Quantum Fluctuations of Polaronic Cloud in a BEC of ultracold atoms - PowerPoint PPT Presentation

Quantum Fluctuations of Polaronic Cloud in a BEC of ultracold atoms Yulia Shchadilova Russian Quantum Center In collaboration with: Fabian Grusdt (University of Kaiserslautern), Eugene Demler (Harvard University), Alexey Rubtsov (RQC, Moscow State University) ~ q mail to: yes@rqc.ru Tuesday, September 16, 14

Plan of the talk I. Introduction II. Model and observables III. Mean field approach IV. Gaussian approach V. Results VI. Summary and Outlook ~ q Tuesday, September 16, 14

a fermion interacting Polaron with a scalar boson field impurity + Lev Landau self-induced polarization 1933 polar semiconductors ionic crystal doped quantum magnets e − QFT high-Tc superconductors ultra-cold atoms Tuesday, September 16, 14

Ultracold atoms Two-component mixtures α = a 2 IB - Adjustable interaction parameter a BB ξ - High mobility P RF Tuesday, September 16, 14

Ultracold atoms. Observables RF spectroscopy: Oscillations in a trap: I ( ω ) Z ω 0 E 0 - polaron mass - polaron binding energy Tuesday, September 16, 14

Frölich Model H = P 2 ⇣ ⌘ e i k · R + X X a † a † ˆ V k a k + ˆ a k 2 M + ˆ ω k ˆ k ˆ − k k k Polaronic frame Impurity problem strongly interacting bosons fermion in effective media P Tuesday, September 16, 14

Polaron frame T. D. Lee, F. E. Low, and D. Pines, P Review 90, 297 (1953). U − 1 H → ˆ ˆ LLP H ˆ U LLP U LLP = e i ~ k ~ a † R P ˆ k ˆ k ˆ a k ! 2 1 ⇣ ⌘ X ~ b † X b † X b † ~ ˆ k ˆ k ˆ ˆ k + ˆ ! k ˆ k ˆ H = P − b k V k b − k b k + + 2 M k k k 12 10 0.8 8 0.6 V k ê V • 6 Ω k 0.4 4 0.2 2 0.0 0 0 1 2 3 4 0 1 2 3 4 k k Tuesday, September 16, 14

Limits ! 2 1 ⇣ ⌘ X ~ b † X b † X b † ~ ˆ k ˆ k ˆ ˆ k + ˆ ! k ˆ k ˆ H = P − b k V k b − k b k + + 2 M k k k nonlinearity polarization 1) Heavy impurity, M → ∞ 2) Weak interactions, V k → 0 ˆ b k → ˆ b k − V k / ω k n k = 0 ˆ Tuesday, September 16, 14

Polarization of phonon modes i (ˆ b † − ˆ Basis of coherent states: b ) | MF i = ˆ D ( β ) | 0 i e β k ( ˆ b † k ) b k − ˆ ˆ Y D ( β ) = α k b † + ˆ b † � β ) | 0 i D E ˆ H (ˆ b, ˆ MF = h 0 | H (ˆ b � β , ˆ b b † ) A. Shashi, et al. PRA 89, 053617 (2014) Tuesday, September 16, 14

Polarization of phonon modes i (ˆ b † − ˆ Basis of coherent states: b ) | MF i = ˆ D ( β ) | 0 i e β k ( ˆ b † k ) b k − ˆ ˆ Y D ( β ) = α k b † + ˆ b † � β ) | 0 i D E ˆ H (ˆ b, ˆ MF = h 0 | H (ˆ b � β , ˆ b b † ) Minimization gives the mean field self-consistent condition: Ω k V k � k = − ⇣ ⌘ ~ P − ~ ~ ! k + k 2 k P ph 2 M − M ~ k | � k | 2 ~ X P ph = k k A. Shashi, et al. PRA 89, 053617 (2014) Tuesday, September 16, 14

Phonon Zero-Point Fluctuations i (ˆ b † − ˆ b ) Basis of squeezed coherent states: Q | GSC i = ˆ D ( β ) ˆ S ( Q ) | 0 i α b † b † kk 0 Q kk 0 ˆ k ˆ 1 ˆ P k 0 − H.c. S ( Q ) = e 2 b † + ˆ ˆ b Tuesday, September 16, 14

Phonon Zero-Point Fluctuations i (ˆ b † − ˆ b ) Basis of squeezed coherent states: Q | GSC i = ˆ D ( β ) ˆ S ( Q ) | 0 i α b † b † kk 0 Q kk 0 ˆ k ˆ 1 ˆ P k 0 − H.c. S ( Q ) = e 2 b † + ˆ ˆ b Polarization + Bogoliubov transformation S † ( Q ) ˆ ˆ D † ( β )ˆ b k ˆ D ( β ) ˆ [cosh Q ] kk 0 ˆ [sinh Q ] kk 0 ˆ b † X X b k 0 + S ( Q ) = β k + k 0 k 0 k 0 Tuesday, September 16, 14

