FGIP-Student Forum セミナーinfo.phys.sci.titech.ac.jp/uploads/info/C1002/I1907_1718...FNCP Interacting bosons in 1D 8/31 Motivation Harmonically conﬁned bosons Extractable entanglement

Francis Norman Paraan（State University of New York, Stony Brook, Phillipine）

日時： 2011 年 3月 29日（火）１５時～１６時３０分

場所：本館２階Ｈ２８４Ａ室

‘Some aspects of the one-dimensional interacting

boson gas ’

• Abstract: In this talk we discuss two recent results involving a one-

dimensional boson gas with contact interactions as modeled by the Lieb-Liniger

hamiltonian. First, we present a perturbation calculation for the ground state

energy and chemical potential of such a gas in the presence of longitudinal

harmonic confinement about the impenetrable boson limit. This result is

compared to that obtained from the Thomas-Fermi formalism. Second, we

quantify the amount of entanglement that can be extracted by certain coarse-

grained measurements on the ground state of this gas when it is confined in a

ring. We demonstrate that the amount of entanglement in these projections

increases monotonically with interparticle repulsion strength.

担当鹿野豊（内線 3893）

名前大学滞在期間受入担当氏名Francis N.C. Paraan State University of New York, 3/17-3/31 鹿野

Stony Brook (Phillippine)

FGIP-Guest student の滞在スケジュール

教員、修士課程大学院生の参加も歓迎します。

東工大基礎・物性物理学専攻「物理学リーダーシップ」

FGIP：Foreign Graduate Student Invitation Program

外国人博士課程大学院生の短期招待・共同研究

FGIP-Student Forum セミナー

http://www.phys.titech.ac.jp/leadership/fgip/FGIP-Student Forum 事務局小林慶鑑（内線２３６９）

MotivationHarmonically confined bosons

Extractable entanglement by projectionsSummary and References

Some aspects of the one-dimensionalinteracting boson gas

Harmonic confinement and extractable entanglement byfixed number projections

Francis N. C. ParaanAdvisor: Vladimir E. Korepin

Department of Physics & AstronomyState University of New York at Stony Brook

29 March 2011Tokyo Institute of Technology

FNCP Interacting bosons in 1D 1/31



Outline

1 MotivationLieb-Liniger modelExperiments

2 Harmonically confined bosonsThomas-Fermi approximationStrong interaction limit

3 Extractable entanglement by projectionsCoarse-grained measurements




Lieb-Liniger modelExperiments

Hamiltonian

With repulsive coupling constant c > 0 [Lieb & Liniger, 1963],

Many-body hamiltonian

H = −N∑

i=1

∂2

∂x2i

+ 2c∑〈i, j〉

δ(xi − xj). (1)

The eigenstates are specified by a set of quasi-momenta {k}that satisfy the Bethe equations.





Exact solution

Bethe ansatz

χ(x) =1N∑{P}

(−1)[P]eikP ·x∏m<n

kPm − kPn − ic sgn(xn − xm). (2)

With periodic boundary conditions:

Bethe equations

e−ikmL = −N∏

n=1

km − kn + ickm − kn − ic

. (3)





Lieb-Liniger (LL) equations

Continuum limit gives LL equations:

Quasi-momentum distribution f (k)

1 + 2c∫ K

−K

f (x) dxc2 + (x− k)2 = 2πf (k), (4)

coupled to

Normalization ∫ K

−Kf (k) dk = ρ, (5)

where ρ = N/L is the number density.





Quasi-momentum distribution

Γ = 0.0340

Γ = 0.1214

Γ = 0.4051

Γ = 4.993Γ = ¥

-3 -2 -1 0 1 2 30.0

0.5

1.0

1.5

2.0

Κ

fHΚL

The distribution of quasi-momenta κ = k/ρ is sharply peaked aboutκ = 0 in the noninteracting limit and uniform in the free fermion limit.

(γ ≡ c/ρ)

[Lieb & Liniger, 1963]





Ground state energy and chemical potential

Ε � Ρ2

Μ � Ρ2

0 20 40 60 80 1000

2

4

6

8

10

Γ

The ground state energy per particle ε = ρ2e(γ) and chemicalpotential µ = ρ2(3e− γe′) saturate to the free fermion values

(e =∫κ2 f̃ (κ) dκ).

