Condensates of p-wave pairs are exact solutions for rotating two-component Bose gases

0.16 MB·English

by  T. Papenbrock

Condensates of p-wave pairs are exact solutions for rotating two-component Bose gases
T. Papenbrock,1,2,3,4 S. M. Reimann,5 and G. M. Kavoulakis6
1Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA
2Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
3GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
4Institut fu¨r Kernphysik, Technische Universita¨t Darmstadt, D-64289 Darmstadt, Germany
5Mathematical Physics, LTH, Lund University, P.O. Box 118, SE-22100 Lund, Sweden
6Technological Educational Institute of Crete, P.O. Box 1939, GR-71004, Heraklion, Greece
2
1
We derive exact analytical results for the wave functions and energies of harmonically trapped
0
two-component Bose-Einstein condensates with weakly repulsive interactions under rotation. The
2
isospin symmetric wave functions are universal and do not depend on the matrix elements of the

n two-body interaction. The comparison with the results from numerical diagonalization shows that
a the ground state and low-lying excitations consists of condensates of p-wave pairs for repulsive
J contact interactions, Coulomb interactions, and therepulsive interactions between aligned dipoles.

3
1 PACSnumbers: 05.30.Jp,03.75.Lm,67.25.-dk,03.65.Fd

]
s
Exact analytical solutions for interacting quantum for repulsive contact interactions, Coulomb forces, and
a
g many-body systems are very rare [1, 2]. However, they repulsive forces between aligned dipoles [21], the ground
- are of tremendous interest since they may provide us state contains a maximum number of isospin-singlet p-
t
n with further insight into the correlations of quantum- wavepairs. ThisLetteralsoexplainstherecentstudyby
a mechanical many-body systems. Even rarer are cases Bargiet al. [19],whofoundbynumericaldiagonalization
u
where the exactly solvable quantum many-body system that the yrast energy of two-component Bose gases is a
q
. canalsopotentiallyberealizedexperimentally. Afamous simple function of angular momentum.
t
a exampleisthecelebratedLaughlinstate[3]anditsgener- We consider a harmonically trapped two-component
m alizations [4] that describe the two-dimensional electron
dilute gasof N bosons ofafirstspecies ,andM bosons
- gas in the quantum Hall regime. In recent years, the of a second species . We assume that↓the interactions
d advanceswithultra-coldatomicquantumgaseshavesig- ↑
n are perturbatively weak and of equal strength between
nificantly broadened the range of experimentally acces-
o intra-speciesandinter-species. Thisspecialcaseofequal
c sible many-body systems that are also solvable exactly. scattering lengths is approximately realized in gases of
[ AnexampleistheinteractingBosegasinonedimension. 87Rb and gases of 23Na [22]. Let us list the conserved
Here, the Bethe ansatz offers an analytic solution to the
2 quantities involved in this problem. Besides the isospin
v Lieb-Liniger model [5], and the fermionization of bosons T, the total angular momentum L, the angular momen-
7 intheTonks-Girardeau[6]limithasbeenobserved[7,8].
tumofthecenterofmassL ,thetotalnumberofbosons
1 c
A M +N and the isospin projection T =(M N)/2
5 Exact analytical solutions exist also for single- ≡ z −
4 component dilute and weakly interacting Bose-Einstein areconserved. Thereisnosimplebasisthatreflectsthese
7. condensatesunderrotation[9–17]. Foralargeclassofre- symmetries simultaneously. In second quantization the
conservationofL,A,T ,andT isstraightforward,while
0 pulsive two-body interactions, the exact ground state of z
1 N bosons at angular momentum L results from project- the conservation of L, M, N, and Lc is most easily ex-
1 pressed in the configuration space within first quantiza-
ing the unique state where L particles carry one unit of
: tion. We will employ both pictures in what follows.
