The Bose polaron- theory and experiments Georg M. Bruun Aarhus - PowerPoint PPT Presentation

The Bose polaron- theory and experiments Georg M. Bruun Aarhus University R. S. Christensen, J. Levinsen & GMB, PRL 115 , 160401 (2015) J. Levinsen, M. M. Parish & GMB, PRL 115 , 125302 (2015) N. B. Jørgensen et al, arXiv:1604.07883

Bose Polaron Mobile impurity interacting with bosonic reservoir • Electrons coupled to phonons • Helium mixtures • High T c superconductors • Elementary particles coupled to the Higgs boson

Nice to have experimental realisation in cold atoms Fermi polaron gave lots of new insights 1.5 c − 3.53 1. Schirotzek et al. , Phys. Rev. Lett. 102 , signal 1 230402 (2009) 0.5 2. Kohstall et al., Nature 485 , 615 (2012) 3. Koschourek et al., Nature 485 , 619 (2012) 0 − 1 0 1 Very recently two independent experimental realisations of the Bose polaron: ❶ N. B. Jørgensen et al. , arXiv:1604.07883 ❷ Ming-Guang Hu et al ., arXiv:1605.00729

This Talk 1. Theory Good understanding, both at weak and strong coupling 2. Experiment First observation of long lived Bose polaron using RF spectroscopy

People Experiment Nils Lars Kristoffer T. Aarhus University Jan Arlt Jørgensen Wacker Skalmstang Theory Rasmus S. Jesper Meera Christensen Levinsen Parish Monash University

Theory Astrakharchik & Pitaevskii, Phys. Rev. A 70 , 013608 (2004) 
 Cucchietti & Timmermans , Phys. Rev. Lett. 96 , 210401 (2006) 
 Mean-field: Kalas & Blume , Phys. Rev. A 73 , 043608 (2006) Bruderer, Bao & Jaksch, Eu. Phys. Lett. 82 , 30004 (2008) Huang & Wan, Chin. Phys. Lett. 26 , 080302 (2009) 
 Tempere et al . , Phys. Rev. B 80 , 184504 (2009) 
 Fröhlich: Castels & Wouters , Phys. Rev. A 90 , 043602 (2014) Grust et al ., Sci. Rep. 5 , 12124 (2015) Vlietinck et al ., New J. Phys. 17 , 033023 (2015) Field theory: Rath & Schmidt, Phys. Rev. A 88 , 053632 (2013) Li & Das Sarma, Phys. Rev. A 90 , 013618 (2014) Variational: Schhaddilova, Schmidt, Grusdt & Demler, arXiv:1604.06469

Perturbation theory k a k + 1 X k a † X V B ( q ) a † k + q a † ✏ B H = k 0 − q a k 0 a k BEC 2 V k k , k 0 , q k c k + 1 X ✏ k c † X V ( q ) c † k + q c † + k 0 − q a k 0 c k a V k k , k 0 , q a B Impurity Impurity-BEC interaction BEC weakly interacting na B3 ≪ 1 ⇒ Bogoliubov theory Perturbation theory V(q): 1. order 2. order

Replace V ( q ) → T v = 2 π a/m r in a consistent way Diagrams like comes from expanding T v 1 − T v Π 11 ( p ) = T v + T 2 T ( p ) = v Π 11 ( p ) + . . . Self-energy in Σ ( p, ω ) = Σ 1 ( p, ω ) + Σ 2 ( p, ω ) + Σ 3 ( p, ω ) + . . . powers of a: (a) 28 third order diagrams 1 2 2 1 1 + = (b) with 2 + = 1 2 2 1 1 2 2 1 (c)

Energy Same structure as ξ + A ( α ) a 2 ξ 2 + B ( α ) a 3 E (0) = a ξ 3 ln( a ∗ / ξ ) Lee-Huang-Yang + Ω Wu-Hugenholz-Pines-Sawada √ √ Ω = 2 π n ξ /m r A (1) = 8 2 / 3 π B (1) = 2 / 3 − 3 / π a ∗ = max( a, a B ) α = m/m B a=a B :  � N = 4 π na 3 √ π ( na 3 ) 1 / 2 + 4(2 32 E √ 3) na 3 ln( na 3 ) 1 + 3 π − m Weakly interacting BEC  � N = 2 π na 15 √ π ( na 3 ) 1 / 2 + 8(4 128 E 3) na 3 ln( na 3 ) √ 1 + 3 π − m

