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Static and Dynamic Properties of Trapped Fermionic Tonks-Girardeau Gases M. D. Girardeau∗ and E. M. Wright† Optical Sciences Center, University of Arizona, Tucson, AZ 85721 (Dated: February 2, 2008) We investigate some exact static and dynamic properties of one-dimensional fermionic Tonks- Girardeau gases in tight de Broglie waveguides with attractive p-wave interactions induced by a 5 Feshbachresonance. Aclosed form solution fortheone-bodydensitymatrix forharmonictrapping 0 isanalyzedintermsofitsnaturalorbitals, withthesurprisingresultthatforodd,butnotforeven, 0 numbers of fermions the maximally occupied natural orbital coincides with the ground harmonic 2 oscillator orbitalandhasthemaximally allowed fermionic occupancyofunity. Theexactdynamics of thetrapped gas following turnoff of thep-waveinteractions are explored. n a PACSnumbers: 03.75.-b,05.30.Jp J 1 1 The spin-statistics theorem, according to which iden- gas, the generalized FB mapping [20, 21, 22] can be ex- ticalparticleswithintegerspinarebosonswhereasthose ploited in the opposite direction to map the “fermionic ] t with half-integer spin are fermions, breaks down if the TGgas”[22,24],aspin-alignedFermigaswithstrong1D f o particles are confined to one or two dimensions. Real- atom-atom interactions mediated by a 3D p-wave Fesh- s ization of this fact had its origin when it was shown by bach resonance, to the ground state of the trapped ideal . at oneofus[1]thatthe many-bodyproblemofhard-sphere Bose gas. In this Letter we study some new and exact m bosonsinonedimensioncanbemappedexactlyontothat static and dynamic properties of fermionic TG gases. ofanidealFermigasviaa Fermi-Bose(FB)mapping. It Spin-aligned Fermi gas: Amagneticallytrapped,spin- - d is now known that this “Fermi-Bose duality” is a very alignedatomicvaporofspin-1 fermionicatomsinatight n 2 general property of identical particles in one-dimension waveguideismagneticallyfrozeninthespinconfiguration o (1D). In recent years this subject has become highly rel- , so the space-spin wave function must be spa- c ↑1 ··· ↑N [ evantthroughexperimentsonultracoldatomicvaporsin tially antisymmetric, s-wave scattering is forbidden, and atomwaveguides[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. the leading interaction effects at low energies are deter- 2 Exploration of these systems is facilitated by tunability mined by the 3D p-wave scattering amplitude. For a 1D v 4 of their interactions by external magnetic fields via Fes- gasof free spin-alignedfermions with harmonic trapping 2 hbach resonances [15]. In fermionic atoms in the same of frequency ω the Hamiltonian is 2 spinstate,s-wavescatteringisforbiddenbytheexclusion 01 pHroiwnceivpelre,athnedypc-awnabveeginreteartalycteinonhsanacreedubsyuaFlelyshnbeagclhigriebsloe-. Hˆ0 = N Hˆ(xp)= N (cid:20)−2~m2 ∂∂x22 + 21mω2x2p(cid:21), (1) 5 Xp=1 Xp=1 p 0 nances,whichhaverecentlybeenobservedinanultracold / atomic vapor of spin-polarized fermions [16]. Additional and the N-fermion ground state ψF0(x1, ,xN) is a t ··· a resonancesareinducedbytighttransverseconfinementin Slater-determinantof the N lowestHO orbitals φn(x)= m an atom waveguide. Of particular interest is the regime e−Q2/2Hn(Q)/π1/4x1o/sc2√2nn! [23], with Hn(Q) the Her- - of low temperatures and densities where transverse os- mite polynomials and Q=x/x where x = ~/mω osc osc d cillator modes are frozen and the dynamics is described isthelongitudinaloscillatorwidth. TheFermi-Bopsemap- n o by an effective 1D Hamiltonian with zero-range interac- pingmethod[1]mapsthisspin-alignedN-fermionground c tions [17, 18], a regime already reached experimentally state to the N-bosongroundstate of a systemof 1D