Rotating ground states of a one-dimensional spin-polarized gas of fermionic atoms with attractive p-wave interactions on a mesoscopic ring
aa r X i v : . [ c ond - m a t . o t h e r] N ov Rotating ground states of a one-dimensional spin-polarized gas of fermionic atomswith attractive p-wave interactions on a mesoscopic ring
M. D. Girardeau ∗ and E. M. Wright
1, 2, † College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA Department of Physics, University of Arizona, Tucson, AZ 85721, USA (Dated: November 15, 2018)The major finding of this paper is that a one-dimensional spin-polarized gas comprised of aneven number of fermionic atoms interacting via attractive p-wave interactions and confined to amesoscopic ring has a degenerate pair of ground states that are oppositely rotating. In any realizationthe gas will thus spontaneously rotate one way or the other in spite of the fact that there is noexternal rotation or bias fields. Our goal is to show that this counter-intuitive finding is a naturalconsequence of the combined effects of quantum statistics, ring topology, and exchange interactions.
PACS numbers: 03.75.-b,05.30.Fk
The rapidly increasing sophistication of experimentaltechniques for probing ultracold gases has caused a shiftof emphasis in theoretical and experimental work in re-cent years, from effective field approaches to more re-fined methods capable of dealing with strong correlations.Such strong correlations arise, for example, in ultracoldgases confined in de Broglie waveguides with transversetrapping so tight that the atomic dynamics is essentiallyone-dimensional (1D) [1], with confinement-induced res-onances [1, 2] allowing Feshbach resonance tuning [3] ofthe effective 1D interactions to very large values. Thishas allowed for the experimental verification [4, 5, 6] ofthe fermionization of bosonic ultracold vapors in suchgeometries predicted by the Fermi-Bose (FB) mappingmethod [7]. The “fermionic Tonks-Girardeau” (FTG) gas[8, 9], a spin-aligned Fermi gas with very strong attractive
1D odd-wave interactions, can be realized by a 3D p-waveFeshbach resonance as, e.g., in ultracold K vapor [10].It has been pointed out [2, 8, 9] that the generalizedFB mapping [2, 8, 9, 11] can be exploited in the oppo-site direction to map the ideal FTG gas with infinitelystrong attractive p-wave interactions to the ideal Bose gas, leading to “bosonization” of many properties of thisFermi system. One of us (MDG) in collaboration withA. Minguzzi showed recently [12] that for an even num-ber N of fermions on a mesoscopic ring, the ideal FTGground state is highly degenerate due to fragmentationof the mapped ideal Bose gas between two macroscopi-cally occupied orbitals with circumferential wave vectors ± π/L , where L is the ring circumference.The purpose of the present paper is to examine howthe ground state properties of a FTG gas on a mesoscopicring are changed for an even number N of fermions whenthe strength of the atom-atom attraction is made finite,although still very large, in which case there exists anFB mapping between the FTG gas and a system of N bosons with repulsive delta-function interactions. Basedon this we have found the remarkable result that whilstmost of the ground state degeneracy of the ideal FTGgas is lifted a twofold degeneracy remains, and these two states have nonzero total angular momentum ± N ~ , cor-responding to BEC of all N particles of the mapped Bosegas into either an orbital of angular momentum ~ or oneof angular momentum − ~ . The lifting of the degener-acy will be given a natural interpretation in terms of themapped Bose system using the famous no-fragmentationtheorem of Nozieres and Saint James [13]. Thus, in anygiven experimental realization with an even number N of fermions the FTG will spontaneously rotate one wayor the other around the ring with equal probability. Thisapparently violates a theorem of F. Bloch [14] requiringthe ground state to be non-rotating, and we shall demon-strate why the theorem fails for our case of fermions ona mesoscopic ring. Ideal FTG gas:
