Cold atoms in 2D optical lattices under staggered rotation - PowerPoint PPT Presentation

Cold atoms in 2D optical lattices under staggered rotation Cristiane MORAIS SMITH Institute for Theoretical Physics, Utrecht University, The Netherlands ��� ��� INSTANS 2008 – p.1/28

Collaborators Lih-King Lim and Andreas Hemmerich INSTANS 2008 – p.2/28

Outline Low-D systems: observation of quantum effects 2DES in a : several interesting quantum phases �✂✁ electron-liquid phases: Laughlin, Moore-Read, Read-Rezayi electron-solid phases: Wigner crystals, bubbles, stripes nematic phases, BEC of excitons in bilayers, etc... INSTANS 2008 – p.3/28

2D Systems: cond-mat Lattice Uniform graphene GaAs, Si-MOSFETs B I + + + + + + 2D electron gas _ _ _ _ _ _ U U L H INSTANS 2008 – p.4/28

2D Systems: cold atoms Lattice Uniform rotating BECs in rotating BECs optical lattices Laser BEC Gaspacho INSTANS 2008 – p.5/28

Cold atoms in optical lattices Sorensen, Demler, Lukin, PRL 94, 086803 (2005) quadrupolar potential + tunneling for bosons FQHE Laughlin state: 95% overlap larger gap than in harmonically trapped BECs because in- teraction energy is larger INSTANS 2008 – p.6/28

✠ ☞ ✑ ✌ ✍ ✏ ✝ ✌ ✓ ☛ ✝ ☛ ✒ ✡ ✠ ✒ ✆ ☎ ✄ ✝ ✄ ✔ ✕ ✝ Cold atoms in optical lattices 3D: M. Greiner et al., Nature 419, 51 (2002) 2D: Phillips group, PRL 100, 120404 (2008) Superfluid-Mott insulator transition Theoretical description: Bose-Hubbard model �✂✁ �✎✍ ✝✟✞ INSTANS 2008 – p.7/28

Cold atoms in optical lattices simulators of cond-mat systems - full control of lattice parameters - load with bosons or fermions - control of interactions (Feshbach resonance) - no disorder generate NEW situations - alternating magnetic fields Haldane 1988: graphene with zero net magnetic field per plaquette QHE: no uniform B, but break time-reversal symmetry INSTANS 2008 – p.8/28

Cold atoms in optical lattices A. Hemmerich and C.M.S., PRL 99, 113002 (2007) Bosons and fermions Staggered rotational field Novel phases INSTANS 2008 – p.9/28

✒ ✒ ✣ ✌ ✕ ✔ ✧ ✓ ✚ ✥ ✣ ✁ ✚ ✕ ✥ ✖ ✆ ✔ ✓ ✓ ✓ ✧ ✢ ✒ ✜ ✒ ✔ ✤ ✚ ✙ ✩ ✓ ✕ ✣ ✄ ✕ ✔ ✧ ✓ ✣ ✣ ✥ ✬ ✓ ✫ ✪ ✩ ✁ ✕ ✖ ✆ ✤ ★ ✕ ✖ ✧ ✏ ✞ ✖ ✎ ✝ ✡ ✒ ✕ ✖ ✆ ✤ ✓ ✒ ✑ ✏ ✍ ✣ ✥ ✝ ✡ ✁ ✠ ✟ ✕ ✞ ✓ ✧ ✖ ✕ ✗ ✞ ✘ ✓ ☛ ✗ ✝ ✛ ✡ ✒ ✕ ✖ ✆ ✔ ✓ ✒ ✙ ✑ ☞ ✝ ✡ ✚ ✠ ✟ ✕ ✞ ✕ ✌ ✏ ✧ Staggered Rotational Field A. Hemmerich and C.M.S., PRL 99, 113002 (2007) How to realize it experimentally? Linearly polarized bichromatic light-field ☛✌☞ ☛✌✘ ✓ ☎✄✝✆ ✓ ✕✔ �✂✁ ✓ ☎✄✝✆ �✂✚ We assume , ✤✦✥ ✤✦✥ ✤✦✥ ✓ ✕✔ INSTANS 2008 – p.10/28

✔ ✆ ✓ ✆ � ★ ✚ ✒ ✕ ✖ ✆ ✆ ✓ ✓ ✒ ✚ ✕ ✚ ✠ ✌ ✔ ✖ ✠ ✓ ✣ ✫ ✞ ✚ ✒ ✕ ✖ ✆ ✔ ✓ ✆ ✒ ✚ ✠ ✁ ✠ ✝ ✁ ✕ ✞ ✁ ✓ ✝ ✞ ✁ ✕ ✞ ✜ ✂ ✁ ✕ ✞ ✁ ✓ ✟ ✕ ✞ ✆ ✖ ✆ ✕ � ✄ ✔ ✁ ✕ ✕ ✖ ✆ ✔ ✓ ✄ � ✖ ✞ ✆ ✆ ✖ ✆ ✔ ✓ ✕ ✌ ✕ ✖ ✓ Staggered Rotational Field A. Hemmerich and C.M.S., PRL 99, 113002 (2007) ✓ ✕✔ ✓ ☎✄✝✆ ✓ ☎✄✝✆ �☎✄ �☎✆ Stationary term Time-dependent term ✓ ✕✔ ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� INSTANS 2008 – p.10/28

