Nascent Superfluidity in Bilayer Two-Dimensional Electron Systems
Rotational spectra of molecules in small Helium clusters: Probing superfluidity in finite systems F....
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Rotational spectra of molecules in Rotational spectra of molecules in small Helium clusters:small Helium clusters:
Probing superfluidity in finite Probing superfluidity in finite systemssystems
F. Paesani and K.B. WhaleyF. Paesani and K.B. Whaley
Department of Chemistry andDepartment of Chemistry and
Pitzer Center for Theoretical ChemistryPitzer Center for Theoretical Chemistry
University of California, Berkeley, CA, 94720University of California, Berkeley, CA, 94720
$$$ NSF $$$ NSF
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Spectroscopy in large Spectroscopy in large 44He dropletsHe droplets
courtesy A.F. Vilesovcourtesy A.F. Vilesov
• 44He droplets: ultracold environment (T≈ 0.15-0.4 He droplets: ultracold environment (T≈ 0.15-0.4 K)K) for high resolution spectroscopyfor high resolution spectroscopy• Free rotation in Free rotation in 44He He
• Rotational diffusion in Rotational diffusion in 33He He
Grebenev, Toennies and Vilesov, Grebenev, Toennies and Vilesov,
Science Science 279279, 2083 (1998), 2083 (1998)
• Superfluidity Superfluidity
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Spectroscopy in small Spectroscopy in small 44He clustersHe clusters
OCS(OCS(44He)He)NN NN22O(O(44He)He)NN
Tang, Xu, McKellar and JägerTang, Xu, McKellar and JägerScience Science 297297, 2030 (2002), 2030 (2002)
Xu, Jäger, Tang and McKellarXu, Jäger, Tang and McKellarPhys. Rev. Lett. Phys. Rev. Lett. 9191, 163401 (2003, 163401 (2003))
Similar results also for :Similar results also for :• COCO22((44He)He)NN
Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 9292, 145503 (2004, 145503 (2004))
• CO(CO(44He)He)NN
McKellar, J. Chem. Phys. McKellar, J. Chem. Phys. 121121, 6868 (2004), 6868 (2004)
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Simulations of doped Simulations of doped 44He clustersHe clusters
• Molecules:Molecules: OCS, NOCS, N22O and COO and CO22
• Methods:Methods: 1. Projection Operator Imaginary 1. Projection Operator Imaginary Time Spectral Evolution (POITSE)Time Spectral Evolution (POITSE) Blume, Lewerenz, Niyaz and Whaley, Phys. Rev. E Blume, Lewerenz, Niyaz and Whaley, Phys. Rev. E 5555, 3664 (1997), 3664 (1997)
Rotational spectrumRotational spectrum
2. Path-Integral Monte Carlo (PIMC)2. Path-Integral Monte Carlo (PIMC) Ceperley, Rev. Mod. Phys. Ceperley, Rev. Mod. Phys. 6767, 279 (1995), 279 (1995)
Superfluid propertiesSuperfluid properties
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Potential Energy SurfacesPotential Energy Surfaces
He-NHe-N22OOChang, Akin-Ojo, Bukowski and Szalewicz, Chang, Akin-Ojo, Bukowski and Szalewicz, J. Chem. Phys. J. Chem. Phys. 119119, 11654 (2003), 11654 (2003)
He-COHe-CO22Yan, Yang and Xie, Yan, Yang and Xie, J. Chem. Phys. J. Chem. Phys. 109109, 10284 (1998), 10284 (1998)
He-OCSHe-OCSPaesani and Whaley, Paesani and Whaley, J. Chem. Phys. J. Chem. Phys. 121121, 4180 (2004), 4180 (2004)
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Monte Carlo methodsMonte Carlo methods
€
HArot =Bx
∂2
∂2 j x
+By
∂2
∂2 j y
+Bz
∂2
∂2 jz
€
