One-dimensional Fermi gas with a single impurity in a harmonic trap: Perturbative description of the upper branch

0.23 MB·English

by  Seyed Ebrahim Gharashi

One-dimensional Fermi gas with a single impurity in a harmonic trap: Perturbative
description of the upper branch
Seyed Ebrahim Gharashi, X. Y. Yin, Yangqian Yan, and D. Blume1
1Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
(Dated: January 20, 2015)
The transition from “few to many” has recently been probed experimentally in an ultracold
5 harmonically confined one-dimensional lithium gas, in which a single impurity atom interacts with
1 a background gas consisting of one, two, or more identical fermions [A. N. Wenz et al., Science
0 342, 457 (2013)]. For repulsive interactions between the background or majority atoms and the
2 impurity, the interaction energy for relatively moderate system sizes was analyzed and found to
n convergetoward thecorresponding expression for an infinitelylarge Fermigas. Motivated bythese
a experimentalresults,weapplyperturbativetechniquestodeterminetheinteractionenergyforweak
J and strong coupling strengths and derive approximate descriptions for the interaction energy for
repulsiveinteractions with varyingstrength between theimpurityand themajority atoms and any
9
1 numberof majority atoms.

PACSnumbers:
]
s
a
g I. INTRODUCTION perturbation theory to harmonically confined systems
-
t and derive approximate solutions whose accuracy can
n
One-dimensionalBoseandFermisystemswithcontact be improved systematically by considering successively
a
u interactions have been studied for many decades now, higher orders in the expansion in the small parame-
q especially in the regime where the systems obey peri- ter. We focus on one-dimensional Fermi gases with a
. odic boundary conditions [1–5]. A large fraction of the single impurity under external harmonic confinement.
t
a eigenstates can be thought of as corresponding to gas- This system is of particular interest since it has been
m like states. A second subset of eigenstates corresponds realized experimentally in Jochim’s cold atom labora-
- to self-bound droplet-like states. These states maintain tory [18, 19]. In the experiments, the impurity is a
d
their bound state character in the absence of periodic lithium atom that occupies a hyperfine state different
n
o boundary conditions, i.e., in free space. In many cases, from the hyperfine state that the majority atoms oc-
c both the gas-like and droplet-like states can be obtained cupy. The trapping geometry is highly-elongatedand ef-
[ analyticallyviatheBetheansatz. TheBetheansatztakes fectively one-dimensional. We willshowthatour pertur-

advantage of the fact that the zero-range nature of the bativeresultsenableustocalculatetheenergyoftheup-
1
v interactions,combinedwiththefactthatparticlesinone per branch,which has beenstudied experimentally, with
9 dimension have to pass through each other to exchange fairly good accuracy for all N over a wide range of cou-
6 positions,allowsonetoidentifyconstantsofmotion. The pling strengths. In addition, our results provide bounds
3 solutionscanthenbe derivedinterms oftheseconstants on the energies in the weakly- and strongly-interacting
4 of motion. A closely related aspect is that a variety of regimes. These bounds can, e.g., be used to assess the
0 one-dimensional systems with two-body contact interac- accuracy of numerical solutions.
.
1 tions are integrable [1, 4]. The remainder of this paper is organized as follows.
0 The solution of the homogeneous system can be ap- Section II introduces the system Hamiltonian and nota-
5 plied to one-dimensional systems under spatially vary- tion. Section III summarizes our perturbative results.
1
ing external confinement via the local density approxi- The perturbative results areanalyzedinSecs. IV andV.
:
v mation [6–10]. This approximation typically provides a Finally, Sec. VI concludes.
i
X highly accurate description for a large number of par-
ticles but not necessarily for a small number of parti-
r II. SYSTEM HAMILTONIAN
a cles. It is thus desirable to derive moreaccuratedescrip-
tions for small one-dimensional systems with two-body
delta-function interactions under external confinement. We consider a single impurity immersed in a one-
Unfortunately, extensions of the Bethe ansatz to inho- dimensional Fermi gas that consists of N identical mass
mogeneous systems are, in general, not known. This m fermions. The mass of the impurity is equal to that
can be understood intuitively by realizing that the rel- of the majority or background particles. The impurity,
ative two-body momentum in inhomogeneous systems is with position coordinate z , interacts with the majority
0
not conserved due to the presence of the spatially vary- particles, with position coordinates z (j = 1, ,N),
j
···
ingconfinement. Correspondingly,harmonicallytrapped through a zero-range two-body potential with strength
one-dimensionalfew-bodysystemshavebeentreatednu- g,
merically by various techniques [7, 11–17].