Optical Setup cold # atoms AOM M Laser BS PZT Two nested Michelson interferometers PZT: piezoelectric transducer M: mirror BS: beam splitter AOM: acousto-optic frequency shifter INSTANS 2008 – p.11/28

✂ ✒ ☎ ✁ ✕ ✔ ✄ ✁ ✒ � ✓ ✁ � ✓ ✍ ✌ ✄ ✕ ✁ ✑ ✝ ✔ ✌ ✁ ✒ ☎ ✞ ✕ ✤ ✄ ✂ ✁ ✎ ✓ ✕ ✞ ✜ ✓ ✥ ✣ ✕ ✁ ✔ ✞ ✕ ✕ ✔ ✄ ✓ ✌ ☎ ✁ � ✞ ✍ ✞ ✛ ✁ ✒ ✞ ✄ ✂ ✁ ✄ ✁ ✕ ✓ ☎ � � ✓ ✔ ✚ ✫ ✞ ✣ ✓ ✜ ✞ ✕ ✞ ☎ ✓ ✌ ✁ ✎ ✍ ✌ ✄ ✂ ✁ ✌ ☞ ✝ ☛ ✓ ✏ ✁ ☛ � ✁ ☞ ☛ � ✟ ✕ ✞ ✞ Time-Dependent Bose-Hubbard Model H.c. ✍ ✡✠ A B A ☎✝✆ B A B ☎✑✏ A where A B ✒✑✓ anisotropic time-varying hopping λ time-varying energy offset INSTANS 2008 – p.12/28

INSTANS 2008 – p.13/28 ✁ ☛ � ☛ ☞ ✁ � ☛ ☛ ✁ ✌ ✌ ✔ ✝ ✑ ✁ ✂ ✄ ✏ ✛ ✍ ✞ � � ✗ ✄ ✁ ✂ ✄ ✁ ☛ ✛ ✞ ✒ ✏ ✒ ✘ ✝ ✙ ✌ � ✞ ✒ ✛ ✁ ☎ ✁ ✝ ✏ ✚ ✚ ✩ ✓ ✔ ✁ ✔ ✚ ✒ ✓ ✜ ✥ ✬ ✒ ✕ ✁ ✓ � ✒ ✁ ✄ ✔ ✒ ✁ ✏ ✒ ✁ ☎ ✚ ✌ ✚ ✚ ✖ ✕ ✓ ✆ ✓ � ✂ ✕ ✁ ✡ ✛ ✝ � ✞ ✌ � ✑ ✓ � ✂ � ✑ ✜ ✔ ✓ � � ✓ ✞ ✕ ✁ � � ✞ ☎ ✂ ✕ � ✂ ✟ ✝ ✒ ✄ ✁ ✕ ✌ ✕ ✓ ✄ ✓ ✕ ✞ ✑ ✓ ✞ ✑ ✖ � � ✓ ✞ ✕ � ✒ ✚ ✄ ✞ ✑ ✒ ✓ ✌ ✕ ✓ � � ✓ ✕ ✞ with ✍✔✎ H.c. ✌ ✁� ✍✏✎ ✍ ✡✠ Effective Hamiltonian ✟✡✠☞☛ Hamiltonian is periodic: ✍✔✎ Effective Hamiltonian Dyson Series : ☎✝✆ eff .

INSTANS 2008 – p.14/28 ✛ ✚ ✎ ✁ ✕ ✍ ✚ ✝ ✓ ✣ ✫ ✞ ✧ ✍ ✣ ✫ ✞ ✍ ✧ ✣ ✫ ✞ ✝ ✌ ✛ ✧ ✁ ✚ ✁ ✝ ☎ ✕ ✙ ✧ ✄ ✘ ✧ ✓ ✛ ✓ ✧ ✝ ☎ ✕ ✙ ✧ ✌ ✘ ✧ ✧ ✣ ✆ ✆ ✫ ✞ ✆ ✞ ✆ ✟ ✆ � ✞ ✁ � ✆ ✝ ✆ ☎ ✄ � � � ☛ ☞ ✒ ✌ ✍ ✧ ✚ ✣ ✫ ✞ ✒ ✌ ✏ ✝ ✁ ✕ ✁ ✒ ☞ ✎ ✒ ✕ ✂ �✡✠ �✡✠ Single-particle spectrum space in Write