Vtot(R) = VA−B(Ri )i=1
N
∑ + VB−B(Rij )i< j
N
∑
• Total interaction potentialTotal interaction potential
€
H=−1
2MA
∇A2 −HA
rot −1
2m B
∇i2 +V tot(R)
i=1
N
∑
• Rotational Hamiltonian of molecule ARotational Hamiltonian of molecule A
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
• Hamiltonian for ABHamiltonian for ABNN systems systems
Projection Operator Imaginary Time Projection Operator Imaginary Time Spectral Evolution methodSpectral Evolution method
€
k ω( ) = φ0 A φnn
∑2
d(E0 −En +ω)
€
Λ k ω( ){ } =k(τ) =φ0 Aexp−(H−E0 )τ[ ]A
+ φ0
φ0 φ0
≡ A+ (0)A(τ)
€
k τ( ) =ΨT Aexp−(H−Eref )τ[ ]A
+ ΨT
ΨT exp−(H−Eref )τ[ ] ΨT
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
• Laplace transformLaplace transform
• In practice: In practice: ΨΨTT ≈ ≈ 0 0 and Eand Erefref ≈ E ≈ E00
ˆ
ˆ ˆˆ ˆ
ˆ ˆ
˜
˜
1. D1. Djjmkmk((,,,,)) = molecular Wigner functions = molecular Wigner functions
2. 2. ,,,, = Euler angles = Euler angles molecule-fixed frame molecule-fixed frame space-fixed frame space-fixed frame
1. m=k=01. m=k=0 2. 2. DDjj
0000((,,,,)) P Pjj(cos(cos) ) j≈Jj≈J eigenfunctions for linear rotorseigenfunctions for linear rotors eigenfunctions for symmetric top rotors eigenfunctions for symmetric top rotors (k=0) (k=0)
• Present calculationsPresent calculations
Projection Operator Imaginary Time Projection Operator Imaginary Time Spectral Evolution methodSpectral Evolution method
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
• Â Â D Djjmkmk((,,,,))
€
ε( J ) τ( ) =−1
τlnk
J( ) τ( )/ c1 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
P k ω( )k τ( ), I ⎡ ⎣ ⎢
⎤ ⎦ ⎥∝P k τ( )k ω( ), I
⎡ ⎣ ⎢
⎤ ⎦ ⎥P k ω( ) I[ ]
€
k(ω) =Λ−1 k τ( ){ }
• Exponential fitExponential fit
€
k( J ) τ( ) = cn exp−En −E0( )τ[ ]n
∑
Projection Operator Imaginary Time Projection Operator Imaginary Time Spectral Evolution methodSpectral Evolution method
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
• Maximum Entropy (based on Bayes’s statistics)Maximum Entropy (based on Bayes’s statistics)
˜
˜ ˜
˜
˜
N=1N=1POITSE: 0.657±0.002 cmPOITSE: 0.657±0.002 cm-1-1
Chang et al.: 0.65297 cmChang et al.: 0.65297 cm-1-1
N=2N=2POITSE: 0.419±0.006 cmPOITSE: 0.419±0.006 cm-1-1
N>2: J=1-3N>2: J=1-3
Rotational Excited States:Rotational Excited States:POITSE calculations for NPOITSE calculations for N22O(O(44He)He)NN
POITSE for N ≤ 2POITSE for N ≤ 2
POITSE for N > 2POITSE for N > 2
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Rotational Excited States:Rotational Excited States:POITSE calculations for NPOITSE calculations for N22O(O(44He)He)NN
Rotational Spectrum for N > 5Rotational Spectrum for N > 5
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
POITSE for N ≤ 2POITSE for N ≤ 2
POITSE for N > 2POITSE for N > 2
Experiment: Tang, Xu, McKellar and Jäger, Science Experiment: Tang, Xu, McKellar and Jäger, Science 297297, 2030 (2002), 2030 (2002)Theory: Paesani, Viel, Gianturco and Whaley, Phys. Rev. Lett. Theory: Paesani, Viel, Gianturco and Whaley, Phys. Rev. Lett. 9090, 073401 (2003), 073401 (2003)
E = BE = BeffeffJ(J+1)J(J+1)
Exp.
Theory
Spectroscopic Constants for OCS(Spectroscopic Constants for OCS(44He)He)NN
Theory vs ExperimentTheory vs Experiment
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Exp.
Theory
POITSE (exact)
Rigid coupling (approx.)
