Molecular impurities interacting with a many-body environment: - - PowerPoint PPT Presentation
Molecular impurities interacting with a many-body environment: dynamics in Helium nanodroplets G. Bighin and M. Lemeshko Institute of Science and Technology Austria SuperFluctuations 2018 San Benedetto del Tronto, September 6th, 2018
Institute of Science and Technology Austria SuperFluctuations 2018 – San Benedetto del Tronto, September 6th, 2018
Definition: one (or a few particles) interacting with a many-body environment. How are the properties of the particle modified by the interaction? O(1023) degrees of freedom.
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Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a solid, atomic impurities in a BEC.
Image from: F. Chevy, Physics 9, 86.
Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange.
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Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a solid, atomic impurities in a BEC.
Image from: F. Chevy, Physics 9, 86.
Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange.
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Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a solid, atomic impurities in a BEC.
Image from: F. Chevy, Physics 9, 86.
Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange.
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Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a solid, atomic impurities in a BEC.
Image from: F. Chevy, Physics 9, 86.
Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange.
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This scenario (with a bosonic bath) can be for- malized in terms of quasiparticles using the po- laron and the Fröhlich Hamiltonian.
Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a solid, atomic impurities in a BEC.
Image from: F. Chevy, Physics 9, 86.
Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange.
3/19
This scenario (with a bosonic bath) can be for- malized in terms of quasiparticles using the po- laron and the Fröhlich Hamiltonian.
Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a solid, atomic impurities in a BEC.
Image from: F. Chevy, Physics 9, 86.
Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange.
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This scenario (with a bosonic bath) can be for- malized in terms of quasiparticles using the po- laron and the Fröhlich Hamiltonian. What about a rotating particle? Can there be a rotating counterpart of the polaron quasiparti- cle? The main difgiculty: the non-Abelian SO(3) algebra describing rotations.
A composite impurity in a bosonic environment can be described by the angulon Hamiltonian1,2,3,4 (angular momentum basis: k → {k, λ, µ}): ˆ H = Bˆ J2
+ ∑
kλµ
ωkˆ b†
kλµˆ
bkλµ
+ ∑
kλµ
Uλ(k) [ Y∗
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ ]
weakly-interacting BEC1.
molecule in any kind of bosonic bath3.
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A composite impurity in a bosonic environment can be described by the angulon Hamiltonian1,2,3,4 (angular momentum basis: k → {k, λ, µ}): ˆ H = Bˆ J2
+ ∑
kλµ
ωkˆ b†
kλµˆ
bkλµ
+ ∑
kλµ
Uλ(k) [ Y∗
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ ]
weakly-interacting BEC1.
molecule in any kind of bosonic bath3. λ = 0: spherically symmetric part. λ ≥ 1 anisotropic part.
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Strong motivation for the study of composite impurities comes from many difgerent fields. Composite impurities are realized as:
‘cage’ in perovskites.
from the electrons to a crystal lattice.
helium nanodroplets.
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Strong motivation for the study of composite impurities comes from many difgerent fields. Composite impurities are realized as:
‘cage’ in perovskites.
from the electrons to a crystal lattice.
helium nanodroplets.
Image from: C. Eames et al, Nat. Comm. 6, 7497 (2015). 5/19
Strong motivation for the study of composite impurities comes from many difgerent fields. Composite impurities are realized as:
‘cage’ in perovskites.
from the electrons to a crystal lattice.
helium nanodroplets.
J.H. Mentink, M.I. Katsnelson, M. Lemeshko, “Quantum many-body dynamics of the Einstein-de Haas efgect”, arXiv:1802.01638 5/19
Strong motivation for the study of composite impurities comes from many difgerent fields. Composite impurities are realized as:
‘cage’ in perovskites.
from the electrons to a crystal lattice.
helium nanodroplets.
Image from: J. P. Toennies and A. F. Vilesov, Angew. Chem.
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Molecules embedded into helium nanodroplets: Gas phase (free) in 4He
Images from: J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004). 6/19
Molecules embedded into helium nanodroplets: Gas phase (free) in 4He
Images from: J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004). 6/19
Molecules embedded into helium nanodroplets: Gas phase (free) in 4He
Images from: J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004).