✖ � ✁ ✔ ✕ ✒ ☎ ✕ ☛ ☞ ✗ ✘ ✂ ✙ ✖ ✔ ✔ ✕ ✒ ✖ ✣ ✆ ✚ ✔ ✕ ✒ ✖ ✖ � ✟ ✕ � ✆ ✁ � � ✁ � ✂ ✄ ☎ ✆ ✂ ✝ ✞ � ✁ ✟ ✠✡ ✄ ☛ ☞ ✌ ✝ ✞ � ✁ ✟ � ✠ ☎ ✍ ✎ ✏ � Bosons: Mean Field Theory ✑✓✒ eff Mott regime Hubbard-Stratonovich field to decouple the hopping term Integrate out the boson fields Effective action (quadratic order in , ) ✛✢✜ ✛✢✜ INSTANS 2008 – p.15/28

✝ ✂ ✌ ✂ ✖ ☞ ✂ ✄ ✕ ✠ ✙ ☎ ✆ ✆ ✎ ✞ ✜ ✞✟ ✄ ✆ ✝ ☎ ✄ ✌ ✆ ☎ ✡ ✍ ✕ ✂ ✁ ✌ ✆ ✄ ✌ ✆ ✠ ✆ ☞ ☛ ✌ ✆ ✆ ☞ ☛ ✠ ☞ ✌ ✆ ✖ ✆ ✞ ✆ ✜ ✛ ✣ ✆ ✜ ✛ ✚ ✜ ✁ ✛ ✂ ☎ ✠ ✟ ✁ ✞ ✂ ✁ � ✄ ✆ ☎ ✛ ✁ ✆ ✜ ✛ ✣ ✆ ✜ ✛ ✚ ✁ ✛ ✖ ✆ ✁ ✄ ✕ ✆ ✁ ✄ ✆ ✄ ✠ Mean Field Theory - Effective Action Real frequencies ( ), Quasi-particle (hole) energy dispersion ✠☛✡ where : energy for creating a quasiparticle-quasihole pair. INSTANS 2008 – p.16/28

✡ ✙ ☎ ✠ ✂ ✠ Phase Diagram L.-K. Lim, C.M.S., and A. Hemmerich, PRL 100, 130402 (2008) ��� �� � ( )-plane: dashed white line INSTANS 2008 – p.17/28

✡ ✝ ☎ ✄ ✟ ✍ ✂ ✂ ☎ � ✖ ✖ ✟ ✂ ✝ ✕ ☎ ✄ ✁ ✍ ✡ ✂ ✁ � ✕ Phase Diagram Bosons: BEC at lowest single-particle state : min at GS is uniform SF : min at GS is finite SF θ = Ar ct an[ W / J] π / 2 st agger ed vor t ex r e d r super f l ui d o d n 2 π / 4 M ot tI nsul at or 1stor der 2nd or uni f or m der super f l ui d J 2 +W 2 ) 1/ 2 2 U / 4( 6 5. 83 INSTANS 2008 – p.18/28

� ✝ ✌ ✝ ✂ ☎ ✁ ✂ ✡ ✍ � ✁ ☎ ✁ ✄ ☎ � ✖ ✄ ✁ ✡ ✠ ✡ ✂ ☎ ✁ ☎ ☎ ✁ ✠ ✍ ✡ ✂ ✎ ✍ ✂ ✡ ✟✠ ✁ ✞ ✝ ✆ ✝ ☎ ✕ ✂ ✁ ✟ � ✌ ✍ ✡ ✁ ✞ ✁ � ✖ ✁ ✕ ✂ ✡ ✞ ✖ ✝ ✆ ✄ ✄ ✞ � ✄ Superfluid Phases Variational mean-field ansatz for the ground state: ✕ ☎✄ ☛✌☞ : : Order parameter changes discontinuously by at Finite momentum SF: analogies with Abrikosov lattice and “FFLO states” INSTANS 2008 – p.19/28

✓ ✒ ✕ ★ ✓ ✕ ✂ ✂ ✕ ✕ ✂ ✓ ✍ ★ ✚ ✕ ✓ ✂ ✓ ✒ ✁ ✕ ✄ ✂ ✓ ✁ ✕ ✂ ✓ ☞ ✁ � ✂ Experimental Detection Momentum distribution: ★ ✆☎ ��� ��� �� � � � �� � � �� � � � �� � � �� � � � �� � � (a) Uniform SF (b) Staggered vortex SF : Fourier transform of Wannier function : structure factor (B: Bravais lattice, P: plaquette) INSTANS 2008 – p.20/28

� ☎ ✂ ✡ ✍ ✂ Fermions in optical lattices At half-filling: anisotropic Dirac cones Graphene under uniaxial pressure At : staggered- flux phase (HTSC) INSTANS 2008 – p.21/28