v angular momentum onto the subspace with zero angular
i momentumforthecenterofmass[14–16]. InthisLetter, Single-particle states φ of the spherical harmonic
X nlm
wegeneralizethisexactsolutiontotheinterestingcaseof oscillator have energies E = ~ω(2n + l + 3/2). We
r nl
a a two-component Bose gas [18, 19], and arrive at a very are interested in low-energetic states at high angular
appealing result: As in the single-component case, the momentum. For weakly perturbative interactions, only
exactsolutionsatangularmomentumLarestateswhere single-particle states with no radial excitation (n = 0)
Lbosonscarryoneunitofangularmomentumeach,and can contribute, and we can limit ourselves to maximally
as before one has to projectout excitations of the center aligned states with m = l. Thus, single-particle states
ofmass. Inthetwo-componentcase,however,isospinen- with single-particle angular momentum l are φ (z) =
0ll
ters as a good quantum number, and eigenstates can be zlexp( z 2/2) and φ (w) = (wlexp( w2/2) for the
0ll
−| | −| |
labeledbythenumberofisospin-singletp-wavepairsthat bosons of the species and , respectively. We denote
↓ ↑
enter the wave function. Isospin was introduced in nu- the coordinates of bosons belonging to the species by
↓
clear physics by Heisenberg [20] and – like ordinary spin z = x +iy (j = 1,...,N) and and by w = u +iv ,
j j j j j j
– is based on SU(2) symmetry of two-component sys- (j =1,...,M)forthe species. Here,xandy (u andv)
↑
tems. Thecomparisonwithnumericalresultsshowsthat are the two Cartesian coordinates perpendicular to the
2
axis ofrotationfor the ( ) species. Inwhat follows,we L=τ,T =0,T =0. The operatorsˆb† yields the angu-
↓ ↑ z 1↑
omit the ubiquitous Gaussians from the single-particle larmomentumL weseek,increasethe number ofbosons
wave functions. Let ˆb† and ˆb† create a boson in the to L+τ, while keeping isospin T = T = (L τ)/2 a
l↓ l↑ z −
state corresponding to zl and wl, respectively. The cor- goodquantumnumber. The applicationoftheoperators
respondingannihilationoperatorsareˆb andˆb ,andthe ˆb† increasethenumberofbosonstoAandkeepsisospin
l↓ l↑ 0↑
creation and annihilation operators fulfill the canonical a good quantum number. Finally, the desired particle
commutation relations for bosons. numbers M and N (i.e. Tz = (M N)/2) results from
Inthecaseofasinglespeciesofbosonsunderrotation, the Tˆ− operators. For repulsive int−eractions the ground
the groundstateatangularmomentumLconsistsessen- state consists of the maximum number of isospin-singlet
tiallyofLbosonscarryingoneunitofangularmomentum pairs (i.e. τ = min(L,N)). Thus, the observation by
each while the remaining N L bosons carry no angu- Bargi et al., together with an adaptation of Hund’s rule
−
lar momentum [12]. Small modifications of this picture for bosons and isospinsymmetry leads to eigenstates (2)
are due to the preservation of the angular momentum consisting of condensates of isospin-singlet p-wave pairs.
of the center of mass. Remarkably, the two-component Notethattheseargumentsareindependentofthedetails
case is somewhat similar, and Bargi et al. [19] numer- of the repulsive interaction. These are the main result
ically found that the ground state consists entirely of of the present Letter. As in the single-component case,
single-particle states with angular momenta l = 0 and minor modifications of this picture are due to the con-
l = 1. For L N M, there are L+1 such states served angular momentum of the center of mass. Let us
≤ ≤
(labeled by the number of bosons of the species that contrastour results to BCSpairing in Fermisystems. In
↓
are in the single-particle state with l = 1). What is BCS theory, a weakly attractive interaction leads to the
the ground state for repulsive interactions in the space formationofCooperpairs(i.e. pairsoffermions intime-
spanned by these states? To address this question for a reversed orbits). The resulting BCS ground state is a
two-component system, we recall the essence of Hund’s condensate of spin-singlet s-wave pairs, where the sym-
rule: For repulsive interactions, the interaction energy metric configuration-space wave function optimizes the
is minimized for wave functions that are antisymmetric interactionenergy. In our case, we deal with bosons and
in position space. In fermionic electron systems such as with repulsive interactions,andthis modifies the picture
atoms and quantum dots, this leads to a symmetric spin accordingly.
wavefunction. Thesamereasoningappliedtothepresent Let us compute the energies of the states (2). We
case of a two-species Bose gas would require the isospin generalize and extend the results of Ref. [14] to the
wavefunctiontobeantisymmetric,too,thusmakingthe case of two-component Bose gases and sketch the main
total wave function symmetric under particle exchange. steps. The Hilbert space (N) of N identical bosons
HL ↓
The operator at total angular momentum L is spanned by products
e (z)e (z) ... e (z) (with λ +λ +...+λ =L)of
Bˆ† 1 ˆb† ˆb† ˆb† ˆb† (1) elλe1mentλa2ry sy·mm·etλrkic polynomi1als 2 k
≡ √2(cid:16) 1↓ 0↑− 0↓ 1↑(cid:17)
creates an isospin-singlet p-wave pair (i.e. L = 1 and eλ(z)≡eλ(z1,...,zN)= X zi1zi2···ziλ .