Residue & Effective Mass Z − 1 = 1 + C ( α ) a 2 a B ξ + D ( α ) a 3 a B ξ 2 m = 1 + F ( α ) a 2 a B ξ + G ( α ) a 3 m ∗ a B ξ 2 p p C (1) = 2 2 / 3 π D (1) ⇡ 0 . 64 F (1) = 16 2 / 45 π G (1) ' 0 . 37 a 2 Condition for Z ≃ 1: a B ξ ⌧ 1 Breaks down for ideal BEC

Variational Theory Multichannel model h i X E k � † k � k + ✏ k c † d † ˆ ✏ d � � k + ⌫ 0 H = k c k + k d k k Bog. modes Impurity Molecule ⇣ ⌘ X X d † d † � � + g √ n 0 k c k + h.c. + g q c q − k b k + h.c. k k , q Introduces effective range r 0 Regularises 3-body problem

Variational wave function ! k + 1 α 0 c † X α k c † − k β † X α k 1 k 2 c † − k 1 − k 2 β † k 1 β † k 2 + γ 0 d † X γ k d † − k β † | ψ i = | BEC i 0 + 0 + k 2 k k 1 k 2 k BEC BEC BEC k 1 − k k − k 1 − k 2 k 2 a − Recovers Efimov spectrum for 1+2 bosons for n 0 → 0 a − ' 9000 r 0 Can always keep |a/r 0 | ≫ 1 
 Even when a - large

Efimov state 3. order pert. � Avoided crossing with Efimov state. - � Mean field Residue small close - �� to unitarity - �� 2 bog. approx. 1 bog. approx. (ladder) � - �� � a = a − � � ��� � � � - �� - � � � - �� - � � Efimov physics suppressed for a - ≫ n 0-1/3 n 1 / 3 a − = − 1 Efimov state “too large” 0

Spectral functions 1 /k n a = − 5 1 /k n a = − 1 6 6 100 TBM1 50 TBM2 0 − 0 . 45 − 0 . 4 − 0 . 35 − 0 . 3 − 0 . 25 Pert. Th. 4 4 E n I 0 ( ω ) TBM1 E n I 0 ( ω ) 300 TBM2 200 Pert. Th. 100 2 2 0 − 0 . 1 − 0 . 08 − 0 . 06 0 0 − 0 . 4 − 0 . 3 − 0 . 2 − 0 . 1 0 0 . 1 0 . 2 − 0 . 15 − 0 . 1 − 0 . 05 0 0 . 05 0 . 1 0 . 15 0 . 2 ω /E n ω /E n Many-body continuum Polaron peak BEC Weak coupling: Variational theory agrees with pert. theory − k k Strong coupling: Pert. theory breaks down. Many-body continuum significant

Theory - bottom lines • Analytical perturbation theory to 3.order in a/ ξ • Polaron well-defined for weak coupling • Strong coupling: Variational ansatz including 3- body Efimov correlations • Significant many-body continuum for strong coupling • Impurity atoms in BEC not the Fröhlich model

Experiment bottom lines Jan Arlt • First realisation of the Bose polaron ( See also JILA group ) • Well-defined polaron both for repulsive and attractive interaction • Many-body continuum dominates at strong coupling • Excellent agreement with theory • 3-body decay has no significant effects

Earlier experiments Impurity in thermal Spethman et al. Phys. Rev. Lett. 109 , 235301 (2012) bose gas: Zipkes et al ., Nature 464 , 388 (2010) Charged or fixed Schmid et al ., Phys. Rev. Lett. 105 , 133202 (2010) Balewski et al ., Nature 502 , 664 (2013) impurities in BEC: Scelle et al ., Phys. Rev. Lett. 111 , 070401 (2013) Impurities in lattice: Ospelkaus et al ., Phys. Rev. Lett. 96 , 180403 (2006) Magnons: Marti et al ., Phys. Rev. Lett. 113 , 155302 (2014)

