatsuch damping was observed in the experiment reportedin Ref. [19]. Finally we remark that the ground state cor-responding to p = 0 should have equal population in allfour spin states and an order parameter value indicatedby the horizontal line in the lower panels of Fig. 5. Inthe absence of any dissipation, as in our simulation, thesystem remains far away from the ground state. V. CONCLUSION
To conclude, we have examined the spin mixing inter-action of a degenerate Fermi gas with four internal spincomponents. It is quite remarkable that in the presence ofan effective quadratic Zeeman field that shifts relativelythe bare energies of spin-( ± / ) and spin-( ± / ) states,the system becomes almost normal. This is analogous tothe effect of the Zeeman field that breaks the symmetry ofthe two spin components in a two-component (spin-1/2)Fermi gas. We have also investigated the spin-mixing dy-namics initiated by quenching the quadratic Zeeman fieldand show that, unlike in the case of a spinor condensate,damping will necessarily occur in a many-body Fermi gasdue to the intrinsic multi-mode nature of fermions. Acknowledgments
This work is supported by the ARO Grant No.W911NF-07-1-0464 with the funds from the DARPA OLE Program, the Welch foundation (C-1669, C-1681),and the NSF. Y. Dong acknowledges financial supportfrom NNSF of China (Grant No. 11274085). [1] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur,S. Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle,Phys. Rev. Lett. , 2027 (1998).[2] B. DeMarco and D.S. Jin, Science , 1073 (1999).[3] A.G. Truscott, K.E. Strecker, W.I. McAlexander, G.B.Partridge, and R.G. Hulet, Science , 2570 (2001).[4] J.M. McNamara, T. Jeltes, A.S. Tychkov, W. Hoger-vorst, and W. Vassen, Phys. Rev. Lett. , 080404(2006).[5] T. Fukuhara, Y. Takasu, M. Kumakura, and Y. Taka-hashi, Phys. Rev. Lett. , 030401 (2007).[6] A. Rapp, et.al., Phys. Rev. Lett. , 160405 (2007).[7] P. Lecheminant, E. Boulat and P. Azaria, Phys. Rev.Lett. , 240402 (2005).[8] K. Rodriguez, et al. , Phys. Rev. Lett. , 050402 (2010).[9] T.L. Ho and S.K. Yip, Phys. Rev. Lett. , 247 (1999).[10] S.K. Yip and T.L. Ho, Phys. Rev. A , 4653 (1999).[11] Congjun Wu, Jiang-ping Hu, and Shou-cheng Zhang,Phys. Rev. Lett. , 186402 (2003).[12] Congjun Wu, Mod. Phys. Lett. B , 1707 (2006).[13] B. J. DeSalvo, M. Yan, P. G. Mickelson, Y. N. Martinezde Escobar, and T. C. Killian, Phys. Rev. Lett. ,030402 (2010).[14] S. Taie, Y. Takasu, S. Sugawa, R. Yamazaki, T. Tsuji- moto, R. Murakami, and Y. Takahashi, Phys. Rev. Lett. , 190401 (2010).[15] A.V. Gorshkov, et.al., Nature Phys. , 289 (2010).[16] M. Hermele, V. Gurarie, and A.M. Rey, Phys. Rev. Lett. , 135301 (2009).[17] M.A. Cazalilla, A.F. Ho, and M. Ueda, New J. Phys. ,103033 (2009).[18] Congjun Wu, Phys. Rev. Lett. , 266404 (2005).[19] J. S. Krauser, et al. , Nature Phys. , 813 (2012).[20] A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod.Phys. , 373 (2006).[21] L. Jiang, L.O. Baksmaty, H. Hu, Y. Chen, and H. Pu,Phys. Rev. A , 061604(R) (2011).[22] C. K. Law, H. Pu and N. P. Bigelow, Phys. Rev. Lett. , 5257 (1998).[23] H. Pu, C. K. Law, S. Raghavan, J. H. Eberly and N. P.Bigelow, Phys. Rev. A , 1463 (1999).[24] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman,and L. You, Phys. Rev. A , 013602 (2005).[25] M.-S. Chang, Q. Qin, W. Zhang, L. You, and M.S. Chap-man, Nature Phys. , 111 (2005).[26] A. Robertson, L. Jiang, H. Pu, W. Zhang, and H. Y.Ling, Phys. Rev. Lett.99