Rotational spec- trum Renormalizated lines (smaller efgec- tive B)
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Dynamical alignment experiments:
alignment as a function of time.
and varying the time between the two pulses, one gets ⟨ cos2 ˆ θ2D ⟩ (t) with: cos2 ˆ θ2D ≡ cos2 ˆ θ cos2 ˆ θ + sin2 ˆ θ sin2 ˆ ϕ
Image from B. Shepperson et al., Phys. Rev. Lett. 118, 203203 (2017). 7/19
Interaction of a free molecule with an ofg-resonant laser pulse ˆ H = Bˆ J2 − 1 4∆αE2(t) cos2 ˆ θ When acting on a free molecule, the laser excites in a short time many rotational states (L ↔ L + 2), creating a rotational wave packet:
Movie
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Efgect of the environment is substantial: free molecule vs. same molecule in He.
Stapelfeldt group, Phys. Rev. Lett. 110, 093002 (2013).
Not even a qualitative
— S t r
g c
p l i n g — O u t
q u i l i b r i u m d y n a m i c s — F i n i t e t e m p e r a t u r e ( B ∼ k
B
T )
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Bosons: laboratory frame (x, y, z) Molecules: rotating frame (x′, y′, z′) defined by the Euler angles (ˆ ϕ, ˆ θ, ˆ γ). ˆ S = e−i ˆ
φ⊗ˆ Λze−iˆ θ⊗ˆ Λye−iˆ γ⊗ˆ Λz
where ⃗ ˆ Λ = ∑
µν b† kλµ⃗
σµνbkλν is the angular momentum of the bosons. The ˆ S transformation takes us to the molecular frame.
(cf. Lee-Low-Pines for the polaron).
B 0 limit. An expansion in bath excitations is a strong coupling expansion.
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Bosons: laboratory frame (x, y, z) Molecules: rotating frame (x′, y′, z′) defined by the Euler angles (ˆ ϕ, ˆ θ, ˆ γ). ˆ S = e−i ˆ
φ⊗ˆ Λze−iˆ θ⊗ˆ Λye−iˆ γ⊗ˆ Λz
where ⃗ ˆ Λ = ∑
µν b† kλµ⃗
σµνbkλν is the angular momentum of the bosons. The ˆ S transformation takes us to the molecular frame.
(cf. Lee-Low-Pines for the polaron).
B → 0 limit. An expansion in bath excitations is a strong coupling expansion.
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Bosons: laboratory frame (x, y, z) Molecules: rotating frame (x′, y′, z′) defined by the Euler angles (ˆ ϕ, ˆ θ, ˆ γ). ˆ S = e−i ˆ
φ⊗ˆ Λze−iˆ θ⊗ˆ Λye−iˆ γ⊗ˆ Λz
where ⃗ ˆ Λ = ∑
µν b† kλµ⃗
σµνbkλν is the angular momentum of the bosons. The ˆ S transformation takes us to the molecular frame.
(cf. Lee-Low-Pines for the polaron).
B → 0 limit. An expansion in bath excitations is a strong coupling expansion.
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✓ S t r
g c
p l i n g — O u t
q u i l i b r i u m d y n a m i c s — F i n i t e t e m p e r a t u r e ( B ∼ k
B
T )
We use a time-dependent variational Ansatz: |ψ⟩ = gLM(t) |0⟩bos |LM0⟩ + ∑
kλn
αLM
kλn(t)b† kλn |0⟩bos |LMn⟩
Lagrangian on the variational manifold defined by |ψ⟩: LT=0 = ⟨ψ|i∂t − ˆ H|ψ⟩ Euler-Lagrange equations of motion: d dt ∂L ∂ ˙ xi − ∂L ∂xi = 0 where xi = {gLM, αkλn}. { ˙ gLM(t) = . . . ˙ αLM
kλn(t) = . . .
S t r
g c
p l i n g O u t
q u i l i b r i u m d y n a m i c s — F i n i t e t e m p e r a t u r e ( B k
B
T )
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We use a time-dependent variational Ansatz: |ψ⟩ = gLM(t) |0⟩bos |LM0⟩ + ∑
kλn
αLM
kλn(t)b† kλn |0⟩bos |LMn⟩
Lagrangian on the variational manifold defined by |ψ⟩: LT=0 = ⟨ψ|i∂t − ˆ H|ψ⟩ Euler-Lagrange equations of motion: d dt ∂L ∂ ˙ xi − ∂L ∂xi = 0 where xi = {gLM, αkλn}. { ˙ gLM(t) = . . . ˙ αLM
kλn(t) = . . .