In this work, we apply standard Raleigh-Schr¨odinger V (z )=gδ(z z ), (1)
2b j0 j 0
−
2
wherez =z z . TheHamiltonianH fortheharmon-
ically cojn0finedj−(N0,1) system then reads 10 (a)
)
ω
N N _h
(
H = Hho(zj)+Hho(z0)+ V2b(zj0), (2) ) / 5
j=1 j=1 N
X X (
E
wherethesingleparticleharmonicoscillatorHamiltonian
H (z) is given by
ho 0
~2 ∂2 1 2
H (z)= + mω2z2; (3)
ho −2m∂z2 2 (b)
)
N
here, ω denotes the angular trapping frequency. The (
F
delta-function interactions in Eq. (2) can be replaced by E 1
a set of boundary conditions on the many-body wave ) /
N
function Ψ(z ,z , ,z ),
0 1 N (
··· ε
∂Ψ ∂Ψ gm
= Ψ , (4) 0
∂zj0(cid:12)zj0→0+ − ∂zj0(cid:12)zj0→0−! ~2 |zj0→0 -4 - _h ω 0a / g 4
(cid:12) (cid:12) ho
where the(cid:12)(cid:12)limits zj0 (cid:12)(cid:12)0+, zj0 0−, and zj0 0
→ → →
are taken while keeping the other N coordinates, i.e.,
z , ,z ,z , ,z and (z +z )/2, fixed.
1 j−1 j+1 N j 0
··· ···
Inthe following,we determine the eigenenergiesE(N)
of the Hamiltonian H for various N. Throughout, we
FIG. 1: (Color online) (a) Solid, dotted, and dashed lines
restrict ourselves to the so-called upper branch. This show the energy of the upper branch for N = 1, 2, and 3
branch can be populated by preparing the system in the as a function of −1/g. The energies of the (1,1) system are
non-interacting limit (g 0+) and by then adiabati- obtained by solving the transcendental equation derived in
→
cally first increasing g to large positive values, then con- Ref. [21]. The energies of the (2,1) and (3,1) system are
tinuing acrossthe confinement-induced resonance[20] to taken from Refs. [14, 15, 22]. (b) Solid, dotted, and dashed
infinitely negative g values and finally increasing g to lines show the interaction energy ǫ(N), normalized by the
Fermi energy E (N), for N = 1, 2, and 3, respectively, as
small negative values. Solid, dotted, and dashed lines F
a function of −1/g. The harmonic oscillator length a is
in Fig. 1(a) show the energy of the upper branch for ho
definedin Eq. (10).
N = 1 [21], 2, and 3 [14, 15, 22], respectively, as a func-
tion of 1/g. For all N, the energy increases monotoni-
−
callyasafunctionofincreasing 1/g. Theupperbranch
− forsystemswithN =1,2,and3majorityparticles. The
corresponds to the ground state of the model Hamilto-
energy E (N) is directly proportional to the number of
nian when g is positive but not when g is negative. For F
majority particles,
negative g, the model Hamiltonian supports molecular-
like bound states. In real cold atom systems, energet- E (N)=N~ω. (6)
F
ically lower-lying molecular states exist even for posi-
tive g. However, it has been demonstrated experimen-
Figure 1(b) shows that the normalized interaction en-
tally [19] that the upper branch can be populated with
ergy depends relatively weakly on the number of parti-
reasonablyhighfidelity for positive g, motivating us—as cles. IndependentofN,wehaveǫ(N)=0forg =0+ and
well as others [7, 12, 13, 16, 17, 23–26]—to investigate
ǫ(N)=E (N)for g = . As canbe readoffFigs.1(a)
F
the properties of the upper branch within a stationary | | ∞
and 1(b), the energy increase of the upper branch is the
zero-temperature quantum mechanics framework. Since
sameonthepositiveg sideasitisonthenegativeg side,
decay to states with molecular character can lead to sig- indicating that ǫ(N) approaches 2E (N) in the g = 0−
F
nificantdepopulationofthe upperbranchfornegativeg,
limit for N = 1 3. We refer to E (N) as the Fermi
F
our primary focus in the following lies on the positive g −
energy of the majority particles. It should be noted,
portion of the upper branch.
however, that the “exact” Fermi energy of the majority
For g = 0+, the energy of the upper branch is equal particlesisE (N) ~ω/2,i.e.,E (N)correspondstothe
to E (N) = (N2+1)~ω/2. We write the energy E(N) F − F
ni leading order term of the Fermi energy of the majority
of the upper branch in terms of the interaction energy
particles in the large N limit.