T = Tz = 0). This pair is antisymmetric in position 1≤i1<...<iλ≤N
(3)
spaceandantisymmetricinisospinspaceandthustotally
Note that e (z) carries λ units of angular momentum.
symmetricunderexchange. Thus,the interactionenergy λ
For two-component mixtures of N and M identical
is minimized for condensates of isospin-singlet pairs. In
bosons with N M, the Hilbert space at total angu-
general, we have N = M, and the ground-state wave
≤
6 lar momentum L (with L M) is the sum
function will also consist of unpaired bosons. The states
≤
N−τ A−L−τ L−τ τ min(L,N)
|χτi≡(cid:16)Tˆ−(cid:17) (cid:16)ˆb†0↑(cid:17) (cid:16)ˆb†1↑(cid:17) (cid:16)Bˆ†(cid:17) |0i (2) X Hλ(N)⊗HL(M−)λ (4)
λ=0
haveangularmomentum L,isospinT =A/2 τ, isospin
projectionT =(M N)/2,andconsistentire−lyofsingle- of direct products of the Hilbert spaces of each compo-
z
particle states with−angular momenta l = 0 and l = 1. nent. Productsofelementarysymmetricpolynomialsare
Here 0 denotesthevacuum,andτ =0,1,...min(L,N) linearly independent and form a basis. In the absence of
is the|niumber of isospin-singlet pairs. The isospin oper- interactions, all states in the Hilbert space are degener-
atorsareTˆ = ∞ (ˆb† ˆb ˆb† ˆb )/2, Tˆ = ∞ ˆb† ˆb , ate. Perturbativelyweakinteractionswillliftthis degen-
and Tˆ2 =Tˆz Tˆ†P+l=T0 (Tl↑ +l↑−1).l↓Thl↓e eigen−valuePs ol=f0Tˆl↓anl↑d eracy.
− − z z z The two-body interaction for a two-component mix-
Tˆ2 are denotes as T and T(T + 1), respectively. Let
z ture of Bose gases can be written as
us understand the state (2) in detail starting from the
right. TheapplicationofthepairoperatorsBˆ†tothevac- V = v Vˆ = v (Aˆ +Bˆ +Cˆ ) . (5)
X m m X m m m m
uum yields a state of 2τ bosons with quantum numbers
m≥0 m≥0
3
Here, v is a matrix element and are, however, not eigenstates of the center-of-mass mo-
m
mentum. Let Pˆ be the projector onto wave functions
Aˆ = (z z )m(∂ ∂ )m , (6) 0
m X i− j zi − zj with zero angular momentum of the center of mass, i.e.
1≤i<j≤N Pˆ ψ(z,w) = ψ(z R,w R) for any wave function
0
Bˆm = 1≤iX<j≤M(wi−wj)m(∂wi −∂wj)m , (7) ψsu(bzs,pwa)c.eTsphaenwneadve−bfyunctio[−1n4s].P0Teλh(uzs),eLth−eλ(szt)ataersePˆin0 χthτe
M | i
Cˆm = X (zi−wj)m(∂zi −∂wj)m (8) twointhian|χ(τ5i)w. iNthotferotmhaEtqt.h(e2se)astraeteeisgednostnaottesdoepfethnedHonamthile-
1≤i≤N
matrix elements v of the interaction.
1≤j≤M m
Tocomputethecorrespondingeigenvalueswemakethe
aretheinteractionsbetweenbosonsofspecies ,between
bosons of species , and the inter-species inter↓action, re- ansatz P0ψL,n for the eigenfunction with
↑
spectively. Byconstruction,theinteractionpreservesan-
min(L,N)
gularmomentumL(i.e. thedegreeofthemonomialwave ψ = c(n)e (z)e (w) . (13)
L,n X λ λ L−λ
functionitisactingon),isoftwo-bodynature,and–due
λ=0
to its translationally invariant form (only differences of
Here, n is an additional label that distinguishes between
coordinatesandderivativesappear)–preservestheangu-
min(L,N)+1differentwavefunctions. Ourresultsbelow
larmomentumL ofthecenterofmass. Furthermore,the
c
suggestthatnisthenumberofisospin-singletpairs. The
interaction(5)isinvariantundertheexchangeofz w
l k
↔ eigenvalue equation VP ψ = E P ψ requires the
andthereforepreservesisospin. Forthe zero-rangedcon- 0 L,n n 0 L,n
tact interaction, we have v =( 1/2)m/m! [14]. coefficients c(n) to fulfill
m λ
−
Theactionoftheoperators(6)onelementarysymmet-
ric polynomials is particularly simple [14] 0 = (LM +λ(2L+N −M −2λ)−εn)c(λn) (14)
Aˆ e (z)=Bˆ e (w)=0 for m 3 , + (N λ)(L λ)c(n) +λ(M L+λ)c(n) .
