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TRANSCRIPT
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
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Outline
1 MotivationLieb-Liniger modelExperiments
2 Harmonically confined bosonsThomas-Fermi approximationStrong interaction limit
3 Extractable entanglement by projectionsCoarse-grained measurements
FNCP Interacting bosons in 1D 2/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
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.
FNCP Interacting bosons in 1D 3/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Lieb-Liniger modelExperiments
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)
FNCP Interacting bosons in 1D 4/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Lieb-Liniger modelExperiments
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.
FNCP Interacting bosons in 1D 5/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Lieb-Liniger modelExperiments
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]
FNCP Interacting bosons in 1D 6/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Lieb-Liniger modelExperiments
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]
FNCP Interacting bosons in 1D 7/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Lieb-Liniger modelExperiments
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.
FNCP Interacting bosons in 1D 8/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Lieb-Liniger modelExperiments
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]
FNCP Interacting bosons in 1D 9/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Lieb-Liniger modelExperiments
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]
FNCP Interacting bosons in 1D 9/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
Harmonically confined bosons
Many-body hamiltonian
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.
FNCP Interacting bosons in 1D 10/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
Harmonically confined bosons
Many-body hamiltonian
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.
FNCP Interacting bosons in 1D 10/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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.
FNCP Interacting bosons in 1D 11/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
TF approximation
∆x
ΡHxL
FNCP Interacting bosons in 1D 12/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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.
FNCP Interacting bosons in 1D 13/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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.
FNCP Interacting bosons in 1D 13/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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).
FNCP Interacting bosons in 1D 14/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
Local chemical potential
∆x
G2 c2
Μloc
FNCP Interacting bosons in 1D 15/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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]
FNCP Interacting bosons in 1D 16/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
Strong interaction limit
Many-body hamiltonian
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]
FNCP Interacting bosons in 1D 17/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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).
FNCP Interacting bosons in 1D 18/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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.
FNCP Interacting bosons in 1D 19/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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.
FNCP Interacting bosons in 1D 20/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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.
FNCP Interacting bosons in 1D 21/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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).
FNCP Interacting bosons in 1D 22/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
Thermodynamic limit
Require an extensive E0 as N →∞
ETG = 12~ωN2 ∝ N ⇒ ωN → constant
⇒√
N/c→ constant. (19)
FNCP Interacting bosons in 1D 23/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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).
FNCP Interacting bosons in 1D 24/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Thomas-Fermi approximationStrong interaction limit
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.
FNCP Interacting bosons in 1D 25/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
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
FNCP Interacting bosons in 1D 26/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
Coarse-grained measurements
Extracting entanglement from the LL gas
Manuscript in preparation: Some slides not available online.
FNCP Interacting bosons in 1D 27/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
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.
FNCP Interacting bosons in 1D 28/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
SummaryAcknowledgmentsReferences
Acknowledgments
• NSF Grant No. DMS-0905744• Foreign Graduate Student Invitation Program• Prof. Akio Hosoya• Y. Shikano and H. Katsura
FNCP Interacting bosons in 1D 29/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
SummaryAcknowledgmentsReferences
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).
FNCP Interacting bosons in 1D 30/31
MotivationHarmonically confined bosons
Extractable entanglement by projectionsSummary and References
SummaryAcknowledgmentsReferences
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)
FNCP Interacting bosons in 1D 31/31