ǫ(N),
One of the main goals of this paper is to derive ex-
pansions for the interaction energy of the upper branch
E(N)=E (N)+ǫ(N). (5)
ni around g = 0+ and g = using standard Raleigh-
| | ∞
Solid, dotted, and dashed lines in Fig. 1(b) show the Schr¨odinger perturbation theory for any N, i.e., for
interaction energies, normalized by the energy E (N), N = 1, , . To this end, we express the interaction
F
··· ∞
3
energies ǫ(0+) and ǫ(|∞|) in the vicinity of g = 0+ and
g = , respectively,inapowerseriesofthe dimension- TABLE I: Coefficients B(k)(N) for various (N,k) combina-
| | ∞ tions. The numbers in parenthesis denote the uncertainty
less interaction parameter γ (for g small) or in a power
series of γ−1 (for g large) [27], | | thatarisesfromevaluatingtheperturbationtheorysumswith
| | a finite energy cutoff. The numbers without errorbars have
been rounded.
kmax
ǫ(0+)(N)= B(k)(N)γk E (N)+ (γkmax+1) (7) k=1 k=2 k=3
" # F O N=1 0.179587 −0.0223551 0.00179230
kX=1 N=2 0.190481 −0.0239838 0.00179523
N=3 0.194409 −0.0244852 0.00177603(1)
and
N=4 0.196423 −0.0247210 0.0017627(1)
N=5 0.197647 −0.0248563 0.0017535(1)
ǫ(|∞|)(N)=
N=6 0.198469 −0.0249435 0.0017470(1)
kmax N=7 0.199059 −0.0250042
1+ C(k)(N)γ−k EF(N)+ (γ−(kmax+1)), (8) N=8 0.199503 −0.0250488
" k=1 # O N=9 0.199849 −0.0250828
X
N=10 0.200126 −0.0251096
wherethe dimensionlessinteractionparameterγ isgiven N=11 0.200353 −0.0251313
by N=12 0.200543 −0.0251491
N=∞ 0.202642 −0.0253303 0.00171100
π g
γ = , (9)
√2N ~ωaho
with TABLE II: Coefficients C(k)(N) for various (N,k) combina-
tions. Thenumbersinparenthesisdenotetheuncertaintythat
~ arisesfromevaluatingtheperturbationtheorysumswithafi-
a = (10) niteenergycutoff. Thenumberswithouterrorbarshavebeen
ho
mω
r rounded.
k=1 k=2 k=3
denotingthe harmonicoscillatorlength. Aswewillshow
N=1 −3.54491 3.85603 34.3007
below, the scaling of the interaction energy by E (N)
F N=2 −3.17245 2.41904(1) 25.38(2)
ensures a smooth connection between the energy shifts
N=3 −3.02854 1.8142(2) 23.78(8)
for finite and infinite N. In Eqs. (7)-(8), the dimension-
N=4 −2.95040
less kth-order perturbation theory coefficients B(k)(N) N=5 −2.90081
andC(k)(N)dependonN andwillbe determinedinthe N=6 −2.86634
next section. N=7 −2.84091
N=8 −2.82133
N=9 −2.80578
III. PERTURBATIVE RESULTS N=∞ −2.66667 0 21.0552
N limit: The impurity problem for the homo-
geneo→us ∞system with positive γ was solved by McGuire (1,1) system: The eigenenergies of the harmonically
in 1965[28]. Within the localdensity approximationthe trapped (1,1) system can be obtained for any γ by solv-
Fermi wave vector is replaced by the wave vector at the ing a simple transcendental equation [21]. Expanding
trapcentersuchthattheinteractionenergyoftheground the transcendental equation around the known eigenen-
state for the harmonically trapped system with N ergies for small and large γ, one obtains power series in
becomes [19] →∞ theinteractionenergy. Invertingtheseseries,oneobtains
analytical expressions for the B(k)(1) and C(k)(1) coeffi-
ǫ( ) γ γ γ 2π γ cients. We find B(1)(1) = π−3/2, B(2)(1) = ln(2)/π3,
E ∞( ) = π2 1− 4 + 2π + γ arctan 2π .(11) B(3)(1) = [π2 9ln(4)2]/(24π9/2), C(1)(1)−= 2π1/2,
F ∞ (cid:20) (cid:18) (cid:19) (cid:16) (cid:17)(cid:21) C(2)(1) = −4π[ln−(2) 1], and C(3)(1) = π3/2[π2− 24
− − − −
ExpandingEq.(11)aroundγ =0+ and γ = ,respec- 9(ln(4) 4)ln(4)]/3. The numericalvalues of these coef-
tively, B(k)( ) and C(k)( ) can be ob|ta|ine∞d for k = ficients−are summarizedin Tables I-II. As in the N
1,2, . We∞findB(1)( )=∞2/π2, B(2)( )= 1/(4π2), case,thesmallandlargeγ seriesforN =1,Eqs.(7)→an∞d
B(3)(···) = 1/(6π4), C∞(1)( ) = 8/3∞, C(2)(− ) = 0, (8), have a finite radius of convergence. Employing the
and C∞(3)( ) = 32π2/15. T∞he num−erical values∞of these techniquesofRef.[29],wefind—usingupto50expansion
coefficient∞s are summarized in Tables I-II. It is readily coefficients—that the small and large γ series converge
shown that the small and large γ series, Eqs. (7) and for γ <1.0745(2) 2π and γ −1 <[1.0745(2) 2π]−1,
| | × | | ×
(8), convergefor γ <2π and γ−1 <(2π)−1, respectively. respectively. Our result for the convergence of the small
Table II shows that C(2)( ) vanishes. We will return γ series is consistent with what is reported in the litera-
to this finding when we dis∞cuss the N-dependence of the ture [30].