[Lieb & Liniger, 1963]





Quasi-1D Bose gas

Under tight transverse harmonic confinement, the potential

3D interaction pseudopotential

U(r) = g3Dδ(r)∂

∂r(r· ) (6)

leads to the effective Q1D potential [Olshanii, 1998]

Q1D interaction potential

U1D(x) = g1Dδ(x). (7)

g1D is a simple function of g3D and the transverse length scale.





Bosons in optical lattices

Haller et al., Science 325, 1224 H2009L

The Q1D equation of state of the confined LL gas has beenmeasured in recent experiments. [Kinoshita et al., 2004]





Bosons in optical lattices

Haller et al., Science 325, 1224 H2009L

The Q1D equation of state of the confined LL gas has beenmeasured in recent experiments. [Kinoshita et al., 2004]




Thomas-Fermi approximationStrong interaction limit

Harmonically confined bosons


H =

N∑i=1

−12∂2

∂x2i

+12

x2i + c

∑〈i, j〉

δ(xi − xj). (8)

Solved in the Thomas-Fermi (TF) approximation by Dunjko etal. (2004) and Ma & Yang (2009).

Difficulty: Translational invariance is broken.





Harmonically confined bosons


H =

N∑i=1

−12∂2

∂x2i

+12

x2i + c

∑〈i, j〉

δ(xi − xj). (8)

Solved in the Thomas-Fermi (TF) approximation by Dunjko etal. (2004) and Ma & Yang (2009).

Difficulty: Translational invariance is broken.





Thomas-Fermi approximation

Assume that the energy can be locally described by theLieb-Liniger energy + the external confining potential.

Ground state energy

E0 =

∫e0(x) +

x2

2ρ(x) dx. (9)

where e0 ≡ c3βζ(β) is the homogeneous ground state energydensity and β = ρ/c = δN/cδx.





TF approximation

∆x

ΡHxL





Local thermodynamics

Fundamental equation

d(e0δx) = −pd(δx) + µlocd(δN) (10)

gives

Local equations of state

p = c3β2ζ ′(β), β = ρ(x)/c, (11)

µloc = c2[ζ(β) + βζ ′(β)]. (12)

Static hydrodynamic equilibrium is explicitly demonstrated.





Local thermodynamics

Fundamental equation

d(e0δx) = −pd(δx) + µlocd(δN) (10)

gives

Local equations of state

p = c3β2ζ ′(β), β = ρ(x)/c, (11)

µloc = c2[ζ(β) + βζ ′(β)]. (12)

Static hydrodynamic equilibrium is explicitly demonstrated.





Equilibrium density profiles

Extremizing the energy functional E0[ρ] at constant N =∫ρ dx

gives

TF density

G2c2

N− x2

2N=µloc

N=

c2

N

[ζ(β) + βζ ′(β)

]. (13)

G2c2 is a Lagrange multiplier that fixes N (or√

N/c).





Local chemical potential

∆x

G2 c2

Μloc





Equilibrium density profiles

N1�2 �c = 20.0

N1�2 �c = 0+

-1.0 -0.5 0.0 0.5 1.0x¢

0.2

0.4

0.6

0.8

1.0

1.2

Ρ¢

[x′ = x/√

2N; ρ′ = ρ/√

2N]

√N/c→ 0+ yields an elliptical Tonks-Girardeau profile,√N/c� 1 yields a parabolic mean field profile. [Ma & Yang, 2010]





Strong interaction limit


H =

N∑i=1

−12∂2

∂x2i

+12

x2i + c

∑〈i, j〉

δ(xi − xj). (14)

Aim: Obtain 1/c corrections to the ground state energy andchemical potential.

[FP & Korepin, 2010]





Fermion-Boson mapping

Bosonic Schrödinger equation

[H0 + cV̂b]Ψb = EbΨb (15)

is mapped to

Fermionic Schrödinger equation

[H0 + c−1V̂ f]Φf = EfΦf, Ψb = AΦf. (16)

A is a unit anti-symmetrizer∏

i<j sgn(xj − xi).





Fermionic pseudopotential

Matrix elements

〈ϕf|c−1V̂ f|φf〉 = −4c

∑i<j

∫lim

rij→0

[∂ϕf∗

∂rij

∂φf

∂rij

]dRij, (17)

with rij = xj − xi and Rij = 12(xj + xi). [Cheon & Shigehara, 1999]

Pseudopotential is a sum of two-particle potential operators.