✓ S t r
g c
p l i n g ✓ O u t
q u i l i b r i u m d y n a m i c s — F i n i t e t e m p e r a t u r e ( B ∼ k
B
T )
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For the impurity: average over a statistical ensamble, with weights WL ∝ exp(−βEL). For the bath: defining the ‘Chevy operator’ ˆ O = gLM(t) |LM0⟩ 1 + ∑
kλn
αLM
kλn(t) |LMn⟩ ˆ
b†
kλn
at T = 0 the Lagrangian is LT=0 = ⟨0|ˆ O†(i∂t − ˆ H)ˆ O|0⟩bos , suggesting that at finite temperature LT = Tr [ ρ0 ˆ O†(i∂t − ˆ H)ˆ O ] where ρ0 is the density matrix for the medium. S t r
g c
p l i n g O u t
q u i l i b r i u m d y n a m i c s F i n i t e t e m p e r a t u r e ( B k
B
T )
[1] A. R. DeAngelis and G. Gatofg, Phys. Rev. C 43, 2747 (1991). [2] W.E. Liu, J. Levinsen, M. M. Parish, “Variational approach for impurity dynamics at finite temperature”, arXiv:1805.10013 12/19
For the impurity: average over a statistical ensamble, with weights WL ∝ exp(−βEL). For the bath: defining the ‘Chevy operator’ ˆ O = gLM(t) |LM0⟩ 1 + ∑
kλn
αLM
kλn(t) |LMn⟩ ˆ
b†
kλn
at T = 0 the Lagrangian is LT=0 = ⟨0|ˆ O†(i∂t − ˆ H)ˆ O|0⟩bos , suggesting that at finite temperature LT = Tr [ ρ0 ˆ O†(i∂t − ˆ H)ˆ O ] where ρ0 is the density matrix for the medium. ✓ S t r
g c
p l i n g ✓ O u t
q u i l i b r i u m d y n a m i c s ✓ F i n i t e t e m p e r a t u r e ( B ∼ k
B
T )
[1] A. R. DeAngelis and G. Gatofg, Phys. Rev. C 43, 2747 (1991). [2] W.E. Liu, J. Levinsen, M. M. Parish, “Variational approach for impurity dynamics at finite temperature”, arXiv:1805.10013 12/19
Comparison of the theory with preliminary experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: I2.
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Comparison of the theory with preliminary experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: I2. Which rotational states are populated as the laser is switched
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Comparison of the theory with preliminary experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: I2. Which rotational states are populated as the laser is switched
Free case: the angular momentum goes to the molecule. In a Helium droplet: the angular momentum goes to the molecule and to the bath. Free molecule In Helium
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Comparison of the theory with preliminary experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: I2. Which rotational states are populated as the laser is switched
Free case: the angular momentum goes to the molecule. In a Helium droplet: the angular momentum goes to the molecule and to the bath.
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Comparison of the theory with preliminary experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: I2. ⟨ cos2 ˆ θ2D ⟩ (t) Laser fluence F measured in J/cm2
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Comparison of the theory with preliminary experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: CS2. ⟨ cos2 ˆ θ2D ⟩ (t)
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Comparison of the theory with preliminary experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: OCS. ⟨ cos2 ˆ θ2D ⟩ (t) Laser fluence F measured in J/cm2, time measured in ps.
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More numerical approach: DiagMC, sampling all diagrams in a stochastic way. = + + + + …+ + + … How do we describe angular momentum redistribution in terms of diagrams? How does the configuration space looks like? Connecting DiagMC and the theory of molecular simulations!
GB and M. Lemeshko, Phys. Rev. B 96, 419 (2017). GB, T.V. Tscherbul, M. Lemeshko, arXiv:1803:07990 16/19
many-body excitations.
interpreted in terms of angulons.
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Lemeshko group @ IST Austria:
Misha Lemeshko Enderalp Yakaboylu Xiang Li Igor Cherepanov Wojciech Rządkowski
Collaborators:
Henrik Stapelfeldt (Aarhus)
Dynamics in He Dynamical alignment ex- periments
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This work was supported by the Austrian Science Fund (FWF), project Nr. P29902-N27.
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Free rotor propagator G0,λ(E) = 1 E − Bλ(λ + 1) + iδ Interaction propagator χλ(E) = ∑
k
|Uλ(k)|2 E − ωk + iδ