Far-from-equilibrium dynamics of molecules in 4 He nanodroplets: a - - PowerPoint PPT Presentation
Far-from-equilibrium dynamics of molecules in 4 He nanodroplets: a quasiparticle perspective Giacomo Bighin Institute of Science and Technology Austria Universitat Politcnica de Catalunya Barcelona, September 18th, 2019 Quantum impurities
4He nanodroplets: a quasiparticle perspective
Institute of Science and Technology Austria Universitat Politècnica de Catalunya — Barcelona, September 18th, 2019
One particle (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 (e.g. a molecule): translational and 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 (e.g. a molecule): translational and rotational degrees of freedom/linear and angular momentum exchange.
3/35
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 (e.g. a molecule): translational and rotational degrees of freedom/linear and angular momentum exchange.
3/35
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 (e.g. a molecule): translational and rotational degrees of freedom/linear and angular momentum exchange.
3/35
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 (e.g. a molecule): translational and rotational degrees of freedom/linear and angular momentum exchange.
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What about a rotating impurity? How can this scenario be realized experimentally?
Strong motivation for the study of composite impurities comes from many difgerent fields. Composite impurities can be 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 can be 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). 4/35
Strong motivation for the study of composite impurities comes from many difgerent fields. Composite impurities can be 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”,
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Strong motivation for the study of composite impurities comes from many difgerent fields. Composite impurities can be 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.
4/35
Strong motivation for the study of composite impurities comes from many difgerent fields. Composite impurities can be 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.
4/35
First part:
molecules in He nanodroplets. Second part: angular momentum, Feynman diagrams and Diagrammatic Monte Carlo.
A molecular impurity embedded into a helium nanodroplet: a controllable system to explore angular momentum redistribution in a many-body environment. Temperature ∼ 0.4K Droplets are superfluid Easy to produce Free of perturbations Only rotational degrees of freedom Easy to manipulate by a laser
Image from: S. Grebenev et al., Science 279, 2083 (1998). 5/35
A molecular impurity embedded into a helium nanodroplet: a controllable system to explore angular momentum redistribution in a many-body environment. Temperature ∼ 0.4K Droplets are superfluid Easy to produce Free of perturbations Only rotational degrees of freedom Easy to manipulate by a laser
Image from: S. Grebenev et al., Science 279, 2083 (1998).
Interaction of a linear molecule with an ofg-resonant laser pulse: ˆ Hlaser = −1 4∆αE2(t) cos2 ˆ θ
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Molecules embedded into helium nanodroplets: rotational spectrum Gas phase (free) in 4He
Images from: J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004). 6/35
Molecules embedded into helium nanodroplets: rotational spectrum Gas phase (free) in 4He
Images from: J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004). 6/35
Molecules embedded into helium nanodroplets: rotational spectrum 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 (Stapelfeldt group, Aarhus University):
alignment as a function of time.
and varying the time between the two pulses, one gets
θ2D
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). 7/35
A simpler example: a free molecule interacting 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:
Image from: G. Kaya et al., Appl. Phys. B 6, 122 (2016).
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). 9/35
Dynamics of isolated I2 molecules
Experiment: Henrik Stapelfeldt, Lars Christiansen, Anders Vestergaard Jørgensen (Aarhus University)
Dynamics of I2 molecules in helium Efgect of the environment is substantial:
F =
unknown origin.
He-DFT?
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The quantum mechanical treatment of many-body systems is always
Polaron: an electron dressed by a field of many-body excitations. Angulon: a quantum rotor dressed by a field of many-body excitations.
Image from: F. Chevy, Physics 9, 86. 11/35
The quantum mechanical treatment of many-body systems is always
Polaron: an electron dressed by a field of many-body excitations. Angulon: a quantum rotor dressed by a field of many-body excitations.
Image from: F. Chevy, Physics 9, 86. 11/35
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
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
weakly-interacting BEC1.
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
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
weakly-interacting BEC1.
in any kind of bosonic bath3. λ = 0: spherically symmetric part. λ ≥ 1 anisotropic part.