Theoretical interpretationTheoretical interpretationTransition from van der Waals complexes to quantum solvationTransition from van der Waals complexes to quantum solvation
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Spectroscopic Constants for OCS(Spectroscopic Constants for OCS(44He)He)NN
Theory vs ExperimentTheory vs Experiment
E = BE = BeffeffJ(J+1)J(J+1)
Experiment: Xu, Jäger, Tang and McKellar, Phys. Rev. Lett. Experiment: Xu, Jäger, Tang and McKellar, Phys. Rev. Lett. 9191, 163401 (2003), 163401 (2003)Theory: Paesani and Whaley, J. Chem. Phys. Theory: Paesani and Whaley, J. Chem. Phys. 121121, 5293 (2004, 5293 (2004)
E = BE = BeffeffJ(J+1) - DJ(J+1) - DeffeffJJ22(J+1)(J+1)22
Distortion ConstantRotational Constant
Exp.
Theory
Spectroscopic Constants for NSpectroscopic Constants for N22O(O(44He)He)NN
Theory vs ExperimentTheory vs Experiment
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Distortion ConstantRotational Constant
Spectroscopic Constants for COSpectroscopic Constants for CO22((44He)He)NN
Theory vs ExperimentTheory vs Experiment
E = BE = BeffeffJ(J+1) - DJ(J+1) - DeffeffJJ22(J+1)(J+1)22
Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 9292, 145503 (2004), 145503 (2004)Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. 9494, 153401 (2005), 153401 (2005)
Exp.
Theory
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Rotational Dynamics in Rotational Dynamics in 44He clustersHe clusters
• 11stst solvation shell solvation shell OCS: BOCS: Beffeff ≈ B ≈ Bdropletdroplet
NN22O, COO, CO22: B: Beff eff > B> Bdropletdroplet
??????
BBeffeff/B/B00 • decrease for small N decrease for small N “ “rigid” couplingrigid” coupling
• turnaround at N ≈ 5-9turnaround at N ≈ 5-9 ??????
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Does this behavior reflect onset of superfluidity ???Does this behavior reflect onset of superfluidity ???
Path-Integral Monte Carlo methodPath-Integral Monte Carlo method
€
O =1
ZdRd ′ R ′ R∫ O R ρ R, ′ R ;( )
• Thermal density matrixThermal density matrix
€
ρ R, ′ R ;( ) = Re−H ′ R
€
ρB R, ′ R ;( ) =1
N!ρ R,P ′ R ;( )
P
∑Bose symmetryBose symmetry
€
=1/ kBT
• Implementation: Implementation: sampling ofsampling of
€
ρ R,P ′ R ;( ) = ...∫ dR1...dRM−1∫ ρ R,R1; τ( )...ρ RM−1,P ′ R ; τ( )
€
R,R1, ...,RM−1,P ′ R{ }
€
τ =/ MBerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
• Thermal averageThermal average
ˆˆ
ffijij= superfluid fraction= superfluid fraction
€
||( ⊥)s =
4m 2kbT
I ||( ⊥)cl
A||( ⊥)A||( ⊥)
Superfluidity in PIMCSuperfluidity in PIMC
€
I ijcl =m dRρ R( ) R2dij −xix j( )∫
IIclcl= classical moment of inertia= classical moment of inertia
A = projected area of a pathA = projected area of a path macroscopic exchangesmacroscopic exchanges (Feynman, 1953)(Feynman, 1953)
ss
€
I ij =∂ Λi
∂ω j
ωj =0
€
I ij =I ijcl(1−f ij
s)
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
• Response to a slow rotation of an external fieldResponse to a slow rotation of an external field
• Superfluid fractionSuperfluid fraction
Superfluidity in PIMCSuperfluidity in PIMC
€
ρ||( ⊥)s R( ) =
4m2NkbT
I ||( ⊥)cl
A||( ⊥) R( ) ⋅A||( ⊥)
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
• Response to a slow rotation of an external fieldResponse to a slow rotation of an external field
• Superfluid densitySuperfluid density
• Superfluid fractionSuperfluid fraction
€
I ij =∂ Λi
∂ω j
ωj =0
€
I ij =I ijcl(1−f ij
s)
€
||( ⊥)s =
4m 2kbT
I ||( ⊥)cl
A||( ⊥)A||( ⊥)
• direct correspondence between fdirect correspondence between f and B and Beffeff
- N ≤ 5: f- N ≤ 5: f negligible negligible decrease of B decrease of Beffeff van der Waals complexes van der Waals complexes - N > 5: increase of f- N > 5: increase of f increase of B increase of Beffeff onset of superfluidity onset of superfluidity - N ≥ 12: saturation of f- N ≥ 12: saturation of f saturation of B saturation of Beffeff
• ff||||≈1 for N ≥ 5 ≈1 for N ≥ 5 negligible component of J on the CO negligible component of J on the CO22 axis axis no Q-branch no Q-branch
Onset of Superfluidity in COOnset of Superfluidity in CO22((44He)He)NN
Rotational ConstantRotational Constant Superfluid FractionSuperfluid Fraction
Exp.