im- : [8,9,10,11,12,13,14]. Transversemodesarestillvirtu- penetrablebosons(TGgas)leadingtoanexplicitexpres- v i ally excited during collisions, leading to renormalization sion for the N-boson ground state [23]. X of the effective 1D coupling constant via a confinement- Here we consider the opposite limit of the “fermionic r induced resonance. This was first shown for bosons [17] TGgas”[22,24],aspin-alignedFermigaswithstrong1D a and recently explained in terms of Feshbach resonances atom-atominteractionsinducedbya3Dp-waveFeshbach associated with bound states in closed, virtually excited resonance [16]. Granger and Blume derived an effective transverseoscillationchannels[19],andrecentlyGranger one-dimensional K-matrix for longitudinal scattering of and Blume [20] have shown that the same phenomenon two spin-alignedfermions confined in a single-mode har- occurs in spin-polarized fermionic vapors. monic atom waveguide [20]. It can be shown [20, 22, 24] that in the low-energy domain the K-matrix can be re- For a system of bosons with hard core repulsive inter- produced, with a relativeerroras smallas (k3), by the actions, a Tonks-Girardeau (TG) gas, the FB mapping O z contact condition relates the ground state for the system to the ground state of a free Fermi gas [23]. We have recently pointed ψ (x =0+)= ψ (x =0 )= aF ψ′ (x =0 ) F jℓ − F jℓ − − 1D F jℓ ± out [22, 24] that in the case of the spin-aligned Fermi (2) 2 where the prime denotes differentiation, Since A2 = 1, all properties expressible in terms of ψ 2, including the density profile, are the same as F aF1D = 6aV2p[1+12(Vp/a3⊥)|ζ(−1/2,1)|]−1 (3) |thos|e ofthe trappedBosegas groundstate Nj=1φ0(xj). ⊥ However, properties not so expressible, Qsuch as the momentum distribution, differ dramatically, as in is the one-dimensional scattering length, a = ~/µω ⊥ ⊥ the case of the usual bosonic TG gas. The ap- is the transverse oscillator length, Vp =pa3p = propriate tool for studying such differences is the limk→0tanδp(k)/k3 is the p-wave “scattering volume” reduced one-body density matrix (OBDM) ρ (x,x′;t)= − 1 [25], ap is the p-wave scattering length, ζ( 1/2,1) = N ψ (x,x , ,x ;t)ψ∗(x′,x , ,x ;t)dx dx . ζ(3/2)/4π = 0.2079... is the Hurwitz ze−ta function F 2 ··· N F 2 ··· N 2··· N − − ThRe normalized natural orbitals satisfy λjuj(x) = evaluated at ( 1/2,1) [26], and µ = m/2 is the reduced ∞ dx′ρ (x,x′;t = 0)u (x′), with eigenfunctions u (x), mass. The ex−pression (3) has a resonance at a nega- −∞ 1 j j jR = 0,1,2,..., and eigenvalues or orbital occupations tive criticalvalueVcrit/a3 = 0.4009 . Inaccordance p ⊥ − ··· 0 λj 1. For a spatially uniform bosonic TG with(2),thelow-energyfermionicwavefunctionsaredis- ≤ ≤ gas of length L the natural orbitals are plane waves continuous at contact, but left and right limits of their u (x) = ei2πxj/L/√L and the distribution of occupa- j derivatives coincide, and following Ref. [21] we assume tions λ reflects a momentum distribution function with j the same here. For an interatomic interaction of short low-momentum peaking and broad wings, whereas for but nonzero range x , the fermionic relative wave func- 0 a free Fermi gas the ground state is a filled Fermi sea tion vanishes at x = 0; Eq. (2) refers to the x 0+ jℓ 0 with λ = 1,j = 0,1,...,N 1, all other occupations → j limit ofthe exteriorwavefunction just outside the range − being zero [13, 14, 23]. Although natural orbitals are of the interaction [21], [22]. much-used in quantum chemistry, the first application The contact condition (2) is generated by the to degenerate gases is due to Dubois and Glyde [30]. 