C(k)(N) coefficients. Weakly-repulsive(N,1)system,N =2,3, : Totreat
···
4
the weakly-interacting system with finite N, N > 1, we These matrix elements are closely related to the bound-
rewrite the system Hamiltonian in second quantization aryconditionrepresentationofthe one-dimensionalodd-
and expand the field operators for the majorityparticles parity pseudo-potential [35, 36]. We evaluate the inte-
and the impurity in terms of single particle harmonic grals in Eq. (13) analytically for N =1 4. The analyt-
−
oscillator states (see, e.g., Ref. [31]). The interaction ical results for N = 1 and 2 read C(1)(1) = 2√π and
−
matrix elements can be evaluated analytically and the C(1)(2) = π/2(81/32). The analytical expressions
−
first-order perturbation theory treatment for positive g for N = 3 and 4 are lengthy and not reported here [37].
p
yields ForlargerN,weperformallbutoneintegrationforeach
oftheperturbationmatrixelementsanalytically. There-
2√NΓ(1/2+N) sultingnumericallydeterminedenergyshiftsareaccurate
B(1)(N)= . (12)
π2N! to morethan 10digits. Table II summarizesthe numeri-
cal values for the coefficient C(1)(N) for N 9 obtained
≤
The first-order energy shift may be interpreted by us. The extension to larger N is, although tedious,
as the leading-order mean-field shift. We find possible in principle.
limN→∞B(1)(N) = 2/π2, which agrees with the coef- To determine the energy shift proportional to γ−2,
ficient obtained by expanding Eq. (11). The evalua- we use second-order perturbation theory. Reference [38]
tion of the second-order energy shift involves the eval- pointed out that the second-order perturbation theory
uation of infinite sums. We find, as expected, that these energy shift of the (1,1) system diverges, thus requiring
sums converge. The reason is that the one-dimensional regularization. Analogous divergencies arise in the per-
delta-function interaction does not, unlike two- or three- turbativetreatmentofone-dimensionalsingle-component
dimensional delta-function interactions [32, 33], require Fermi gases with generalized delta-function interactions
any regularization if used in standard perturbation the- (see, e.g., Ref. [35]) and that of one-dimensional Bose
oryapproaches. Wedidnotfindacompactanalyticalex- gases with effective range dependent zero-range interac-
pression applicable to all N for the second-order energy tions. In the following, we discuss the impurity problem
shift. For N = 1 and 2, we have B(2)(1) = ln(2)/π3 with N = 2 and 3. To evaluate the second-order energy
−
andB(2)(2)=[ 9+6√3+3ln(2+√3) 12ln(2)]/(4π3). shifts, we need to know the complete set of eigenstates
− −
For larger N, the expressions are lengthy. The numer- of the (2,1) and (3,1) systems with g = . For the
ical values for N 12 are listed in Table I. Table I (2,1) system, we use the analytical wa|v|e fun∞ctions from
≤
also summarizes the numerically determined values for Ref. [39] and evaluate the integrals analytically. For the
the third-order coefficients B(3)(N) for N = 2 6. The (3,1)system,wederivecompactformsfortheeigenstates
B(3)(N) coefficient increases slightly as N chan−ges from using spherical coordinates and evaluate the relevant in-
1 to 2, and then decreases monotonically as N increases tegrals analytically. We then evaluate the second-order
further. ThenumericallydeterminedB(3)(N)coefficients perturbation theory sums numerically, imposing an en-
for N =2 6 approach the N = coefficient smoothly ergycutoff onthe relative energyof the intermediate (or
− ∞
if plotted as a function of 1/N. virtual)states that are being summed over. The second-
Strongly-interacting (N,1) system, N = 2,3, : The orderenergyshiftisfoundtocontainpowerlawdivergen-
···
strongly-interacting regime has been treated perturba- ciesintheenergycutoff. Thesedivergenciesarecanceled
tively at leading order, i.e., at order 1/γ, for N 8 [26] through the introduction of a counterterm and the con-
≤
(note, though, that only the coefficients for N 4 were stant (and physically meaningful) part is extracted with
≤
reported explicitly, i.e., in equation or numerical form). high precisionby a regularizationscheme similar to that
Toderivetheseresults,thetwo-bodyinteractionforlarge developedforharmonicallytrappedbosons[40]. TableII
g ismodeledbyimposingthetwo-bodyboundarycondi- reports the resulting second-order perturbation theory
| |
tion on the many-body wave function when the distance coefficients with errorbars. Our perturbative coefficients
betweenthe unlike particlesapproacheszero[23,26,34]. are consistent with the coefficients obtained by fitting
Since the groundstate eigenenergy for g = is degen- the (2,1) and (3,1) energies reported in Refs. [15, 22] to
| | ∞
erate,the perturbationshiftis obtainedbydiagonalizing a polynomial in γ−1. For the N = 2 and 3 systems, we
the perturbation matrix constructed using the degener- extend the above treatment to the third order (see Ta-
ate states for g = . For the many-body states Ψ and bleII).Thesethird-ordercalculationsrequiretheevalua-
α
∞
Ψ , the perturbation matrix element V reads [23] tionofmatrixelementsV betweenexcitedstates. Since
β αβ αβ
the third-order perturbation expression is more involved
~4 thanthesecond-orderperturbationexpression,ourthird-
V =
αβ −m2g × order result has a larger errorbar than our second-order
result [41].
