1 Polaron Seminar, AG Widera AG Fleischhauer, 05/06/14 Introduction to polaron physics in BECs Fabian Grusdt Physics Department and Research Center OPTIMAS, University of Kaiserslautern, Germany Graduate School of Materials Science in Mainz, Kaiserslautern, Germany
2 Motivation Q: What is a polaron? A: a long-lived quasiparticle compromised of an impurity dressed by phonons. e phonon
3 Motivation Polaron s in BECs BECs single impurity Anderson et al., Science 269 (1995) z y x q Frese et al, PRL 85 (2000)
4 Outline Fröhlich Hamiltonian in a BEC derivation from first principles conditions for the approach Polaron properties (MF polaron theory) ground state energy
5 Bogoliubov-Fröhlich Hamiltonian derivation from first principles see e.g. Tempere et al., PRB 80 (2009) Bruderer et al., PRA 76 (2007)
6 Fröhlich Hamiltonian Microscopic model: Z apple H = d 3 ~r ˆ r 2 (~r) + g BB ˆ (~r) 2m B 2 ˆ(~r) ˆ(~r) Z apple r + d 3 ~r ˆ 2 (~r) 2M + g IB ˆ (~r) ˆ(~r) ˆ(~r). +traps or lattices 87 Rb 133 Cs pseudo-potentials: V IB (~r) =g IB (~r) ˆ(~r) ˆ(~r) Bose field impurity field
7 Fröhlich Hamiltonian BEC - Bogoliubov theory Hamiltonian ˆ(~r) =L 3/2 X ~k H B = X ~k BEC ˆ0 = p N 0 e i ~r ˆ~k ˆ ˆ~k 2 2m B + 1 L 3 X, 0, discrete set of modes [ ˆ, ˆ 0 ]=, 0 g BB 2 ˆ + ˆ 0 ˆ~k 0 ˆ weakly interacting BEC H B = E 0 + X ~k ~ ˆ k 2 ˆ~k + g BBN 0 2m B 2L 3 X 6=0 2 ˆ ˆ~k + ˆ ˆ + ˆ ˆ ~k
8 Fröhlich Hamiltonian Bogoliubov transformation ˆ~k = cosh ~k â ~k sinh ~k â, phonons: â ~k, â Bogoliubov phonons: 8 H B = E 0 0 + X ~k k â â ~k 6 4 BEC characterization: BEC 10 18...10 21 m 3 1µm c 1mm/s 2 0 0 1 2 3 k = ck p 1+( k) 2 /2
9 Fröhlich Hamiltonian Boson - impurity interaction H IB = g IB L 3 Z d 3 ~r ˆ (~r) X ~k, ~k 0 e i ( 0 ) ~r ˆ ˆ~k 0 ˆ(~r) BEC ˆ0 = p N 0 2 H IB = g IB L 3 4N 0 + X ~k6=0 e p i ~r N ˆ 0 ~ +h.c. + X k, 0 6=0 e i( 0 ) ~r ˆ ˆ~k 0 3 5 BEC mean field shift phonon-impurity scattering phonon-phonon scattering
Fröhlich Hamiltonian H IB = g IB L 3 2 4N 0 + X ~k6=0 e p i ~r N ˆ 0 ~ +h.c. + X k, 0 6=0 e i( 0 ) ~r ˆ ˆ~k 0 3 5 Phonon-impurity scattering ˆ~k = cosh ~k â ~k sinh ~k â, H a = Z d 3 ~r ˆ (~r) X ~k6=0 Vk disc e i ~r â ~k +â ~ ˆ(~r) k scattering amplitude V disc k = g IB L 3 p N0 k 2 /2m B 2g BB BEC + k 2 /2m B 1/4 modified by Bogoliubov mixing angles ~k
11 Fröhlich Hamiltonian H IB = g IB L 3 2 4N 0 + X ~k6=0 e p i ~r N ˆ 0 ~ +h.c. + X k, 0 6=0 e i( 0 ) ~r ˆ ˆ~k 0 3 5 continuum limit L 1 N 0 L 3 = BEC â ~k scattering amplitude 2 L 3/2 â( ) [â( ), â ( 0 )] = ( 0 ) V disc k = g IB L 3 p N0 k 2 /2m B 2g BB BEC + k 2 /2m B 1/4 0.5 1 (c) V k = g IB p BEC (2 ) 3/2 ( k) 2 2+( k) 2 1/4 0 0 1 2 3
12 Fröhlich Hamiltonian Phonon-phonon scattering X ˆ ˆ~k N ph 1 X L 3 k< = 1 (2 ) 3 (2 ) 3 L 3 X k< 1 (2 ) 3 H IB = g IB L 3 Z 3 2 4N 0 + X ~k6=0 d 3 3 e p i ~r N ˆ 0 ~ +h.c. + X k hh a-a i hh a i hh a-a i g IB N ph 3 hh a i g IB p BEC 1 s, 0 6=0 N ph 3 e i( 0 ) ~r ˆ ˆ~k 0 3 5 condition for Fröhlich model BEC N ph 3