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We apply a canonical transformation 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.
transformation for the polaron. Bosons: laboratory frame (x, y, z) Molecule: rotating frame (x′, y′, z′) defined by the Euler angles (ˆ ϕ, ˆ θ, ˆ γ). laboratory frame rotating frame
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Result: a rotating linear molecule interacting with a bosonic bath can be described in the frame co-rotating with the molecule by the following Hamiltonian: ˆ H = ˆ S−1ˆ Hˆ S = B(^ L − ˆ Λ)2 +
ωkˆ b†
kλµˆ
bkλµ +
Vkλ ˆ b†
kλ0 + ˆ
bkλ0
bosons.
the B 0 limit. An expansion in bath excitations is a strong coupling expansion.
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Result: a rotating linear molecule interacting with a bosonic bath can be described in the frame co-rotating with the molecule by the following Hamiltonian: ˆ H = ˆ S−1ˆ Hˆ S = B(^ L − ˆ Λ)2 +
ωkˆ b†
kλµˆ
bkλµ +
Vkλ ˆ b†
kλ0 + ˆ
bkλ0
bosons.
the B 0 limit. An expansion in bath excitations is a strong coupling expansion.
Compare with the Lee-Low-Pines Hamiltonian ˆ HLLP =
k kˆ
b†
kˆ
bk 2 2mI +
ωkˆ b†
kˆ
bk + g V
ˆ b†
k′ˆ
bk′
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Result: a rotating linear molecule interacting with a bosonic bath can be described in the frame co-rotating with the molecule by the following Hamiltonian: ˆ H = ˆ S−1ˆ Hˆ S = B(^ L − ˆ Λ)2 +
ωkˆ b†
kλµˆ
bkλµ +
Vkλ ˆ b†
kλ0 + ˆ
bkλ0
bosons.
U in the B → 0 limit. An expansion in bath excitations is a strong coupling expansion.
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We describe dynamics using a time-dependent variational Ansatz, including excitations up to one phonon: |ψLM(t)⟩ = ˆ U(gLM(t) |0⟩bos |LM0⟩ +
αLM
kλn(t)b† kλn |0⟩bos |LMn⟩)
Lagrangian on the variational manifold defined by |ψLM⟩: LT=0 = ⟨ψLM|i∂t − ˆ H|ψLM⟩ Euler-Lagrange equations of motion: d dt ∂L ∂ ˙ xi − ∂L ∂xi = 0 where xi = {gLM, αLM
kλn}. We obtain a difgerential system
gLM(t) = . . . ˙ αLM
kλn(t) = . . .
to be solved numerically; in αkλµ the momentum k needs to be discretized.
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Comparison with experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: I2.
0.5 0.7 0.5 0.7 0.5 0.7 0.5 0.7 50 100 150 200 250 300 350 0.5 0.7 0.9 14.2 J/cm2 7.1 J/cm2 5.0 J/cm2 2.8 J/cm2 Falign=1.4 J/cm2 I2 in helium droplets a1 a5 a3 a2 a4 t (ps)
<cos2
2D >Generally good agreement for the main features in experimental data:
growing in amplitude as the laser fluence is increased.
periods are visible.
change much with fluence.
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0.5 0.6 0.5 0.6 0.5 0.6 0.5 0.6 0.5 0.6 0.5 0.7 0.5 0.7 0.5 0.7 50 100 150 200 250 300 350 0.5 0.7 c9 c8 c7 c6 c5 c4 c3 c2 c1 14.2 J/cm2 10.6 J/cm2 7.1 J/cm2 5.0 J/cm2 2.8 J/cm2 1.4 J/cm2 1.1 J/cm2 0.7 J/cm2 Falign=0.4 J/cm2 CS2 in helium droplets t (ps)
<cos2
2D >Comparison with experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: CS2.
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0.5 0.6 0.5 0.6 0.5 0.6 0.5 0.6 0.5 0.6 0.5 0.6 0.5 0.6 50 100 150 200 250 300 350 0.5 0.6 0.7 d8 d7 d6 d5 d4 d3 d2 d1 7.1 J/cm2 14.2 J/cm2 5.0 J/cm2 2.8 J/cm2 2.1 J/cm2 1.4 J/cm2 1.1 J/cm2 Falign=0.7 J/cm2 J/cm2 OCS in helium droplets t (ps)
<cos2
2D >Comparison with experimental data from Stapelfeldt group, Aarhus University, for difgerent molecules: OCS.
a couple of cases where one weak
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they sometimes disappear? What about the decay?
momentum redistribution: microscopical insight on the problem!
levels) are afgected by the many-body environment.
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they sometimes disappear? What about the decay?
momentum redistribution: microscopical insight on the problem!
levels) are afgected by the many-body environment.