N ∂Ψ∗ ∂Ψ∗
α α δ(z ) Thecalculationsofthe second-andthird-orderenergy
j0
j=1Z ···Z ∂zj0(cid:12)zj0→0+ − ∂zj0(cid:12)zj0→0−! × shifts can, in principle, be extended to larger N. To do
X (cid:12) (cid:12) so,twochallengesneedtobeovercome. First,anefficient
(cid:12) (cid:12)
∂Ψβ (cid:12) ∂Ψβ (cid:12) dz dz dz (.13) method to generate the complete set of eigenstates at
0 1 N
∂zj0(cid:12)(cid:12)zj0→0+ − ∂zj0(cid:12)(cid:12)zj0→0−! ··· |g|=∞hastobedevised. Second,anefficientschemefor
(cid:12) (cid:12)
(cid:12) (cid:12)
5
TABLEIII:Fittingcoefficientsb(k)fork=2and3. Fork=2 -2.7
j
and 3, we used j =6 and 4, respectively.
max k=2 k=3 N) -3
j=0 −0.0253304 1.71100×10−3 (1)C(
j=1 0.0019591 2.23905×10−4 -3.3
j=2 0.0033477 6.59881×10−6 (a)
j=3 −0.0116972 −3.67025×10−4 -3.6
j=4 0.0379414 2.64073×10−4
4
j=5 −0.0684440 (b)
j=6 0.0486346 3
)
N
(2
2)
evaluating the matrix elements andinfinite perturbation (C1
theory sums has to be developed. This is not pursued
0
here.
35
(c)
IV. FITTING THE B(k)(N) AND C(k)(N)
COEFFICIENTS )30
N
(
3)
Tables I-II suggest that the coefficients B(k)(N) and (C25
C(k)(N) change,for fixedk, smoothly with N. This mo-
tivates us to write 20
0 0.2 0.4 0.6 0.8 1
jmax 1 j 1/N
B(k)(N)= b(k) (14)
j N
j=0 (cid:18) (cid:19)
X
and
FIG. 2: (Color online) Symbols show the coefficients (a)
C(k)(N)=jmaxc(k) 1 j. (15) C(1)(N), (b) C(2)(N), and (c) C(3)(N) as a function of 1/N.
j N The solid lines show our fits with j =6, 3, and 3, respec-
j=0 (cid:18) (cid:19) max
X tively.
It should be noted that the expressions (14)-(15) reduce
tob(k) andc(k),respectively,inthe N limit. Inthe
follo0wing, th0e parameters b(jk) and c(jk→) a∞re obtained by TkA=B1L,E2,IaVn:dF3i,ttwinegucsoedeffijcient=s6c,(jk3),faonrdk3,=re1s,p2e,catinvdely3.. For
fitting the coefficients B(k)(N) and C(k)(N) for fixed k. max
k=1 k=2 k=3
WestartwithB(2)(N). WefitEq.(14)totheB(2)(N)
j=0 −2.66667 0.00000 21.05520
values for N = 1 80 (the values for N = 1 12 are
− − j=1 −1.40749 7.06739 8.80915
reported in Table II), varying j from 2 20. We find
max j=2 1.78704 −5.70589 −5.07455
−
that the most reliable fit is obtained for jmax = 12 j=3 −4.21746 2.49453 9.51090
−
13. In this case, the fitting parameter b(2) differs from j=4 7.33094
0
1/(4π2) [the resultobtainedby expandingEq.(11)] by j=5 −7.13604
l−ess than 10−8. This suggests that not only the k = j=6 2.76476
1 coefficient (see discussion above) but also the k = 2
coefficient connects smoothly with the infinite N result.