To set the stage we consider the idealFTG gas in general and on a ring. The ideal FTG gasconsists of spin-aligned fermions with infinitely-strongzero-range attractions which are a zero-width, infinitedepth limit of a square well of depth V and width 2 x ,with the limit taken such that V x → ( π ~ ) / m eff where m eff is the effective mass of the colliding pair[2, 8, 9]. It causes odd-wave scattering (1D analog of 3Dp-wave scattering) with 1D scattering length a D = −∞ ,with the result that all energy eigenstates Ψ F of the FTGgas are obtained from corresponding ideal Bose gas statesΨ B by the mapping Ψ B → Ψ F = A Ψ B , where A is thesame mapping function used in the original solution ofthe TG gas [7]: A ( x , · · · , x N ) = Q ≤ j<ℓ ≤ N sgn( x j − x ℓ )where the sign function sgn( x ) is +1 ( −
1) if x > x < x → x j − x ℓ = 0, so the discon-tinuity is an illusion of the zero-range limit [2, 8, 9, 11].The exact ground state isΨ F ( x , · · · , x N ) = A ( x , · · · , x N ) N Y j =1 φ ( x j ) , (1)where φ ( x ) is the lowest single-particle orbital for thegiven boundary conditions, illustrating the mapping fromthe ideal Bose gas ground state Ψ B = Q Nj =1 φ ( x j )to the fermionic FTG ground state Ψ F . The physicalmeaning of this mapping is clarified by looking at the so-lution of the two-fermion problem both inside and outsidethe square well, before passing to the limit x →
0. De-noting the relative coordinate by x = x − x , the exte-rior solution ( | x | > x ) in the ideal FTG limit (1D scat-tering length a D → −∞ ) represents a zero-energy scat-tering resonance, with wave function +1 for x > x <
0, and the interior solution ( | x | < x ) fittingsmoothly onto this is sin( κx ) with κ = p mV / ~ = π/ x . The corresponding mapped Bose gas state is+1 everywhere outside the well (ideal Bose gas groundstate), but inside the well it is sin( κ | x | ). Since thisvanishes with a cusp at x = 0, physical consistencyrequires the presence of a zero-diameter hard core inter-action added to the square well. The mapped Bose gasis then not truly ideal, but rather a TG gas with super-imposed attractive well , whose nontrivial interior wavefunction becomes invisible in the zero-range limit, simu-lating an ideal Bose gas insofar as the energy and exteriorwave function are concerned. The required impenetrablecore is physically quite reasonable, since the atoms havea strong short-range Pauli exclusion repulsion of their in-ner shells, whose diameter is effectively zero and strengthinfinite on length and energy scales appropriate to ultra-cold gas experiments.
Ideal FTG gas on a ring:
Here we review the discus-sion of Girardeau and Minguzzi for fermions on a ring ofcircumference L [12]. For an odd number N of fermionsEq. (1) is the correct FTG ground state, with φ = 1 / √ L the trivial and unique ground state orbital, a plane-wavewith zero wave vector k and hence zero angular momen-tum. However, if N is even we encounter a difficulty.For example, for N = 2 and 0 < x < L the mappingfunction A = sgn( x − x ) is -1 for x = 0 + ǫ but +1for x = L − ǫ , where ǫ → F isnot periodic but instead antiperiodic, violating the re-quirement that the fermion wave function be single val-ued. More generally, A ( x , · · · , x N ) is antiperiodic forall even N , leading to the same difficulty. This problemis easily repaired by taking the Bose ground state Ψ B to be the lowest anti periodic state, a fragmented BECwith wN atoms in the orbital e iπx j /L and (1 − w ) N in e − iπx j /L with 0 ≤ w ≤