Theory
T= 0.15 KT= 0.15 K
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
ParallelParallel
PerpendicularPerpendicular
Onset of Superfluidity in COOnset of Superfluidity in CO22((44He)He)NN
Superfluid FractionSuperfluid Fraction
Total DensityTotal Density
Perpendicular Superfluid DensityPerpendicular Superfluid Density
Parallel Superfluid DensityParallel Superfluid Density
COCO22((44He)He)55
• parallel:parallel: localized in the global minimumlocalized in the global minimum• perpendicular: perpendicular: zero superfluid densityzero superfluid density
T= 0.15 KT= 0.15 K
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
ParallelParallel
PerpendicularPerpendicular
COCO22((44He)He)99
• parallel:parallel: diffused around the global minimumdiffused around the global minimum• perpendicular: perpendicular: extending along the COextending along the CO22 axis axis
Onset of Superfluidity in COOnset of Superfluidity in CO22((44He)He)NN
Parallel Superfluid DensityParallel Superfluid Density
Perpendicular Superfluid DensityPerpendicular Superfluid Density
Total DensityTotal Density
Superfluid FractionSuperfluid Fraction
T= 0.15 KT= 0.15 K
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
ParallelParallel
PerpendicularPerpendicular
COCO22((44He)He)1313
• parallel:parallel: diffused around the global minimumdiffused around the global minimum• perpendicular: perpendicular: extending along the COextending along the CO22 axis axis
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Parallel Superfluid DensityParallel Superfluid Density
Perpendicular Superfluid DensityPerpendicular Superfluid Density
Total DensityTotal Density
Onset of Superfluidity in COOnset of Superfluidity in CO22((44He)He)NN
Superfluid FractionSuperfluid Fraction
T= 0.15 KT= 0.15 K
ParallelParallel
PerpendicularPerpendicular
COCO22((44He)He)1717
• parallel:parallel: diffused around the global minimumdiffused around the global minimum• perpendicular: perpendicular: extending along the COextending along the CO22 axis axis
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Onset of Superfluidity in COOnset of Superfluidity in CO22((44He)He)NN
Superfluid FractionSuperfluid Fraction
T= 0.15 KT= 0.15 K
Parallel Superfluid DensityParallel Superfluid Density
Perpendicular Superfluid DensityPerpendicular Superfluid Density
Total DensityTotal Density
ParallelParallel
PerpendicularPerpendicular
Linear responseLinear response
Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 9292, 145503 (2004), 145503 (2004)Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. 9494, 153401 (2005), 153401 (2005)
Exp.
POITSE
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Onset of Superfluidity in COOnset of Superfluidity in CO22((44He)He)NN
€
I ⊥ =I ⊥cl(1−f⊥
s )
€
Beff =1
2I ⊥
Exp.
POITSE
Linear response
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005
Onset of Superfluidity in COOnset of Superfluidity in CO22((44He)He)NN
Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 9292, 145503 (2004), 145503 (2004)Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. 9494, 153401 (2005), 153401 (2005)
Linear responseLinear response
€
I ⊥ =I ⊥cl(1−f⊥
s )
SummarySummary
• Calculations of rotational excitations in Calculations of rotational excitations in 44He clustersHe clusters very good agreement with experimentsvery good agreement with experiments
• Calculations of superfluid propertiesCalculations of superfluid properties onset of superfluidity in COonset of superfluidity in CO22((44He)He)NN
direct relation to Bdirect relation to Beffeff
insight to local contributionsinsight to local contributions to superfluid densityto superfluid density
Paesani, Kwon and WhaleyPaesani, Kwon and WhaleyPRL PRL 9494, 153401 (2005), 153401 (2005)
BerkeleyBerkeleyUniversity of CaliforniaUniversity of California
60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005