Table III reports the results of our fit to the B(2)(N) Symbols in Figs. 2(a)-2(c) show the C(k)(N) coeffi-
coefficientswithN =1 80and byapolynomialwith cients with k =1,2, and 3, respectively, as a function of
− ∞
j =6. 1/N. Our fits to these data (see Table II) using poly-
max
As mentioned earlier,the B(3)(1) coefficient is slightly nomials with j =6,3, and 3 are shown by solid lines
max
smaller than the B(3)(2) coefficient. The B(3)(N) coeffi- (see Table IV for the coefficients).
cients for N 2, however, decrease monotonically. This The discussion so far has focused on the coefficients
motivatesus≥tofittheB(3)(N)coefficientswithN =2 6 B(k)(N)andC(k)(N)withk=1 3. Itis,ingeneral,not
and byapolynomialwithjmax =4. Thefitcoefficien−ts feasibletoextendthe perturbativ−ecalculationstohigher
∞
are reported in Table III. It can be seen that the coef- k for arbitrary N. However, for N = 1 and , the co-
ficient b(3) agrees with the coefficient B(3)( ) reported efficients with larger k can be obtained readil∞y. We find
in Table0I. We believe that our fit provides∞an accurate that B(k)(1) decreases monotonically with increasing k
description of the 6<N < coefficients. (we c|hecked |this for k 50). The C(k)(1) coefficient
∞ ≤ | |
6
increases monotonically with increasing k for k <37; for
(a)
k 37,weobservesmallnon-monotonicoscillations. For 1.1
≥
N = ,wefindthattheB(k)( )withk evenandk 4
vanish∞while the B(k)( ) wi∞th k odd decrease mo≥no- 2)
| ∞ | ρ(
tonically with increasing k (again, we checked this for
k 50). Similarly, the C(k)( ) with k even and k 2 1
≤ ∞ ≥
vanishwhilethe C(k)( ) withkoddincreasemonoton-
| ∞ |
ically with increasing k. Assuming a linear change with
1/N,interpolatingbetweenB(k)(1)andB(k)( )andbe-
tweenC(k)(1)andC(k)( )fork >3yieldses∞timatesfor 1.1 (b)
∞
the finite N, N > 1, coefficients. While rough, these es-
)
timatesmightprovideareasonablemeanstoconnectthe 3
(
ρ
weakandstrongperturbationtheorylimitsforquantities
such as those shown in Figs. 3 and 4. 1
We cannot accurately estimate the radius of conver-
genceofthesmallandlargeγexpansionsfor1<N < .
∞
However,the fact thatthe radiusofconvergenceis given
(c)
by γ <1.0745(2) 2πforN =1andγ <2π forN = 1.1
for|th|e small γ ser×ies and by γ −1 < [1.0745(2) 2π]−∞1
for N = 1 and γ−1 < (2π)−1| f|or N = for th×e large ∞)
∞ ρ(
γ series suggests two speculations: First, a convergent
1
series can be found for any γ and N. Second, the radius
of convergence of the small γ series is approximately 2π
for all N. Figures 3 and 4, which are discussed in the -1 -0.5 0 0.5
-1/γ
next section, are consistent with these speculations.
V. DISCUSSION
FIG. 3: (Color online) Solid lines show the quantity ρ(N)
This sectioncomparesthe perturbativeenergyexpres-
as a function of −1/γ for (a) N = 2, (b) N = 3, and (c)
sionswiththenumericallydeterminedenergiesoftheup-
N = ∞, respectively. For comparison, dotted, dashed, and
perbranch. Figure1(b)showsthatthescaledinteraction
dash-dotted lines show the perturbative results for ρ(N) ac-
energy ǫ(N)/EF(N) depends weakly on N if plotted as counting for terms up to order γ0, γ1 and γ2, respectively,
a function of ~ωaho/g. The dependence on N is even in the weakly-interacting regime and accounting for terms
weakerwhent−heinteractionstrengthisparameterizedby up to order γ−1, γ−2 and γ−3, respectively, in the strongly-
γ asopposedtog. Tobenchmarktheapplicabilityofthe interacting regime.