1D interaction pseudopotential operator [24, 29] In the case of the longitudinally trapped vîFnt = g1FD 1≤j<ℓ≤Nδ′(xjℓ)∂ˆjℓ where ∂ˆjℓψ = fermionic TG gas, the OBDM can be eval- p(1li/n2g)[∂coxjnψst|xajn=txPgℓ+1FD−=∂xℓ~ψ2|axF1jD=x/ℓµ−][2w0i,th24e,ffe2c7t]i.veT1hDisccoaun- uρ1a(txed,x′;itn =clo0s)ed=foNrmφ0(xf)oφr0(xa′l)l[F(Nx,x[′3)1]N]−1yiewldhienrge be compared with the bosonic 1D coupling constant F(x,x′) = ∞ sgn(x x′′)sgn(x′ x′′)φ2(x′′)dx′′ = cgo1BDup=lin−g~c2o/nµstaaB1nDt[i2s0γ,F24=]. −Thmeg1dFDimne/n~s2i.onNleostsefetrhmatiotnhiec W1−e|hearfv(eQn′)u−mRe−err∞fi(cQal)ly|, wdiia−tghoenraf(liQze)dththe−eerOroBrDfu0Mncttoioonb[t3a2i]n. density n is in the numerator, whereas it is in the de- the natural orbitals and occupations for a variety of N. nominator of the bosonic analog γB =mg1BD/n~2. In Figs. 1(a) and (b) we show the occupations λj versus Fermionic TG gas: The fermionic TG gas regime orbital number j for N = 2 and N = 3, respectively, is γF 1 valid in the neighborhood of the reso- as representative of our findings for even and odd ≫ nance where the denominator of Eq. (3) vanishes. numbers of atoms. (The solid line between data points The 1D scattering length is invariant under the FB is included as an aid to the eye). For N =2 we see that mapping [1, 20, 21, 22] ψB = A(x1, ,xN)ψF with the occupancy distribution shows a staircase structure ··· A(x1,··· ,xN) = 1≤j<ℓ≤Nsgn(xℓ −xj) the “unit an- where λ2q ≃ λ2q+1,q = 0,1,2..., the occupancies being tisymmetricfunctiQon”employedintheoriginaldiscovery monotonically decreasing λ > λ . Furthermore, the q q+1 of fermionization [1], so we shall henceforth indicate it occupancy distribution extends well beyond the Fermi by a1D = aF1D = aB1D. Since Vp must be negative to sea for the corresponding free Fermi gas shown as the achieve a resonance, the fermionic TG regime with at- dashed line, reflecting the strongly interacting nature of tractive p-wave interactions a1D →−∞ and g1FD →−∞ the fermionic TG gas . For N = 3 shown in Fig. 1(b) is reached as Vp approaches the resonance through neg- a similar staircase structure is seen with the exception ative values, implying an interaction-free exterior wave that λ = 1, that is the natural orbital u (x) has the 0 0 function. More generally, the spin-aligned Fermi gas maximal occupancy allowed by the exclusion principle, maps to the Lieb-Liniger Bose gas [28] with fermionic and λ λ ,q =0,1,2.... 2q+1 2q+2 ≃ and bosonic coupling constants inversely related [21] ac- ThenaturalorbitalsarestructurallysimilartotheHO dcoimrdeinnsgiotnolegs1BsDco=up−lin~g4/cµo2ngs1FtDan>ts0γ.BT=hemcgo1BrDre/snp~o2nd>in0g ojribsiteaqlusaalntdosthhaerenutmhebeprroopfefiretyldthnaotdetsh.eToorbeixtaalmniunme btheer and γF =−mg1FDn/~2 >0 satisfy γBγF =4 [22]. properties of the maximally occupied natural orbital we Natural orbitals: The N-atom ground state used a Rayleigh-Ritz variationalprocedure with a Gaus- of the longitudinally trapped fermionic TG gas siantrialsolutionu (x)=exp( x2/2w2)/π1/4√wtoob- 0 − maps to the trapped ideal Bose gas ground state tainthecorrespondingoccupancyλ (w)numerically. We 0 (γB = 0) for which all N atoms are Bose-Einstein have found that for odd N the maximal occupancy ob- condensed into the lowest HO orbital φ0(x): tained by varying w is λ0(w = xosc) = 1 for w = xosc, ψ (x , ,x ;t = 0) = A(x , ,x ) N φ (x ). sothatthe maximallyoccupiednaturalorbitalu (x) co- F 1 ··· N 1 ··· N j=1 0 j 0 Q 3 (a) (a) 1 0.7 0.6 0.8 0.5 λOccupation j00..46 Scaled density00..34 0.2 0.2 0.1 00 2 4 6 8 10 −020 −10 0 10 20 Natural orbital number j Q=x/xosc (b) 0.7 (b) 1 0.6 0.5 λOccupation j000...468 Scaled density000...234 0.1 0.2 0 −20 −10 0 10 20 Q=x/x osc 0 0 2 4 6 8 10 Natural orbital number j FIG. 2: Scaled density profile xoscρ(Q,t = T/2) after half a period for a) N =2 and N =8 (dashed line), and b) N =3 FIG. 1: Occupation distribution