Experimental procedure | 1 i = | F = 1 , m F = � 1 i ω RF BEC of 39 K in |1 ⟩ ω 0 ∆ | 2 i = | F = 1 , m F = 0 i RF flip ≤ 10% to |2 ⟩ } 3-body loss Wait for a while TOF Count # |1 ⟩ remaining as f n Remaining atoms of detuning Δ = ω 0 - ω RF Independent of wait time ⇒ k n a = − 0 . 84 loose 100% of |2 ⟩ atoms ⇒ lost |1 ⟩ atoms = 3 × created | 2 ⟩ atoms ∆ /E n

|1 ⟩ - |1 ⟩ :a B Scattering lengths 400 200 |1 ⟩ - |2 ⟩ resonance a (in units of a 0 ) B 0 = 113 . 83G 0 ∆ B = − 15 . 93G a bg = − 45 . 24 a 0 − 200 R ∗ ' 60 a 0 Lysebo and Veseth, PRA 81 , 032702 (2010) − 400 0 50 100 150 200 B (G) Experiment takes place here. a B ≃ 9a 0

Advantages of RF flipping out of BEC ❶ Perfect spatial overlap between impurities and BEC ❷ Selectively probe only k=0 polarons ❸ Simple theoretical interpretation ˙ N 2 = − 2 Ω 2 Im D ( ω ) X a † X a † D ( t � t 0 ) = � i θ ( t � t 0 ) h [ k 0 2 ( t 0 ) a k 0 1 ( t 0 )] i k 1 ( t ) a k 2 ( t ) , k k 0 Bogoliubov theory: D ( ω ) = n 0 G 2 ( k = 0 , ω ) RF probes k=0 impurity ˙ N 2 ∝ A ( k = 0 , ω ) = − 2Im G 2 ( k = 0 , ω ) spectral function: Contrast with Fermi gas or + + thermal Bose gas vertex corrections

Generic Physics E n = k 2 k n = (6 π 2 n ) 1 / 3 n 2 m

Experiment Theory 1.5 a b 1 1 0.5 n h ∆ / E 0 0.5 ¯ − 0.5 − 1 0 − 1.5 1 bog. − 4 − 2 0 2 4 − 4 − 2 0 2 4 1 / k a 1 / k a n n 1.5 c − 3.53 d − 0.62 e − 0.04 f 1.6 g 4.39 signal 1 0.5 0 − 1 0 1 − 1 0 1 − 2 0 2 − 2 0 2 − 1 0 1 h ∆ / E ¯ n ✮ Clear shift away from ω 0 ✮ Excellent agreement between experiment and 2 bog. theory (trap averaging important!) ✮ Well-defined polaron for weak coupling ✮ Many-body continuum dominates for strong coupling

Trap averaging & Fourier broadening E n I ( ω ) 10 1 /k n a = − 5 1 /k n a = − 2 5 ω /E n − 0 . 5 − 0 . 4 − 0 . 3 − 0 . 2 − 0 . 1 0 . 1 0 . 2 Fourier width =0.15 E n BEC

Energy Width 0.8 0.6 1.5 0.4 Pert. theory 0.2 1 n σ / En E / E − 4 − 2 0 2 4 ¯ 0 − 0.2 0.5 2 bog. approx. − 0.4 − 0.6 0 − 4 − 2 0 2 4 − 2 0 2 1 / k a n 1 / k a 1 bog. approx. n ✮ Remarkable agreement between experiment and theory (some problems at strong repulsion) ✮ Pert. theory explains data for weak coupling ⇒ well defined polaron Γ ∝ n 2 0 a 4 ✮ 3-body decay not needed weak coupling to explain width Γ ∝ E n unitarity Makotyn et al ., Nat. Phys. 10 , 116 (2014)

Conclusions ❶ Good theoretical understanding of Bose polaron both for weak and for strong coupling ❷ Experimental observation of Bose polaron for the first time R. S. Christensen, J. Levinsen & GMB, PRL 115 , 160401 (2015) J. Levinsen, M. M. Parish & GMB, PRL 115 , 125302 (2015) N. B. Jørgensen et al, arXiv:1604.07883