1. The fermionic ground stateΨ F = A Ψ B is then properly periodic, but is ( N + 1)-fold degenerate and conveniently labeled by a quantumnumber ℓ z = ( w − ) N = 0 , ± , ± , · · · , ± N relatedto the eigenvalue L z of orbital angular momentum by L z = ℓ z ~ . Since N is even, L z takes on all integral valuesfrom − N ~ to N ~ , in spite of the half-integral angularmomenta of the two orbitals, and the fermionic groundstates have the same angular momentum spectrum sincethe total angular momentum operator commutes with the mapping function A .The above discussion leading to rotating and degener-ate ground states clearly relies on the mesoscopic natureof the ring as it relies on the discreteness of the angularmomentum. In contrast, Bloch’s “no-rotation” theorem[14] is based on a Galilean transformation treating themomentum as a continuous variable and therefore holdsstrictly only in the thermodynamic limit; that is whyour mesoscopic system can violate Bloch’s theorem. Fur-thermore, the physics underlying why FTG gases withan even number of fermions on a ring can have rotatingground states is also revealed in the above discussion:It is a consequence of the combined effects of quantumstatistics, which is implicit in the use of the mappingfunction A ( x , · · · , x N ) to incorporate the anti-symmetryof the fermion wave function, and the ring topology whichnecessitates a non-zero angular momentum so that theunderlying Bose ground state can exhibit the antiperiodicboundary conditions needed to counter the antiperiodic-ity of A . The same general argument applies also for thecase of finite interactions since the problem can still beapproached using the same mapping function A (see be-low), albeit that the mapped Bose problem is no longerideal. However, for the ideal FTG it is still possible toconsider combinations of the degenerate ground statesthat have zero angular momentum and to address thiswe need to consider the effects of finite interactions. FTG with finite interactions:
It has been realized in re-cent years that the FB mapping method used to exactlysolve the TG gas [7] and FTG gas [8, 9] is not restrictedto the TG case of point hard core boson-boson repul-sion and the FTG case of infinite zero-range fermion-fermion attraction. In fact, exactly the same unit anti-symmetric mapping function A ( x , · · · , x N ) used therealso provides an exact mapping between bosons withdelta-function repulsions g B D δ ( x j − x ℓ ) of any strength,i.e., the Lieb-Liniger (LL) model [15], and spin-alignedfermions with attractive interactions of a generalizedFTG form with reciprocal fermionic coupling constant g F D . The 1D scattering length a D < g B D = 2 ~ /m | a D | , g F D = 2 ~ | a D | /m , and g B D g F D = ~ m [2, 8, 9, 11]. The physical meaning ofthis mapping is illustrated by Fig.1, which shows, for thecase N = 2, how the mapping converts a FTG attractiveinteraction to the LL delta function interaction, whichgenerates cusps in the wave function in accord with theLL contact conditions [15]. The potential, both for Ψ F and Ψ B , consists of a deep and narrow square well ofwidth 2 x and depth V plus a TG impenetrable core at x = 0, as discussed previously for the ideal FTG limit a D → −∞ . The point core has no effect on Ψ F , whichalready vanishes at x = 0 due to antisymmetry, but it FIG. 1: Two-particle fermionic wave function Ψ F and mappedbosonic wave function Ψ B as a function of x for fixed x . Thepotential for both Ψ F and Ψ B is a deep and narrow squarewell plus a point hard core, but in the zero-range limit itseffect on Ψ B outside the well is the same as that of v B = g B D δ ( x ). is necessary for Ψ B for consistency of the mapping. Thegeneralized FTG well involves going to the zero widthlimit x → V → ∞ with V x held constant as for the FTG limit a D → −∞ ,but now at a slightly smaller value, such that the pa-rameter κ in the interior wave function sin( κx ) scalesas κ = π x − π | a D | as x → B as a LL deltafunction g B D δ ( x ); this is the physical meaning of themapping from the spin-aligned Fermi gas to the LL Bosegas. Here we will use this mapping to treat the caseof spin-aligned fermions with strong but finite attractionvia mapping to an LL gas with weak repulsions.The relative strengths of the FTG attractive p-waveinteractions and the LL repulsive interactions are quan-tified in the dimensionless coupling coefficients γ F = mg F D n ~ and γ B = mg B D ~ n , respectively, where n = NL isthe linear atomic density, and γ F γ B = 4. The ideal FTGgas is realized in the limit a D → −∞ so that γ F → ∞ and γ B →
