A diagrammatic approach to composite, rotating impurities. G. Bighin and M. Lemeshko Institute of Science and Technology Austria SuperFluctuations 2017 – San Benedetto del Tronto, September 7th, 2017
Impurity problems particles) interacting with a many-body environment. How are the properties of the particle modified by the interaction? degrees of freedom... Quasiparticle description? 2/21 Definition: one (or a few ( 10 23 ) Still O
Impurity problems particles) interacting with a many-body environment. How are the properties of the particle modified by the interaction? degrees of freedom... Quasiparticle description? 2/21 Definition: one (or a few ( 10 23 ) Still O
From impurities to quasiparticles 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. 3/21
From impurities to quasiparticles Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. 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. 3/21 Most common cases: electron in a
From impurities to quasiparticles Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a Image from: F. Chevy, Physics 9 , 86. Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange. 3/21 solid, atomic impurities in a BEC.
From impurities to quasiparticles Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a Image from: F. Chevy, Physics 9 , 86. Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange. 3/21 This scenario can be formalized in terms of quasiparticles using the polaron: an elec- tations. tron dressed by a field of many-body exci- solid, atomic impurities in a BEC.
From impurities to quasiparticles Structureless impurity: translational degrees of freedom/linear momentum exchange with the bath. Most common cases: electron in a Image from: F. Chevy, Physics 9 , 86. Composite impurity: translational and internal (i.e. rotational) degrees of freedom/linear and angular momentum exchange. 3/21 This scenario can be formalized in terms of quasiparticles using the polaron: an elec- tations. tron dressed by a field of many-body exci- solid, atomic impurities in a BEC.
From impurities to quasiparticles Structureless impurity: translational the non- The main difficulty: particle? What about a rotating particle? Can there tations. quasiparticles using the polaron: an elec- This scenario can be formalized in terms of 3/21 exchange. freedom/linear and angular momentum internal (i.e. rotational) degrees of Composite impurity: translational and Image from: F. Chevy, Physics 9 , 86. Most common cases: electron in a momentum exchange with the bath. degrees of freedom/linear Abelian SO(3) algebra describing rotations. tron dressed by a field of many-body exci- solid, atomic impurities in a BEC. be a rotating analogue of the polaron quasi-
The angulon phonons 4 Y. Shchadilova, ”Viewpoint: A New Angle on Quantum Impurities” , Physics 10 , 20 (2017). 3 M. Lemeshko, Phys. Rev. Lett. 118 , 095301 (2017). 2 R. Schmidt and M. Lemeshko, Phys. Rev. X 6 , 011012 (2016). 1 R. Schmidt and M. Lemeshko, Phys. Rev. Lett. 114 , 203001 (2015). bath 3 . molecule in any kind of bosonic • Phenomenological model for a in a weakly-interacting BEC 1 . • Derived rigorously for a molecule • Linear molecule. molecule-phonon interaction A composite impurity in a bosonic environment can be described by the 4/21 J 2 molecule angulon Hamiltonian 1 , 2 , 3 , 4 (angular momentum basis: k → { k , λ, µ } ): [ ] ∑ ∑ ω k ˆ k λµ ˆ φ )ˆ φ )ˆ ˆ B ˆ b † λµ (ˆ θ, ˆ b † k λµ + Y λµ (ˆ θ, ˆ Y ∗ H = + + U λ ( k ) b k λµ b k λµ ���� k λµ k λµ � �� � � �� �
The angulon bath 3 . molecule-phonon interaction • Linear molecule. • Derived rigorously for a molecule in a weakly-interacting BEC 1 . • Phenomenological model for a molecule in any kind of bosonic This talk: toy po- phonons tential. Can be connected to real PESs 3 . 1 R. Schmidt and M. Lemeshko, Phys. Rev. Lett. 114 , 203001 (2015). 2 R. Schmidt and M. Lemeshko, Phys. Rev. X 6 , 011012 (2016). 3 M. Lemeshko, Phys. Rev. Lett. 118 , 095301 (2017). 4 Y. Shchadilova, ”Viewpoint: A New Angle on Quantum Impurities” , Physics 10 , 20 (2017). A composite impurity in a bosonic environment can be described by the 4/21 J 2 molecule angulon Hamiltonian 1 , 2 , 3 , 4 (angular momentum basis: k → { k , λ, µ } ): [ ] ∑ ∑ ω k ˆ k λµ ˆ φ )ˆ φ )ˆ ˆ B ˆ b † λµ (ˆ θ, ˆ b † k λµ + Y λµ (ˆ θ, ˆ Y ∗ H = + + U λ ( k ) b k λµ b k λµ ���� k λµ k λµ � �� � � �� �
• Ultracold molecules and from the electrons to a Composite impurities and where to find them • Electronic excitations in Chem. Int. Ed. 43 , 2622 (2004). Image from: J. P. Toennies and A. F. Vilesov, Angew. crystal lattice. • Angular momentum transfer Rydberg atoms. ions. Strong motivation for the theoretical study of composite impurities renormalization). rotational constant (rotational spectra, helium nanodroplets comes from many different fields. Composite impurities are realized as: 5/21 • Molecules embedded into
• Ultracold molecules and from the electrons to a Composite impurities and where to find them Rydberg atoms. Chem. Int. Ed. 43 , 2622 (2004). Image from: J. P. Toennies and A. F. Vilesov, Angew. phase Gas crystal lattice. • Angular momentum transfer • Electronic excitations in Strong motivation for the theoretical study of composite impurities ions. renormalization). rotational constant (rotational spectra, helium nanodroplets comes from many different fields. Composite impurities are realized as: 5/21 • Molecules embedded into in 4 He
• Ultracold molecules and from the electrons to a crystal lattice. inertia) fective rotational lines (higher ef- Renormalizated trum Rotational spec- Chem. Int. Ed. 43 , 2622 (2004). Image from: J. P. Toennies and A. F. Vilesov, Angew. phase Gas Composite impurities and where to find them Strong motivation for the theoretical study of composite impurities • Angular momentum transfer Rydberg atoms. • Electronic excitations in ions. renormalization). rotational constant (rotational spectra, helium nanodroplets comes from many different fields. Composite impurities are realized as: 5/21 • Molecules embedded into in 4 He
from the electrons to a Composite impurities and where to find them • Electronic excitations in Phys. Rev. A 94 , 041601(R) (2016). B. Midya, M. Tomza, R. Schmidt, and M. Lemeshko, crystal lattice. • Angular momentum transfer Rydberg atoms. ions. Strong motivation for the theoretical study of composite impurities renormalization). rotational constant (rotational spectra, helium nanodroplets comes from many different fields. Composite impurities are realized as: 5/21 • Molecules embedded into • Ultracold molecules and
from the electrons to a Composite impurities and where to find them ions. Pfau group, Nature 502 , 664 (2013). crystal lattice. • Angular momentum transfer Rydberg atoms. • Electronic excitations in 5/21 Strong motivation for the theoretical study of composite impurities renormalization). rotational constant (rotational spectra, helium nanodroplets comes from many different fields. Composite impurities are realized as: • Molecules embedded into • Ultracold molecules and
Composite impurities and where to find them Strong motivation for the theoretical study of composite impurities comes from many different fields. Composite impurities are realized as: helium nanodroplets (rotational spectra, rotational constant renormalization). ions. • Electronic excitations in Rydberg atoms. • Angular momentum transfer crystal lattice. 5/21 • Molecules embedded into • Ultracold molecules and from the electrons to a
Path integral description for the angulon PI description of a composite, rotating impurity PIs for rotations PIs for struc- tureless impurities Main reference : GB and M. Lemeshko, Phys. Rev. B 96 , 085410 (2017) 6/21
Path integral description for the angulon The path integral in QM describes the transition amplitude between two states with a weighted average over all trajectories, S is the classical action. 7/21 ∫ � ⟨ ⟩ D x e i S [ x ( t )] G ( x i , x f ; t f − t i ) = = x f , t f � x i , t i
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