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 Universitt Heidelberg, May 23th, 2019 Quantum impurities One particle (or a few particles)
4He nanodroplets: a quasiparticle perspective
Institute of Science and Technology Austria Universität Heidelberg, May 23th, 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: 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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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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What about a rotating particle? How can this scenario be realized?
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). 4/28
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.
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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). 5/28
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 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). 6/28
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). 6/28
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 ⟨ 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). 7/28
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).
Very noticeable difgerences in the timescales and in the approach to equilibrium. An intriguing puzzle: not even a qualitative understanding. Monte Carlo? He-DFT?
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Dynamics of isolated I2 molecules
Experiment: Henrik Stapelfeldt, Lars Christiansen, Anders Vestergaard Jørgensen (Aarhus University)
Dynamics of I2 molecules in helium Striking difgerences between the two cases:
F = ∫ dt I(t) is changed.
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Dynamics of isolated I2 molecules
Experiment: Henrik Stapelfeldt, Lars Christiansen, Anders Vestergaard Jørgensen (Aarhus University)
Dynamics of I2 molecules in helium Striking difgerences between the two cases:
F = ∫ dt I(t) is changed.
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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 )
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. 11/28
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. 11/28
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.
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.
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 + ∑
kλµ
ωkˆ b†
kλµˆ
bkλµ + ∑
kλ
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 + ∑
kλµ
ωkˆ b†
kλµˆ
bkλµ + ∑
kλ
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 = ( P − ∑
k kˆ
b†
kˆ
bk )2 2mI + ∑
k
ωkˆ b†
kˆ
bk + g V ∑
k,k′
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 + ∑
kλµ
ωkˆ b†
kλµˆ
bkλµ + ∑
kλ
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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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 + ∑
kλµ
ωkˆ b†
kλµˆ
bkλµ + ∑
kλ
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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✓ 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 describe dynamics using a time-dependent variational Ansatz, including excitations up to one phonon: |ψLM(t)⟩ = ˆ U(gLM(t) |0⟩bos |LM0⟩ + ∑
kλn
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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We describe dynamics using a time-dependent variational Ansatz, including excitations up to one phonon: |ψLM(t)⟩ = ˆ U(gLM(t) |0⟩bos |LM0⟩ + ∑
kλn
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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✓ 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 )
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 [ ρ0 ˆ O†(i∂t − ˆ H)ˆ O ] A couple of additional details:
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 16/28
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 [ ρ0 ˆ O†(i∂t − ˆ H)ˆ O ] A couple of additional details:
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 16/28
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 [ ρ0 ˆ O†(i∂t − ˆ H)ˆ O ] A couple of additional details:
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 16/28
✓ 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 )
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 rotational level structure is modified by the helium medium: one gets rotational constant renormalisation (B → B∗) and centrifugal distortion (D):
constant is renormalized B → B∗.
centrifugal correction D[L(L + 1)]2 becomes relevant.
quadratic spectrum: detachment.
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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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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 Transition probability under a Gaussian pulse Wfi = |Vfi|2 ℏ2 exp ( −σ2ω2
fi
) where ωfi ≡ (Ef − Ei) /ℏ and σ is the pulse duration. The distortion creates a gap afuer 20GHz, so that transitions afuer the gap are strongly suppressed.
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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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iii) Efgect of superfluid helium on angular momentum dynamics: it prevents the rotational energy of the molecule from increasing as rapidly as it would in the gas phase.
Time evolution of the molecular angular momentum, in helium (red) and in the gas phase (blue). 24/28
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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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, Phys. Rev. Lett. 121, 165301 (2018). 26/28
Lemeshko group @ IST Austria:
Misha Lemeshko Enderalp Yakaboylu Xiang Li Igor Cherepanov Wojciech Rządkowski
Collaborators:
Henrik Stapelfeldt (Aarhus) Richard Schmidt (MPI Garching)
Dynamics in He Dynamical alignment experiments
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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
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