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The Fourier transform of the measured alignment cosine ⟨cos2 ˆ θ2D⟩(t) is dominated by (L) ↔ (L + 2) interferences. How is it afgected when the level structure changes? EL+2 − EL
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The Fourier transform of the measured alignment cosine ⟨cos2 ˆ θ2D⟩(t) is dominated by (L) ↔ (L + 2) interferences. How is it afgected when the level structure changes? EL+2 − EL 20Ghz corresponds to 50ps
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i) Is this the full story? Can the observed dynamics be explained only by means of renormalised rotational levels?
Red dashed lines (only renormalised levels) vs. solid black line (full many-body treatment).
ii) How long does it take for a molecule to equilibrate with the helium environment and form an angulon quasiparticle? This requires tens of ps; which is also the timescale of the laser!
Approach to equilibrium of the quasiparticle weight |gLM|2 and of the phonon populations ∑
k |αkλµ|2.
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i) Is this the full story? Can the observed dynamics be explained only by means of renormalised rotational levels?
Red dashed lines (only renormalised levels) vs. solid black line (full many-body treatment).
ii) How long does it take for a molecule to equilibrate with the helium environment and form an angulon quasiparticle? This requires tens of ps; which is also the timescale of the laser!
Approach to equilibrium of the quasiparticle weight |gLM|2 and of the phonon populations ∑
k |αkλµ|2.
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With a shorter 450 fs pulse, same molecule (I2), the strong oscillatory pattern is absent:
Image from: B. Shepperson et al., Phys. Rev. Lett. 118, 203203 (2017).
alignment in the laboratory frame, providing access to the structure of highly excited rotational states.
in the molecular alignment; an explanation requires a many-body theory of angular momentum redistribution.
dynamics of angulon quasiparticles, shedding light onto many-particle dynamics of angular momentum at femtosecond timescales.
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Back to the angulon Hamiltonian: ˆ H = Bˆ J2
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
Perturbation theory/Feynman diagrams:
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Back to the angulon Hamiltonian: ˆ H = Bˆ J2
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
Perturbation theory/Feynman diagrams: = + + + + . . . How does angular momentum enter this picture?
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Back to the angulon Hamiltonian: ˆ H = Bˆ J2
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
Perturbation theory/Feynman diagrams: Fröhlich polaron
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Back to the angulon Hamiltonian: ˆ H = Bˆ J2
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
Perturbation theory/Feynman diagrams: Angulon
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Back to the angulon Hamiltonian: ˆ H = Bˆ J2
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
Perturbation theory/Feynman diagrams: Angulon How does angular momentum enter here?
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Feynman rules Each free propagator
λi µi
Each phonon propagator
λi µi
Each vertex (−1)λi ⟨λi| |Y(λj)| |λk⟩
λj λk µi µj µk
Usually momentum conservation is enforced by an appropriate labeling. Not the same for angular momentum, j and λ couple to |j − λ|, . . . , j + λ.
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Feynman rules Each free propagator
λi µi
Each phonon propagator
λi µi
Each vertex (−1)λi ⟨λi| |Y(λj)| |λk⟩
λj λk µi µj µk
Diagrammatic theory of angular momentum (developed in the context of theoretical atomic spectroscopy)
from D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, “Quantum Theory of Angular Momentum”. 24/35
Let us use the Feynman diagrams! First order self-energy: Dyson equation : =
angulon quantum rotor many-body field
Finally the spectral function allows for a study the whole excitation spectrum of the system: Aλ(E) = − 1 π Im Gλ(E + i0+) Equivalent to a simple, 1-phonon variational Ansatz (cf. Chevy Ansatz for the polaron) |ψ⟩ = Z1/2
LM |0⟩ |LM⟩ +
jm
βkλjCLM
jm,λµb† kλµ |0⟩ |jm⟩ 25/35
Let us use the Feynman diagrams! First order self-energy: Dyson equation : =
angulon quantum rotor many-body field
Finally the spectral function allows for a study the whole excitation spectrum of the system: Aλ(E) = − 1 π Im Gλ(E + i0+) Equivalent to a simple, 1-phonon variational Ansatz (cf. Chevy Ansatz for the polaron) |ψ⟩ = Z1/2
LM |0⟩ |LM⟩ +
jm
βkλjCLM
jm,λµb† kλµ |0⟩ |jm⟩ 25/35
Spectral function: Aλ(E)
What about higher orders? = + + + + . . . + + + . . . + + . . . At order n: n integrals, and higher angular momentum couplings (3n-j symbols).