perturbative expressions we analyze the interaction en-
ergyofthe systemwithN majorityatomsby comparing
where the coefficients ρ(w)(N) are determined by the
with that of the (1,1) system. Specifically, we consider k
the quantities ρ(N), B(l)(N) and B(l)(1) with l k+1. Expanding Eq. (16)
≤
in the strongly-interacting (large γ ) regime, we find
| |
ǫ(N)/E (N)
ρ(N)= F , (16) ρ(N)=1+ρ(s)(N)γ−1+ρ(s)(N)γ−2+
ǫ(1)/E (1) 1 2
F
ρ(s)(N)γ−3+ (γ−4), (19)
and δ(N,N′), 3 O
where the coefficients ρ(s)(N) are determined by the
k
ǫ(N)/E (N) ǫ(1)/E (1) C(l)(N) and C(l)(1) with l k. Solid lines in Figs. 3(a)-
δ(N,N′)= F − F . (17) ≤
ǫ(N′)/E (N′) ǫ(1)/E (1) 3(c) show the quantity ρ(N) for N = 2, 3, and , re-
F − F spectively. The solid lines are obtained using th∞e nu-
For finite N, ρ(N) reduces to ǫ(N)/[Nǫ(1)], i.e., ρ(N) merical (2,1) and (3,1) energies and the semi-analytical
tells one the interaction energy per particle, normalized (1,1) and ( ,1) energies. For γ 0+ (i.e., for
∞ →
bythe interactionenergyofthe(1,1)system. Thequan- 1/γ ), the quantity ρ(N) approaches the con-
− → −∞
tity δ(N,N′) can alternatively be written as [ρ(N) stantρ(w)(N)=B(1)(N)/B(1)(1)[seethehorizontaldot-
1]/[ρ(N′) 1]. − ted line0s in Figs. 3(a)-3(c)], which increases monotoni-
−
ExpandingEq.(16)intheweakly-interacting(smallγ) callyfrom1.0607to 1.1284asN goes from2 to . This
∞
regime, we find portionoftheinteractionenergycanbeinterpretedasthe
mean-field contribution. Inclusion of the next order cor-
ρ(N)=ρ(w)(N)+ρ(w)(N)γ+ρ(w)(N)γ2+ (γ3),(18) rection[theρ(w)(N)γ term]andthe nexttwocorrections
0 1 2 O 1
7
[the ρ(w)(N)γ and ρ(w)(N)γ2 terms] yields the dashed 0.48
1 2
anddash-dottedlinesinFigs.3(a)-3(c). Thedash-dotted
linesprovideafairlyaccuratedescriptionofthe quantity
)
∞
ρ(N) for −1/γ . −0.4. For |γ| → ∞, the leading-order 2,
γ-dependent term [see the (non-horizontal) dotted lines δ(0.44
inFigs.3(a)-3(c)]increasesmonotonicallyfrom0.3725to (a)
0.8782as N changes from2 to . Inclusion of the next-
∞
order correction and the next two corrections yields the
dashed and dash-dotted lines in Figs. 3(a)-3(c). It can
0.64
be seen that the dash-dotted lines provide a fairly accu-
rate description of the quantity ρ(N) for 1/γ & 0.15. )
− − ∞
This value is close to the expected radius of convergence 3,
oftheinteractionenergy[recall,theradiusofconvergence δ(
0.6
is 1/γ = (1.0745 2π)−1 0.148 for the (1,1) system]. (b)
× ≈
Combining the perturbative descriptions for small and
large γ , the expansions provide a fairly accurate de-
| |
scription of the interaction energy for the system with
N majority particles, normalized by that for the (1,1)
system, over a wide range of interaction strengths γ. )0.73
3
Expanding Eq. (17) in the weakly-interacting regime, 2,
(
we find δ
(c)
δ(N,N′)=δ(w)(N,N′)+δ(w)(N,N′)γ+ 0.72
0 1
δ2(w)(N,N′)γ2+O(γ3), (20) -1.5 -1 -1/γ -0.5 0
where the coefficients δ(w)(N,N′) are determined by the
k
B(l)(N),B(l)(N′),andB(l)(1)withl k+1. Expanding
≤
Eq. (17) in the strongly-interacting regime, we find
′
δ(N,N′)=δ0(s)(N,N′)+δ1(s)(N,N′)γ−1+ F−I1G/γ..4:T(hCeolsoorliodnlliinnee)isThfoerq(uaa)nNtity=δ2(Na,nNd N)a′s=a∞funacntidonthoef
δ(s)(N,N′)γ−2+ (γ−3), (21) circlesarefor(b)N =3andN′ =∞and(c)N =2andN′ =
2 O 3. In the large γ regime, the uncertainty of the numerically
where the coefficients δ(s)(N,N′) are determined by the determined (3,1) energies leads to appreciable uncertainties
k inδ(2,3)andδ(3,∞)[seetheerrorbarsinFigs.4(b)and4(c)].