λj versus natural orbital and N =9 (dashed line). number j for a) N = 2, and b) N = 3. In each case the dashed line shows the corresponding result for a free Fermi gas. natural orbitals for the non-interacting gas ρ (x,x′;t>0)= C φ (x)φ (x′)e−i(n−m)ωt, (4) 1 mn n m Xm,n incides with the groundstateφ (x) of the single-particle 0 where C = λ dxu (x)φ (x) dxu (x)φ (x), HO. We have verified this numerical result by proving nm j j n j m j analytically, by tedious rearrangement and summation and the symmetrPy of thRe orbitals dictaRtes that Cnm is non-zeroonlyfor n andm both evenorbothodd. Thus, of series, that λ = 1 assuming u (x) = φ (x) for odd 0 0 0 theOBDMevolutionfollowingtheturnoffoftheinterac- N, an exact and surprising result. In contrast, for even tions is periodic in time with period T = π/ω. We have N the maximum occupancy is obtained numerically for solvedtheequationofmotionfortheOBDMnumerically w = x /2, the occupancy increasing with N but re- osc to obtain the exact dynamics of the initial fermionic TG maininglessthatunity,andcomparisonofthenumerical gas, and here we present results for the evolution of the resultsandGaussiansolutionshowsthatitisanapproxi- density profile ρ(x,t)=ρ (x,x;t). mationto the maximallyoccupiednaturalorbital,albeit 1 Figure 2(a) shows the scaled density profile a very good one. x ρ(Q,t = T/2) after half a period, when the osc Dynamics following turnoff of interactions: We next on-axis density reaches its lowest value, for N = 2 and examine the exact dynamics of the fermionic TG gas N =8(dashedline). Ineachcaseweseethatthedensity that ensue when the interactions are suddenly turned expands and the center (Q = 0) develops a minimum. off at t = 0 by detuning the external magnetic field. This may be intuited as follows: Before the interactions For t < 0 the OBDM is given by that for the fermionic are turned off the width of the density profile is x , osc ∼ TG gas ρ (x,x′;t < 0) = λ u (x)u (x′). For later narrower than the width for a trapped free Fermi gas 1 j j j j times, when the interactioPns are turned off, the evo- xosc√N. Thisdifferenceinwidthsarisesfromthe fact lution of the OBDM is given by i~(∂/∂t)ρ (x,x′;t) = ∼that the Fermi degeneracy pressure that sets the width 1 [Hˆ(x) Hˆ(x′)]ρ (x,x′;t) [33], where Hˆ(x) is the single- of the free gas in conjunction with the HO trapping po- 1 − particleHOHamiltonian. Eventhoughthisequationhas tentialis counteredbythe attractivep-waveinteractions nointeractionsthesubsequentdynamicsareadirectcon- inthefermionicTGgas. However,whentheinteractions sequenceofthestrongmany-bodycorrelationspresentin are turned off the Fermi degeneracy pressure, which the initial OBDM. Formally,the solution for the OBDM increases with N, acts to broaden the density profile. can be written in terms of the HO orbitals that serve as For times t > T/2 the density profile begins to contract 4 again under the action of the harmonic single particle † Electronic address: [email protected] potential, the initial condition being recovered at t=T. [1] M. Girardeau, J. Math. Phys. 1, 516 (1960); M.D. Gi- Figure 2(b) showsthe correspondingresults forN =3 rardeau,Phys.Rev.139,B500(1965),particularlySecs. 2,3, and 6. and N =9 (dashed line), and similar broadening is seen [2] M. Key et al.,Phys. Rev.Lett. 84, 1371 (2000). at t = T/2. In this case, however, a density maximum [3] D. Mu¨ller et al.,Phys. Rev.Lett. 83, 5194 (1999). is maintained at the center of the trap. The persistence [4] N.H. Dekkeret al.,Phys.Rev. 