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A feasible plan? Notice the logarithmic scale: exponentially rare, since they are exponentially more difgicult to compute. For monster stufg, like a 303-j symbol taking 2.3 years to compute, see: C. Brouder and G. Brinkmann, Journal of Electron Spectroscopy and Related Phenomena 86, 127 (1997).
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A feasible plan? Notice the logarithmic scale: exponentially rare, since they are exponentially more difgicult to compute. For monster stufg, like a 303-j symbol taking 2.3 years to compute, see: C. Brouder and G. Brinkmann, Journal of Electron Spectroscopy and Related Phenomena 86, 127 (1997).
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Numerical technique for summing all Feynman diagrams1. More on this later... = + + + + …+ + + … Up to now: structureless particles (Fröhlich polaron, Holstein polaron), or particles with a very simple internal structure (e.g. spin 1/2). Molecules2? Connecting DiagMC and molecular simulations!
2GB, T.V. Tscherbul, M. Lemeshko, Phys. Rev. Lett. 121, 165301 (2018).
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Hamiltonian for an impurity problem: ˆ H = ˆ Himp + ˆ Hbath + ˆ Hint Green’s function G(τ) = + + + + . . . = all Feynman diagrams DiagMC idea: set up a stochastic process sampling among all diagrams1. Configuration space: diagram topology, phonons internal variables, times, etc... Number of variables varies with the topology! How: ergodicity, detailed balance w1p(1 → 2) = w2p(2 → 1) Result: each configuration is visited with probability ∝ its weight.
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Hamiltonian for an impurity problem: ˆ H = ˆ Himp + ˆ Hbath + ˆ Hint Green’s function G(τ) = + + + + . . . = all Feynman diagrams DiagMC idea: set up a stochastic process sampling among all diagrams1. Configuration space: diagram topology, phonons internal variables, times, etc... Number of variables varies with the topology! How: ergodicity, detailed balance w1p(1 → 2) = w2p(2 → 1) Result: each configuration is visited with probability ∝ its weight.
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Works in continuous time and in the thermody- namic limit: no finite-size efgects or systematic errors.
We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
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We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
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We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
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We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
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We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
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We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
30/35
We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
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We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
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We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length will be proportional to G . One can fill a histogram afuer each update and get the Green’s function.
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We need updates spanning the whole configuration space: Add update: a new arc is added to a diagram. Remove update: an arc is removed from the diagram. Change update: modifies the total length of the diagram. Result: the time the stochastic process spends with diagrams of length τ will be proportional to G(τ). One can fill a histogram afuer each update and get the Green’s function.
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Moving particle: linear momentum circulating on lines. Rotating particle: angular momentum circulating on lines. It gets weirder... Down the rabbit hole of angular momentum composition!
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Moving particle: linear momentum circulating on lines. Rotating particle: angular momentum circulating on lines. It gets weirder... Down the rabbit hole of angular momentum composition! ⃗ k and ⃗ q fully deter- mine ⃗ k − ⃗ q j and λ can sum in many difgerent ways: |j−λ|, . . . j+λ
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Moving particle: linear momentum circulating on lines. Rotating particle: angular momentum circulating on lines. It gets weirder... Down the rabbit hole of angular momentum composition!
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Moving particle: linear momentum circulating on lines. Rotating particle: angular momentum circulating on lines. It gets weirder... Down the rabbit hole of angular momentum composition! The phonon takes away ⃗ q1 momen- tum... ...and gives back ⃗ q1 momentum The phonon does not subtract an- gular momentum from the impurity... ...but gives back two quanta!
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Moving particle: linear momentum circulating on lines. Rotating particle: angular momentum circulating on lines. It gets weirder... Down the rabbit hole of angular momentum composition!
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The configuration space is more complex... and bigger! We need difgerent updates. Shufgle update: select
1-particle- irreducible component, shufgle the momenta couplings to another allowed configuration.
The ground-state energy of the angulon Hamiltonian obtained using DiagMC1 as a function of the dimensionless bath density, ˜ n, in comparison with the weak-coupling theory2 and the strong-coupling theory3. The energy is obtained by fitting the long-imaginary-time behaviour of Gj with Gj(τ) = Zj exp(−Ej τ). Inset: energy of the L = 0, 1, 2 states.