Phonon Zero-Point Fluctuations i (ˆ b † − ˆ b ) Basis of squeezed coherent states: Q | GSC i = ˆ D ( β ) ˆ S ( Q ) | 0 i α b † b † kk 0 Q kk 0 ˆ k ˆ 1 ˆ P k 0 − H.c. S ( Q ) = e 2 b † + ˆ ˆ b Polarization + Bogoliubov transformation S † ( Q ) ˆ ˆ D † ( β )ˆ b k ˆ D ( β ) ˆ [cosh Q ] kk 0 ˆ [sinh Q ] kk 0 ˆ b † X X b k 0 + S ( Q ) = β k + k 0 k 0 k 0 Gaussian statistics: D E ˆ b · ˆ b = β · β + cosh Q sinh Q D E D E = β · β + sinh 2 Q ˆ b † · ˆ ˆ b = β b Tuesday, September 16, 14

Phonon Zero-Point Fluctuations ∂ h H i ∂ h H i ∂ Q kk 0 = 0 = 0 ∂α k An approximate ground state solution: - use Taylor series for averages Q 2 - considering only the terms up to in this average Tuesday, September 16, 14

Phonon Zero-Point Fluctuations ∂ h H i ∂ h H i ∂ Q kk 0 = 0 = 0 ∂α k An approximate ground state solution: - use Taylor series for averages Q 2 - considering only the terms up to in this average ✓ ◆ ✓ k 0 q ◆ Ω k + kk 0 Q kk 0 + kk 0 M β q β k 0 Q kq + kq X M β k β k 0 + M + Ω k 0 M β q β k Q qk 0 = 0 q V k � k = − ! k + k ν M − 1 ⇣ ⌘ ~ νλ k λ P − ~ ~ k P ph − 2 M Numerical solution is without further approximations | � k | 2 � kk 0 + Q 2 ⇣ ⌘ X ~ ~ P ph = k kk 0 kk 0 Tuesday, September 16, 14

Phonon Zero-Point Fluctuations Solution without approximations: ✓ ◆ ✓ k 0 q ◆ Ω k + kk 0 Q kk 0 + kk 0 M β q β k 0 Q kq + kq X M β k β k 0 + M + Ω k 0 M β q β k Q qk 0 = 0 q Denote: Q kk 0 = − 1 α k α k 0 η k,k 0 ~ k ~ M k 0 Ω k + 2 M + Ω k 0 Rewrite: ↵ 2 ↵ 2 Z Z ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ q q ~ k · ~ q · ~ ~ d 3 q d 3 q ⌘ ( k, k 0 ) = ⌘ ( q, k 0 ) ⌘ ( k, q ) ~ k · ~ k 0 k 0 q − − M Ω k,q M Ω q,k 0 Introduce: ↵ 2 Z q ~ d 3 q F ( k ) = ⌘ k,q ~ q M Ω k,q An equation for F to be solved iteratively: α 2 α 2 Z Z X X d 3 k 0 k 0 d 3 k 0 k 0 M Ω k,k 0 k 0 µ k 0 M Ω k,k 0 F µ ( k 0 ) k 0 F λ ( k ) = ( k µ − F µ ( k )) k µ λ − λ µ µ Tuesday, September 16, 14

Polaron ground state energy 23 Na 6 Li M/m B ≈ 0 . 26 0.6 diagMC  MF Gaussian 0.4 RG     E p  0.2     0.0  0.0 0.2 0.4 0.6 0.8 1.0 Α diagMC: Vlietinck et. al, arXiv:1406.6506 (2014) Gaussian (this work): Y.E.S., Grusdt, Demler, Rubtsov, in preparation (2014) RG: Grusdt, Y.E.S., Rubtsov, Abanin, Demler., in preparation (2014) Tuesday, September 16, 14

Total energy I ( ω ) Z E = g IB n 0 + h H i ω M/m B =0.26316, q=0, � 0 =2000/ � , n 0 = � − 3 0 E 0 10 8 E 0 [c/ � ] 6 diagMC, Vlietinck et al. 4 RG 23 Na 6 Li variational MF 2 Feynman, Vlietinck et al. M/m B ≈ 0 . 26 0 0 0.2 0.4 0.6 0.8 1 � Tuesday, September 16, 14

Polaron mass 23 Na 6 Li ≡ M δ v p M δ P = 1 − P ph M/m B ≈ 0 . 26 P . M p 4.0 MF 3.5 Gaussian RG 3.0 2d order PT M p ê M 2.5 2.0 1.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Α Tuesday, September 16, 14

Coherence factor 23 Na 6 Li D E ˆ k ˆ k 0 ˆ b k ˆ b † b † b k 0 M/m B ≈ 0 . 26 g (2) ( k, k 0 ) = n k n k 0 α = 1 Bunching: 20 q=p g (2) > 1 q= 0 15 MF g 2 H k,kcos q L Coherent: 10 g (2) = 1 Antibunching 5 (squeezing): 0 g (2) < 1 0 2 4 6 8 10 k ê x Tuesday, September 16, 14

Light/heavy impurities ~ q M/m B 23 Na 6 Li 0 . 26 50 87 Rb 40 K 0 . 46 0 6 Li 2 6 Li 0 . 5 − 50 7 Li 6 Li 0 . 86 − 100 0 10 20 30 40 87 Rb 133 Cs 1 . 53 23 Na 40 K 1 . 74 87 Rb 40 K 2 . 12 Tuesday, September 16, 14

Summary and outlook Quantum fluctuations are significant in case of light or/and strongly interacting impurities. Signatures of entanglement can be captured experimentally. - Subsonic -- supersonic transition - Non-homogeneous BEC ~ - Anisotropic interactions of dipolar gases q - Real-time dynamics Tuesday, September 16, 14