TOPOLOGICAL SUPERFLUIDS IN OPTICAL LATTICES

Transcription

1 TOPOLOGICAL SUPERFLUIDS IN OPTICAL LATTICES Pietro Massignan Quantum Optics Theory Institute of Photonic Sciences Barcelona QuaGATUA (Lewenstein) 1

2 in collaboration with Maciej Lewenstein Anna Kubasiak Anna Sanpera 2

3 Phase transitions Landau: most phases of matter may be classified by the symmetries they break translational (solids) rotational (magnets) gauge (superfluids) BUT: some materials possess distinguishable phases without breaking symmetries (QH and QSH effect) Topological phase transitions! 3

4 Topological properties : stretching, bending : cutting, joining 4

5 Topological properties : stretching, bending : cutting, joining Concern the whole system (non-local) Characterized by integer numbers Robust 4

6 A topological insulator normal insulator Hg-Te quantum well d Hg: Mercury Te: Telluride CdTe E1 H1 HgTe CdTe CdTe HgTe CdTe H1 E1 Phase transition at d=dcrit: normal-to-topological insulator very large resistance c RESISTANCE ( Ω ) MΩ 16 10kΩ independent of d, when d>dcrit h 12 2e 2 RESISTANCE (k Ω ) GATE VOLTAGE (V) GATE VOLTAGE (V) quanta of conductance Qi & Zhang, Physics Today

7 A topological insulator normal insulator Hg-Te quantum well d Hg: Mercury Te: Telluride CdTe E1 HgTe CdTe CdTe HgTe CdTe H1 d>dcrit: topological insulator H1 E1 very large resistance b c RESISTANCE ( Ω ) ENERGY (ev) WAVENUMBER (Å 1) WAVENUMBER (Å 1) MΩ RESISTANCE (k Ω ) GATE VOLTAGE (V) GATE VOLTAGE (V) ENERGY (ev) kΩ h 2e 2 edge states Hg-Te has strong spin-orbit coupling 2 quanta of conductance (independent of d, when d>dcrit) Qi & Zhang, Physics Today

8 interesting..., but where? exotic condensed matter systems (quantum wells, bismuth antimony alloys, Bi2Se3 crystals,...) ν=5/2 FQH state (Pfaffian) ultracold atoms? (talks by Sa de Melo, Le Hur, Morais-Smith, Eckardt, Hemmerich,...) 7

9 Outlook of the talk 2D p-wave fermionic SF 2D s-wave fermionic SF with n n and spin-orbit coupling 8

10 Why 2D? In 2D particles need not to be either bosons/fermions, but may have anyonic statistics ( anyons: any phase under exchange of two particles ) braiding In particular, the statistics can be non-abelian (the exchange of two particles must be described by a matrix) c a b b c a σ1σ2 (a b c) σ1σ2 (a b c) Non-Abelian anyons are the main ingredient for topological quantum computation Nayak, Simon, Stern, Freedman, and Das Sarma, RMP

11 Why fermions? Bosons are not particularly suited, as they condense in the lowest available energy state. EF On the contrary, fermions have to due to obey the Pauli principle. 10

12 Why fermions? Bosons are not particularly suited, as they condense in the lowest available energy state. EF On the contrary, fermions have to due to obey the Pauli principle. By changing the number of particles, we are able to investigate the interesting excitations, and the system becomes sensitive to the global (topological) properties of the band structure. 10

13 2D p-wave SF 11

14 12

15 Not yet observed.. try with ultracold atoms? 12

16 A stable p-wave SF? 3-body losses at a p-wave Feshbach resonance 13

17 A stable p-wave SF? 3-body losses at a p-wave Feshbach resonance Ultracold proposals: dissipation-induced stability in optical lattices (1,2) (i.e., how to get no losses from large losses) time-dependent, staggered lattices (3,4) RF dressing of 2D fermionic polar molecules leads to long-range interactions ( r -3 ) and high TC (5,6) super-exchange interactions in Bose-Fermi mixtures (7,8,9) 13 1:Han, Chan, Yi, Daley, Diehl, Zoller & Duan, PRL :Roncaglia, Rizzi & Cirac, PRL :Lim, Morais-Smith & Hemmerich, PRL :Lim, Lazarides, Hemmerich & Morais-Smith, EPL :Cooper & Shlyapnikov, PRL :Levinsen, Cooper & Shlyapnikov, PRA :Lewenstein, Santos, Baranov & Fehrmann, PRL :Dutta & Lewenstein, arxiv: & PRA :Massignan, Sanpera & Lewenstein, PRA 2010

18 Bose-Fermi mixture 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) 14

19 Bose-Fermi mixture UBF 0 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) 14

20 Bose-Fermi mixture UBF 0 UBF>0 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) 14

21 Bose-Fermi mixture UBF 0 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) UBF>0 UBF<0 composite fermions 14

22 Bose-Fermi mixture 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) Lewenstein, Santos, Baranov & Fehrmann, PRL

23 Bose-Fermi mixture UBF 0 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) Lewenstein, Santos, Baranov & Fehrmann, PRL

24 Bose-Fermi mixture UBF 0 UBF>0 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) composite fermions Lewenstein, Santos, Baranov & Fehrmann, PRL

25 Bose-Fermi mixture UBF 0 UBF>0 UBF<0 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) composite fermions Lewenstein, Santos, Baranov & Fehrmann, PRL

26 Bose-Fermi mixture UBF 0 UBF>0 UBF<0 Attractive 1) UBB>0 2) Strong coupling: tb, tf UBB, UBF (bosons in n=1 Mott state) composite fermions Lewenstein, Santos, Baranov & Fehrmann, PRL