1GB, T.V. Tscherbul, M. Lemeshko, Phys. Rev. Lett. 121, 165301 (2018).
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The ground-state energy of the angulon Hamiltonian obtained using DiagMC1 as a function of the dimensionless bath density, ˜ n, in comparison with the weak-coupling theory2 and the strong-coupling theory3. The energy is obtained by fitting the long-imaginary-time behaviour of Gj with Gj(τ) = Zj exp(−Ej τ). Inset: energy of the L = 0, 1, 2 states.
1GB, T.V. Tscherbul, M. Lemeshko, Phys. Rev. Lett. 121, 165301 (2018).
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coupled angular momenta.
efgects or systematic errors.
(2018).
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Lemeshko group @ IST Austria:
Misha Lemeshko Enderalp Yakaboylu Xiang Li Igor Cherepanov Wojciech Rządkowski
Collaborators:
Henrik Stapelfeldt (Aarhus) Richard Schmidt (MPI Garching) Timur Tscherbul (Reno)
Dynamics in He Dynamical alignment experiments DiagMC
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This work was supported by a Lise Meitner Fellowship of the Austrian Science Fund (FWF), project Nr. M2461-N27. These slides at http://bigh.in/talks
For the impurity: average over a statistical ensamble, weights ∝ exp(−βEL). For the bath: the zero-temperature bosonic expectation values in L are converted to finite temperature ones1,2. LT=0 = ⟨0|ˆ O†(i∂t − ˆ H)ˆ O|0⟩bos − → LT = Tr
O†(i∂t − ˆ H)ˆ O
wavefunction is then
LM LM
focused.
the nuclear spin.
[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
For the impurity: average over a statistical ensamble, weights ∝ exp(−βEL). For the bath: the zero-temperature bosonic expectation values in L are converted to finite temperature ones1,2. LT=0 = ⟨0|ˆ O†(i∂t − ˆ H)ˆ O|0⟩bos − → LT = Tr
O†(i∂t − ˆ H)ˆ O
wavefunction is then |Ψ⟩ =
LM |ψLM⟩
focused.
the nuclear spin.
[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
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
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
weakly-interacting BEC1.
in any kind of bosonic bath3.
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
+
ωkˆ b†
kλµˆ
bkλµ
+
Uλ(k)
λµ(ˆ
θ, ˆ ϕ)ˆ b†
kλµ + Yλµ(ˆ
θ, ˆ ϕ)ˆ bkλµ
weakly-interacting BEC1.
in any kind of bosonic bath3. λ = 0: spherically symmetric part. λ ≥ 1 anisotropic part.
We apply a canonical transformation 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.
transformation for the polaron. Bosons: laboratory frame (x, y, z) Molecule: rotating frame (x′, y′, z′) defined by the Euler angles (ˆ ϕ, ˆ θ, ˆ γ). laboratory frame rotating frame
For the impurity: average over a statistical ensamble, weights ∝ exp(−βEL). For the bath: the zero-temperature bosonic expectation values in L are converted to finite temperature ones1,2. LT=0 = ⟨0|ˆ O†(i∂t − ˆ H)ˆ O|0⟩bos − → LT = Tr
O†(i∂t − ˆ H)ˆ O
wavefunction is then
LM LM
focused.
the nuclear spin.
[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
For the impurity: average over a statistical ensamble, weights ∝ exp(−βEL). For the bath: the zero-temperature bosonic expectation values in L are converted to finite temperature ones1,2. LT=0 = ⟨0|ˆ O†(i∂t − ˆ H)ˆ O|0⟩bos − → LT = Tr
O†(i∂t − ˆ H)ˆ O
wavefunction is then |Ψ⟩ =
LM |ψLM⟩
focused.
the nuclear spin.
[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
For the impurity: average over a statistical ensamble, weights ∝ exp(−βEL). For the bath: the zero-temperature bosonic expectation values in L are converted to finite temperature ones1,2. LT=0 = ⟨0|ˆ O†(i∂t − ˆ H)ˆ O|0⟩bos − → LT = Tr
O†(i∂t − ˆ H)ˆ O
wavefunction is then
LM LM
focused.
the nuclear spin.
[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
✓ 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 )
Some additional considerations:
LM |ψLM⟩
abundances, due to the nuclear spin.