m λ m µ ≥ − − λ+1 − λ−1
Cˆ e (z)e (w)=0 for m 3 ,
m λ µ Here, ε enters the energy eigenvalue
≥
Aˆ e (z)=Bˆ e (w)=Cˆ R=0 for m 1 . (9)
m 1 m 1 m v
≥ 0
E =A(A 1) +ALv +2v ε . (15)
Here, R 1 (e (z)+e (w)) denotes the center of mass. n − 2 1 2 n
≡ A 1 1
Equations(9)showthatonlytheterms0 m 2ofthe
≤ ≤ Note that the eigenvalue problem (14) does not depend
Hamiltonian (5) are of interest when acting on products
on the matrix elements of the two-body interaction. For
e (z)e (w). For the operators V and V we find
λ µ 0 1 the solution of the eigenvalue problem, we make the
Vˆ = A(A 1)/2 , Vˆ =A(Lˆ Lˆ ) , ansatz
0 1 c
− −
N M n
Lˆ ≡ Xzi∂zi +Xwj∂wj , cλ(n) =Xβkλk . (16)
i=1 j=1 k=0
Lˆ 1  N M z w ∂ ∂  . (10) Here, we concealed the fact that the coefficients βk also
c ≡ A X X i k zj wl depend on n. We insert the ansatz (16) into Eq. (14)
i,j=1k,l=1  and compare the coefficients of λm, m = 0,...n + 2.
Only Vˆ is truly non-trivial, and This yields
2
Vˆ2eλ(z)eµ(w)=2(λN +µM +2λµ)eλ(z)eµ(w) εn =AL−n(A+1−n) , (17)
+ 2(N λ+1)(µ+1)e (z)e (w)
− λ−1 µ+1 which enters the energy (15). The coefficients βk,k < n
+ 2(M −µ+1)(λ+1)eλ+1(z)eµ−1(w) , arerecursivelydefinedintermsofβn (whichsetsthenor-
2A(N λ+1)e (z)e (w)R malization). The quantum number n acquires the values
λ−1 µ
− −
n = 0,1,2,...min(L,N), and the lowest energy is ob-
2A(M µ+1)e (z)e (w)R . (11)
λ µ−1
− − tained for n=min(L,N).
Equation (11) shows that the set Figure 1 shows the energy spectrum of a two-
Rne (z)e (w) with (12) component Bose-Einstein condensate with N = 4 and
λ L−λ−n
M≡{ } M = 8 particles per species, respectively, as a func-
0 λ min(L,N) , 0 n L λ
≤ ≤ ≤ ≤ − tion of the angular momentum L. The broken lines
spans a subspace in Hilbert space at angular momen- connect states with energies En from Eq. (15) for fixed
tum L that is left invariant by Vˆ . This subspace con- n = 0,1,2,... (from top to bottom). The ground state
2
tains the states e (z)e (z) which are linear combina- has n = min(L,N). Note that c(0) = 1 solves the
λ L−λ λ
tions of the eigenstates (2). The states e (z)e (z), eigenvalue problem (14) and yields the state ψ =
λ L−λ L,0
4
e (z ,...,z ,w ,...,w ). Thisstateistotallysymmet- ingtwo-componentBosegaseswithweakrepulsiveinter-
L 1 N 1 M
ric under the exchange of any particles, has maximum actions are condensates of isospin-singlet p-wave pairs,
isospin T = A/2 and contains no isospin-singlet p-wave andwederivedanalyticalexpressionsfortheenergiesand
pairs. For L N, the ground state (13) with n=L has the corresponding wave functions. The wave functions
≤
coefficientsc(L)/c(L) =( 1)λ(M L+λ)!(N λ)!/[(M are universalas they do not depend on the details of the
L)!N!]andcaλnbe0rewritt−enasSˆ −L (z w−). Hereth−e two-body interaction. Numerical computations demon-
symmetrization operator Sˆ ensuQresk=t1he ksy−mmketry under strate that these eigenstates are the ground states and
some of the low-lying excitations for the contactinterac-
exchangeofbosonsofeachspecies,andtheantisymmetry
tion,theCoulombinteraction,andrepulsiveinteractions
between the two species in position space is evident. We
between aligned dipoles.
thus believe that the label n of the energies (15) has to
be identified with the number τ ofisospin-singletp-wave
pairs of the eigenstates Pˆ χ with χ from Eq. (2).
0 τ τ
| i | i
70 Acknowledgments
60 WethankS.Bargi,J.Cremon,andW.Nazarewiczfor
discussions. WealsothankJ.Cremonforassistancewith
the numerical work. This work was partly supported
50
by the U.S. Department of Energy under Grant No.
Eint DE-FG02-96ER40963,by the Alexander von Humboldt-
Stiftung, and by the Swedish Research Council.
40
30
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