27 Effective Fermi-Hubbard model super-exchange tunneling t (tbtf)/ubf H = t <i,j> c i c j U 2 <i,j> n i n j µ i nearest-neighbor interaction (super-exchange) n i U>0 16

28 Effective Fermi-Hubbard model super-exchange tunneling t (tbtf)/ubf H = t <i,j> c i c j U 2 <i,j> n i n j µ i nearest-neighbor interaction (super-exchange) n i U>0 BCS approach: introduce BdG operators γ n = i u n (i)c i + v n (i)c i Self-consistent p-wave gap equation ij = Uc i c j = U E n >0 u n(i)v n (j) tanh En 2k B T 16

29 Spectrum (homogeneous system) 2D chiral (px±ipy) SF: E(k) = ξ(k) 2 + h (k) 2 with ξ = 2t[cos(k x a) + cos(k y a)] µ and 2 h = 0 [sin 2 (k x a)+sin 2 (k y a)] Linear dispersion at the Dirac cones μ=-4t μ=0 μ=4t Two distinguishable topological phases for filling F<1/2 and F>1/2 17 Kou and Wen, PRB 2009

30 Spectrum with vortex Energy E 0 of the quasi-majorana mode Interaction strength U/t Strong pairing µ=-4t Weak pairing Δ0 t 10nK (super-exch.) Fermionic filling F Low-lying spectrum: En nω0 (n=0,1,2,...) The eigenstate with E0 Δ0 is a Majorana fermion. Particle-hole symmetry of the BdG eqs.:. Then, if {E n,ψ n } { E n,σ 1 ψ n} E 0 =0, u 0 = v 0 P.M., A. Sanpera & M. Lewenstein, PRA 2010 γ 0 = γ 0 18

31 2D s-wave SF with n n and spin-orbit coupling 19

32 Synthetic gauge fields for neutral atoms Theory: Jaksch&Zoller, NJP 2003 Osterloh et al., PRL 2005 Gerbier&Dalibard, NJP 2010 Bermudez et al., PRL 2010 (TRI Top. Ins.) adiabatic Raman passage adiabatic control of superpositions of degenerate dark states spatially varying Raman coupling Raman-induced transitions to auxiliary states in optical lattices REVIEW: Artificial gauge potentials for neutral atoms J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, RMP

33 a field moving fast.. NIST: Synthetic magnetic fields for ultracold neutral atoms, Nature (2009) A synthetic electric force acting on neutral atoms, Nature Phys. (2011) Spin-orbit-coupled Bose-Einstein condensates, Nature (2011) Observation of a superfluid Hall effect, PNAS (2012) Peierls Substitution in an Engineered Lattice Potential, PRL (2012) (theory) Chern numbers hiding in time-of-flight images, PRA (2011) ICFO & Hamburg & Dresden: Tunable Gauge Potential for Neutral Spinless Particles in Driven Optical Lattices, PRL (2012) (method independent of the internal structure of the atoms!!) Munich: Experimental realization of strong effective magnetic fields in an optical lattice, PRL (2011)

34 PRL webpage in Aug Shanxi Univ. & MIT 22

35 Synthetic gauge fields for neutral atoms,q=kx-q/2>,q=kx+q/2> spin-orbit gap increasing intensity of Raman lasers spin flip momentum kick, i.e., spin-orbit coupling 23

36 fermions in synthetic gauge fields External non-abelian gauge fields yield a fictitious spin-orbit coupling c i =(c i,c i ) complex hoppings = Peierl s phases H 0 = t i c i+ˆx eiσ yα c i + c i+ŷ eiσ xβ c i +h.c. 24

37 Add attractive interactions BCS superfluid Sato, Takahashi & Fujimoto, PRL 2009 Sau Jay, Lutchyn, Tewari and Das Sarma, PRL 2010 strong imbalance topological states Time-reversal and spin-rotation invariances are destroyed by the Zeeman and SO terms as a consequence our BCS Hamiltonian belongs to the most general symmetry class D (Altland&Zirnbauer, PRB 1997) its topological phases are indexed in terms of an integer number 25

38 Spectrum on a cylinder (open b.c. along x) , 0.6, 0.6, Ch=1 k y ky ky ky Imbalance h=μ -μ k y k y , Ch= ky k y 0.2 ky k y edge Gap closing: h k0 = 2 + k 2 0 states 26

39 Topological phases h /t /t h=μ -μ Chern numbers easy to calculate! (see J. Bellissard, condmat/ ) Gap closing at (k 0, h) : Δ=t α=π/4 μ=-0.5t[ cos(α) + cos(β) ] β H eff (k,h)=e(k,h)+σ f(k,h) CN( h) = sign{det[j f (k 0, h)]}. A. Kubasiak, P.M. & M. Lewenstein, EPL

40 Spin imbalance vs. pair breaking 3.0 without SO coupling: analytic CC limit ( hcc = Δ0/ 2 ) h 0 Π, h Π0 4 Ch= ht with SO coupling: 1.5 self-consistent calculation of Δ from the BCS gap equation ht h CC h 00 Ch= α=β=π/4 0.5 μ=-3t A. Kubasiak, 4 P.M. 5 & M. 6 Lewenstein, 7 EPL Vt 28 Vt 0 0 Ch=0

41 Conclusions Ultracold SF fermions possess non-trivial topological phases Optical lattices stabilize p-wave SF FQH P. M., A. Sanpera & M. Lewenstein, PRA(R) 2010 fermions in non-abelian gauge fields A. Kubasiak, P. M. & M. Lewenstein, EPL