C(l)(N), C(l)(N′),andC(l)(1)withl k+1. Thequan-
≤ For comparison, dotted, dashed, and dash-dotted lines show
tity δ(N,N′) is shown by the solid line in Fig. 4(a) for ′
theperturbativeresultsforδ(N,N )accountingfortermsup
(N,N′)=(2, )andbydotsinFigs.4(b)-4(c)for(3, ) toorderγ0,γ1,andγ2,respectively,intheweakly-interacting,
∞ ∞
and (2,3), respectively. We observe that the quantity small |γ| regime and accounting for terms up to order γ−1,
δ(N,N′) changes only slightly as 1/γ goes from γ−2, and γ−3, respectively, in the strongly-interacting, large
− −∞
to 0; this is particularly true for δ(2,3) [see Fig. 4(c)]. |γ| regime.
The limiting values [see the dotted lines in Figs. 4(a)-
4(c)] are given by δ(w)(N,N′) and δ(s)(N,N′), respec-
0 0
tively. Dashed lines include the next order correction in thelargeγ regimetomeaningfullycomparewiththeper-
the weakly- and strongly-interacting regimes, and dash- turbative results.
dotted lines include the next two corrections. In the
weakly-interacting regime, the dash-dotted lines provide
afairlygooddescriptionofthe quantityδ(N,N′). Inthe VI. CONCLUSION
strongly-interactingregime,however,the validity regime
oftheperturbativeexpressionsisquitesmall. Forδ(2,3),
This paper considered the upper branch of a non-
e.g., the expansion coefficients are δ(s)(2,3) = 0.7213,
0 interacting harmonically trapped one-dimensional Fermi
δ1(s)(2,3) = 0.0694(3), and δ2(s)(2,3) = −2.31(12), where gas with a single impurity. Zero-rangetwo-body contact
the numbers in brackets denote the errorbars due to the interactions with strength g were assumed between the
uncertaintiesofthesecond-andthird-orderperturbation majority atoms and the impurity. This system consti-
theorycoefficients. Thefactthat δ(s)(2,3) δ(s)(2,3) tutes one of the simplest mesoscopic systems accessible
| 2 |≫| 1 |
isresponsibleforthe turn-aroundofthe dash-dottedline to experiment and theory. On the experimental side, it
for large positive γ. We note that the errorbar of the hasbeendemonstratedbythe Heidelberggroupthatthe
quantities δ(3, ) and δ(2,3), obtained from the numer- upperbranchofthemodelHamiltoniancanbeemulated
∞
ical energies [see dots in Figs. 4(b)-4(c)], is too large in reliably using ultracold atoms [18, 19]. On the theory
8
side, various numerical and analytical techniques have a program can be carried through explicitly beyond the
beenapplied[7,11–17,23–26]. Thispaperpursuedaper- leading-order correction. (iii) The radii of convergence
turbative approach, which determined expansions of the of the series were reported for N = 1 and . (iv) The
energy of the upper branch in the weakly- and strongly- behavior of the expansion coefficients in the∞series in γk
interacting regimes for various N. In the cases where and γ−k with k > 3 was discussed. (v) The perturba-
we were not able to obtain general N expressions for a tive expressionswerebenchmarkedandfound to provide
fixed order in the perturbative expansion, approximate a reliable description over a wide range of interaction
expressions applicable to all N were obtained through strengths.
fits. Through comparison with accurate numerical few-
The results presented in this work can be used to cal-
body energies, the perturbative expressions were shown
culateperturbativeexpressionsforthecontactandother
to provide a satisfactory description for a wide range of
observables. Moreover, the second- and third-order re-
interaction strengths.
sultsintheγ−1seriesallowonetoassesstheapplicability
The main results of this work are: (i) We determined
regime of effective spin models [26, 34, 42].
an expansion for the energy of the upper branch of a
one-dimensional harmonically trapped Fermi gas with a Acknowledgement: We thank G. Zu¨rn for email cor-
single impurity in the weakly-repulsive regime up to or- respondence that motivated this work and suggested to
der γ3, applicable to any system size. (ii) We deter- analyze the quantity δ(N,N′) shown in Fig. 4. We also
mined an expansion for the energy of the upper branch thank N. Zinner for bringing Ref. [38] to our attention,
in the strongly-interacting regime up to order γ−3, ap- and P. Johnson and E. Tiesinga for helpful discussions
plicable to any system size. While the idea to treat the on the numerical regularization of diverging sums. Sup-
coupling strength 1/γ as a small parameter is not new, port by the National Science Foundation through grant
our work provides an explicit demonstration that such number PHY-1205443is gratefully acknowledged.
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for state 1 and coefficients (1−α2)1/2 and α for state and 708~ω for the second- and third-order perturbation
theory calculations. For the N = 3 system, we used en-
2. An analogous analysis shows that the true adiabatic
ergycutoffsof190~ω and80~ωforthesecond-andthird-
wave functions of the (2,2) system are linear combina-
order perturbation theory calculations.
tions of Eqs. (11) and (12) of the supplemental material
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state with larger slope and α and (1−α2)1/2 for the