13 Fröhlich Hamiltonian condition for Fröhlich model (alternative derivation: Bruderer et.al., PRA 2007) g IB 3 2c/ typical numbers (priv.comm., Farina) =1.3µm, c =0.4mm/s, a IB = 34nm g IB 2 /c =0.2 Fröhlich Hamiltonian: Z H = d 3 k â ( )â( )+g IB BEC + Z r + d 3 ~r ˆ 2 Z (~r) 2M + d 3 ~ h kvk e i ~r â( )+â ( )i ˆ(~r)
14 Model parameters polaron energy see Tempere et al., PRB 80 (2009) Rath & Schmidt, PRA 88 (2013) Shashi et al., PRA in press (2014)
15 Model parameters units length scale healing length: time scale via speed of sound: /c energy scale ~c/ ~ =1 mass scale boson mass m B free parameters parameter typical value weak coupling strong coupling BEC density BEC 1...10 3 small large impurity-boson scattering length a IB 10 2...10 3 small large impurity mass M 10 1...10 1 large small van-der-waals length `vdw 10 3 large small
16 Polaron energy Infinite mass limit H = g IB BEC + M 1 Z classical impurity d 3 k k â â + V k â ~k +â integrable model Û = Y ~k exp ~k â ~k ~k â ~k Û â ~k Û =â ~k + ~k Û ĤÛ = g IB BEC + Z d 3 k k â â ~k V 2 k k
17 Polaron energy ground state energy momentum cut-off E 0 = g IB BEC 4 Z 0 dk k 2 V k 2 k UV-scaling k k 2 V k 1 Z dk k 2 V 2 Z k dk 1= 1 1 k UV divergent?
18 Polaron energy Lippmann-Schwinger equation remember pseudo potential k(~r) = (0) k (~r)+eikr r f(k)+o(r 2 ) low-energies: universal f(k) = a IB 1+ika IB for k. 1/`vdW =: see e.g. Bloch et al., RMP (2008) strategy: use pseudo potential V (~r) =g IB (~r) with same scattering length 1 = 2 a IB g IB 1 m B + 1 M + 2 see e.g. Rath & Schmidt, PRA (2013)
19 Polaron energy ground state energy g IB = 2 a IB m B apple 1+a IB 2 + O(a2 IB) E 0 = g IB BEC 4 Z 0 dk k 2 V k 2 k E 0 =2 a IB BEC m B Z d 3 V 2 k k +4 a2 IB m B UV convergent
20 Summary Fröhlich Hamiltonian in a BEC ˆ, ˆ ˆ, â~k condition: ˆ0 = p N 0 g IB 3 2c/ Z H = d 3 k â ( )â( )+g IB BEC + Z r + d 3 ~r ˆ 2 Z (~r) 2M + d 3 ~ h kvk e i ~r â( )+â ( )i ˆ(~r) Polaron properties, localized impurity ground state energy 1 = 2 a IB g IB 1 m B + 1 M + 2
21 Related work experiments: Schirotzek et al., PRL 102 (2009) Scelle et al., PRL 111 (2013) Fukuhara et al., Nature Phys. 9 (2013) theory: Tempere et al., PRB 80 (2009) Bruderer et al., PRA 76 (2007) NJP 10 (2008) Cucchietti & Timmermans, PRL 96 (2006) Casteels et al., Laser Phys. 21 (2011) Shashi et al., to appear in PRA (2014) Observation of the Fermi polaron Deep lattice Bose polaron Spin-impurity polarons in a BEC Feynman path integral treatment Strong coupling treatment (lattice) Strong coupling treatment (continuum) RF spectra of Fröhlich polaron in BEC
22 Thanks to Eugene Demler Aditya Shashi Dmitry Abanin Shashi et al., arxiv:1401.0952 (2014) Grusdt et al., in preparation Grusdt et al., in preparation Radio frequency spectroscopy of polarons in ultracold Bose gases Bosonic lattice polaron Bloch oscillations Renormalization group treatment of polarons in ultracold Bose gases
and thanks for your attention 23