First order perturbation

With ΦfTG = N!−1/2 det[ψn(xm)], Slater determinant of oscillator

orbitals,

δE = 〈ΦfTG|c

−1V̂ f|ΦTGf〉

=1c

√2π3

N−1∑l=1

Γ(l− 12)

Γ(l + 1)

l−1∑k=0

(l− k)2Γ(k − 12)

Γ(k + 1)3F2

[32 ,−k,−l32−k, 3

2−l; 1]

(18)

where E0 ≈ ETG + δE = 12~ωN2 + δE.





Asymptotic behavior of perturbation

1 5 10 50 100 500 10000.2

0.5

1.0

2.0

5.0

10.0

N

-c∆

E�N

2

Exact result (solid) and asymptotic large N behavior (dashed).

Correction behaves as δE/ETG ∼ 2α0√

N/c, with α0 ≈ −0.408.





Two particles

ETG

0 5 10 15 20 25 301.0

1.2

1.4

1.6

1.8

2.0

2.2

c

E0

Exact ground state energy (solid) and first order 1/c result (dashed).





Thermodynamic limit

Require an extensive E0 as N →∞

ETG = 12~ωN2 ∝ N ⇒ ωN → constant

⇒√

N/c→ constant. (19)





Thermodynamic limitAsymptotic 1

2~ωN2 → ETG forms

E0 ≈ ETG[1 + 2α0/g], g = c/√

N (20)

µ ≈ µTG[1 + 52α0/g], (21)

0.0 0.2 0.4 0.6 0.8 1.00.20

0.25

0.30

0.35

0.40

0.45

0.50

N1�2�c º 1�Γ

E�N

2

Thomas-Fermi result (solid) and first order 1/c result (dashed).





Summary of results

• We obtained 1/c corrections to ground state energy andchemical potential of the harmonically confined LL gas.

• For a consistent extensive energy as N →∞, we musthave

√N/c→ constant.

• Open problems:• Second-order perturbation requires knowledge of form

factors.




Coarse-grained measurements

Extracting entanglement from the LL gas

Aim: Measure entanglement obtained from fixed number purestate projections of the LL ground state.

(with J. Molina-Vilaplana, V. E. Korepin, and S. Bose)

A

B




Coarse-grained measurements

Extracting entanglement from the LL gas

Manuscript in preparation: Some slides not available online.




SummaryAcknowledgmentsReferences

Summary

• We quantified the entanglement extractable from fixednumber projections of the LL ground state.

• Balanced fixed number projections yield optimumentanglement.

• Open problem:• Fixed number projections are not eigenstates of the LL

hamiltonian→ non-trivial time evolution.





Acknowledgments

• NSF Grant No. DMS-0905744• Foreign Graduate Student Invitation Program• Prof. Akio Hosoya• Y. Shikano and H. Katsura





References

T. Cheon and T. Shigehara, Phys. Rev. Lett. 82, 2536 (1999).

V. Dunjko, V. Lorent, and M. Olshanii, Phys. Rev. Lett. 86, 5413 (2001).

T. Kinoshita, T. Wenger and D. S. Weiss, Science 305, 1125 (2004).

E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963).

Z.-Q. Ma and C. N. Yang, Chin. Phys. Lett. 26, 120506 (2009).

Z.-Q. Ma and C. N. Yang, Chin. Phys. Lett. 27, 020506 (2010).

J. Molina-Vilaplana, S. Bose, and V. E. Korepin, Int. J. Quantum. Inf. 6, 739(2008).

M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).

F. N. C. P. and V. E. Korepin, Phys. Rev. A 82, 065603 (2010).





Slater-Condon Rule for two-particle potentials

Let V̂ = 12∑

k 6=l v̂(k, l) andΨ be a Slater determinant of orbitals φi.

vklmn = 〈φkφl|v̂|φmφn〉, (22)

〈Ψ|V̂|Ψ〉 =12

∑k,l

vklkl − vkllk. (23)


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FGIP-Student Forum セミナーinfo.phys.sci.titech.ac.jp/uploads/info/C1002/I1907_1718...FNCP Interacting bosons in 1D 8/31 Motivation Harmonically conﬁned bosons Extractable entanglement