0. Here we are interested in the limit of fi-nite but still very strong p-wave attractive interactions,or the limit γ F >> γ B <<
1. Moreover, thisis precisely the limit where the ground state propertiesof the LL Hamiltonian can be accurately captured us- ing mean field theory in which all N bosons are assumedto occupy the same single particle orbital φ that is de-termined self-consistently via the solution of the Gross-Pitaevskii (GP) equation [16]. The N -particle bosonwave function is then written in second quantized form as | Ψ B i = (ˆ a † ) N | i / √ N !, with ˆ a = R dxφ ∗ ( x ) ˆ ψ ( x ), ˆ ψ † ( x )and ˆ ψ ( x ) being boson creation and annihilation field op-erators. Lieb et al. [17] have studied the parameter spacein which a system of N bosons with repulsive interactionsin a 3D trap with tight transverse binding may be treatedas an effective 1D LL gas, and the GP approach should bevalid in the limit γ B < /N , the same criterion applyingto our ring trap with tight transverse binding.In order to address fragmentation of the ground stateof the FTG gas in the presence of finite p-wave inter-actions, here we adopt a more general Hartee-Fock ap-proach to the mapped LL problem in the limit γ B ≪ φ ± ( x ) = e ± iπx/L / √ L of the GPequation with antiperiodic boundary conditions and en-ergies E ± = ( π ~ ) mL N + gL N ( N −
1) where g ≡ g B D . Theexact second-quantized LL Hamiltonian isˆ H = Z L dx (cid:26) − ~ m ˆ ψ † ( x ) ∂ ∂x ˆ ψ ( x ) + g ψ † ( x )] [ ˆ ψ ( x )] (cid:27) , (2)and we evaluate its expectation value in the ( N +1) states | Ψ B ( w ) i = (ˆ a † + ) wN (ˆ a †− ) (1 − w ) N p ( wN )![(1 − w ) N ]! | i , (3)where ˆ a ± = R dxφ ∗± ( x ) ˆ ψ ( x ). The states in Eq. (3) rep-resent fragmented states of the two orbitals φ ± ( x ) insecond-quantized form, and are the analogue of the pre-viously discussed fragmented states Ψ B associated withthe ideal FTG gas. The energy expectation values of theLL Hamiltonian (2) taken with respect to the fragmentedstates (3) are given by E ( w ) = ( π ~ ) mL N + g L [ − N w +2 N w + N − N ] . (4)The energy assumes a maximum at w = , represent-ing a fragmented state with zero total angular momen-tum and equal occupations N of the orbitals φ ± , andthe energy minima are at w = 1 and w = 0, the un-fragmented states with all N atoms in either φ + or φ − ,energies E (1) = E (0) = ( π ~ ) mL N + g L N ( N − ± N ~ . These two rotating stateslie lower than the nonrotating state w = by an amount g L N , the extra exchange energy arising from fragmen-tation of the state w = , in accordance with the no-fragmentation theorem [13], which can be regarded as aphysical mechanism forcing the ground LL state to berotating. Since the total energy and total angular mo-mentum operators commute with the mapping function A , it follows that the FTG gas has two degenerate groundstates with opposite angular momenta ± N ~ , and this isthe main result of this paper.It is instructive to compare these mean field resultswith exact solutions of the LL Bethe ansatz equations[15] for N = 2 and antiperiodic boundary conditions.These states have the form Ψ( x , x ) = e ik x e ik x − e iθ e ik x e ik x in the sector 0 < x < x < L , and areequal to Ψ( x , x ) when 0 < x < x < L . For γ B ≪ k = − k and k = k where kL = π + ǫ with 0 < ǫ ≪
1, and we find ǫ = γ B π + O ( γ B ). Its energy is 2 (cid:0) πL (cid:1) + γ B L + O ( γ B ), whichagrees to order γ B L with the mean field energy E ( ) ofEq. (4), with LL units ~ = 2 m = 1 and g = 2 c = γ B L .The twofold degenerate ground states Ψ ± have k L = ± π − ǫ and k L = ± π + ǫ with 0 < ǫ ≪
1, and wefind ǫ = √ γ B + O ( γ B ) and energy 2 (cid:0) πL (cid:1) + γ B L , whichagain agrees with the mean field result E (0) = E (1) ofEq. (4) to order γ B L . We conclude that our mean fieldsolutions capture both the qualitative and quantitativefeatures of the exact antiperiodic LL solutions for γ B ≪