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Rev.Lett. 92, 190401 (2004). this maintains the on-axis density peak for odd N. The [12] F. Schrecket al., Phys.Rev.Lett. 87, 080403 (2001). persistence of a central peak may also be related to the [13] B. Paredes, et al.,Nature429, 277 (2004). [14] T. Kinoshita, T. Wenger, and D.S. Weiss, Science 305, fact that for odd N the maximally occupied natural or- 1125 (2004). bital u (x) for the fermionic TG gas with interactions is 0 [15] J.L. Roberts et al.,Phys. Rev.Lett. 86, 4211 (2001). identicaltothe HO orbitalφ (x)whichservesasa natu- 0 [16] C.A. Regal, C. Ticknor, J.L. Bohn, and D.S. Jin, Phys. ralorbitalwithno interactions. Thus, the scaleddensity Rev. Lett.90, 053201 (2003). profile associated with this maximally occupied orbital [17] M. Olshanii, Phys. Rev.Lett. 81, 938 (1998). is given by x ρ (Q)= C x φ2(Q) = exp( Q2)/√π, [18] D.S. Petrov, G.V. Shlyapnikov, and J.T.M. Walraven, which has a omscax0imum on0-0axoiss,c in0tegrates to−unity and Phys. Rev.Lett. 85, 3745 (2000). [19] T. Bergeman, M. Moore, and M. Olshanii, Phys. Rev. represents the centrally pinned particle, and persists af- Lett. 91, 163201 (2003). ter the interactions are turned off. For the N = 3 [20] B.E.GrangerandD.Blume,Phys.Rev.Lett.92,133202 simulation in Fig. 2(b) x ρ(0,T/2) 0.6, whereas osc (2004). ≃ xoscρ0(0)=0.56,showingthatthecentralpeakislargely [21] T. Cheon and T. Shigehara, Phys. Lett. A 243, 111 accountedforbythemaximallyoccupiednaturalorbital. (1998) and Phys.Rev.Lett. 82, 2536 (1999). For largeroddN the centralmaximumhas a largercon- [22] M.D. Girardeau and M. Olshanii, tribution from higher natural orbitals but the peak still arXiv:cond-mat/0309396 (2003). [23] M.D. Girardeau, E.M. Wright, and J.M. Triscari, Phys. persists; see the dashed line in Fig. 2(b). Rev. A 63, 033601 (2001). In summary, the OBDM of a harmonically trapped [24] M.D. Girardeau, Hieu Nguyen, and M. Olshanii, Optics fermionicTGgashasbeenanalyzedintermsofits natu- Communications 243, 3 (2004). ralorbitalsandoccupations. Foranoddnumberofatoms [25] H. Suno, B.D. Esry, and C.H. Greene, Phys. Rev. Lett. the maximally occupied natural orbital was found to co- 90, 053202 (2003). incide with the single-particle HO ground state. Due to [26] E.T. Whittaker and G.N. Watson, A Course of Modern therarityofexactresultsforstronglyinteractingsystems Analysis (Cambridge University Press, 1952), pp. 265- 269. it is important to suggest observable consequences. We [27] M.G. Moore, T. Bergman, and M. Olshanii, Journal de haveexploredtheexactdynamicsfollowingturnoffofthe Physique IV 116, 69 (2004) (Proceedings of Euro Sum- p-wave interactions, finding that for odd N the on-axis mer School on Quantum Gases in Low Dimensions, Les densityremainsamaximumasthe cloudexpandsdue to Houches, April 2003). Fermi degeneracy pressure, in contrast to even N where [28] ElliottH.LiebandWernerLiniger,Phys.Rev.130,1605 the density develops a minimum. (1963). This work was initiated at the ITAMP, Harvard- [29] M.D. Girardeau and M. Olshanii, Phys. Rev. A 70, 023608 (2004). Smithsonian Center for Astrophysics, in part supported [30] J.L. Dubois and H.R. Glyde, Phys. Rev. A 68, 033602 by the National Science Foundation, and at the Be- (2003). nasque,Spain2002workshopPhysicsofUltracoldDilute [31] ScottA.Bender,KevinD.Erker,andBrian E.Granger, AtomicGases. ItalsobenefitedfromtheQuantumGases arXiv:cond-mat/0408623 (2004). program at the Kavli Institute for Theoretical Physics, [32] M. Abramowitz and I.A. Stegun (eds.), Handbook of University of California, Santa Barbara, supported in Mathematical Functions (U.S. Government Printing Of- part by NSF grant PH99-0794. The work of MDG is fice, 1964), pp.297 ff. [33] J.W. Negele, Revs.Mod. Phys. 54, 913 (1982) p. 937. supportedbytheOfficeofNavalResearchgrantN00014- 99-1-0806 through a subcontract from the University of Southern California. ∗ Electronic address: [email protected]