1, and in particular, the exact LL ground state is twofolddegenerate and rotating, with angular momenta ± N ~ .A potential scheme to measure the angular momen-tum of the rotating gas is the non-destructive approachdiscussed in Ref. [18], which employs Raman transitionsbetween hyperfine levels of the atoms due to oppositelycircularly polarized laser fields that propagate along adirection in the plane of the atomic ring. The spatial ab-sorption image for the circular polarizations can dependon the angular momentum state of the gas since the rota-tion gives rise to a Doppler shift that modifies the proba-bility of the Raman transition. Then if the gas is rotatingthe absorption image will show an asymmetry with re-spect to the axis passing through the center of the ring,since the atoms on either side of the axis will be mov-ing in opposite directions, experience different Dopplershifts, and hence produce different absorptions. Ring dark solitons:
So far we have tacitly assumed thatthe density profile of the ground state of the FTG gas,and the underlying LL gas, is spatially homogeneous. Re-laxing this assumption raises the possibility that a non-rotating ground state may be realized in the form of aring dark soliton. Dark solitons are real valued solutions φ DS ( x ) of the GP equation [19], neglecting any overallphase factor, which have an inhomogeneous density pro-file and a phase change of π at the position where the den-sity touches zero. Thus dark solitons have the requiredantiperiodicity for the mapped Bose solutions. Detailsof how to calculate the ring dark solitons for finite inter-actions are given exhaustively in Ref. [19], and for theideal FTG the dark soliton solution of the GP equationwith ( g = 0) degenerates into φ DS ( x ) = q L sin( πx/L ),and is degenerate with the previous rotating plane-wave solutions. When the ring dark solitons ( g >
0) arise theyalways have energies higher than the rotating plane-wavesolutions, so that although they are non-rotating theyrepresent excited states. An intuitive way to convey thisis that the healing length associated with the ring darksolitons always engenders a larger kinetic energy penaltythan the wavevectors ± π/L associated with the rotatingground states.In summary, we predict that the ground state of anFTG gas with an even number of fermions on a ringand finite p-wave interactions is doubly degenerate androtating. We have further shown that the rotation isa macroscopic manifestation of the strong microscopicmany-body correlations present in the gas. Observationof the spontaneous rotation of an FTG gas on a ringwould constitute a striking validation of the underlyingstrong correlations. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] M. Olshanii, Phys. Rev. Lett. , 938 (1998).[2] B.E. Granger and D. Blume, Phys. Rev. Lett. , 133202(2004).[3] J.L. Roberts et al. , Phys. Rev. Lett. , 4211 (2001).[4] B. Paredes, et al. , Nature , 277 (2004); T. Kinoshita,T.R. Wenger, and D.S. Weiss, Science , 1125 (2004).[5] T. Kinoshita, T.R. Wenger, and D.S. Weiss, Phys. Rev.Lett. , 190406 (2005).[6] T. Kinoshita, T.R. Wenger, and D.S. Weiss, Nature ,900 (2006).[7] M. Girardeau, J. Math. Phys. , 516 (1960); M.D. Gi-rardeau, Phys. Rev. , B500 (1965), Secs. 2, 3, and6.[8] M.D. Girardeau and M. Olshanii, cond-mat/0309396.[9] M.D. Girardeau, Hieu Nguyen, and M. Olshanii, OpticsCommunications , 3 (2004).[10] C. Ticknor, C.A. Regal, D.S. Jin, and J.L. Bohn, Phys.Rev. A , 042712 (2004).[11] T. Cheon and T. Shigehara, Phys. Lett. A , 111(1998) and Phys. Rev. Lett. , 2536 (1999).[12] M.D. Girardeau and A. Minguzzi, Phys. Rev. Lett. ,080404 (2006).[13] P. Nozieres and D. Saint James, J. Phys (Paris) , 1133(1982).[14] D. Bohm, Phys. Rev. , 502 (1949).[15] E.H. Lieb and W. Liniger, Phys. Rev. , 1605 (1963).[16] See, for example, E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2 (Pergamon Press, Oxford,1989)pp. 85-118.[17] E.H. Lieb, R. Seiringer, and J. Yngvason, Phys. Rev.Lett. , 150401 (2003); see REGION 2, p. 3 of thispaper.[18] K. P. Marzlin, W. Zhang, and E. M. Wright,Phys. Rev.Lett. , 4728 (1997).[19] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Phys.Rev. A62