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Quantum simulation of 2D topological physics in a 1D array of optical cavities Xi-Wang Luo (罗希望) Work with Zheng-Wei Zhou (周正威) & Xingxiang Zhou (周幸祥) Key lab of Quantum Information, USTC 冷原子物理博士生论坛 长春 Outline • Introduction – Orbital angular momentum of photon, degenerate cavity • 1D simulator – Cavity chain, 2D gauge field • 2D physics – QHE, edge state, topological phase transition • Summary Orbital angular momentum (OAM) Helical phase Wave front Spatial light modulator e il Laguerre-Gaussian modes p: radial mode index Vertical field : e ikz Quantum number l : OAM Unlimited Science 340, 28 (2013) Our idea: Quantum simulation. Gauge field & Topological physics How? Science 338, 2 (2012) Degenerate cavity Degenerate condition: Unit ray matrix E1 ( x1 , y1 ) G( x1, y1; x0 , y0 ) E0 ( x0 , y0 ) G depends on : ray matrix [A,B;C,D] Delta function : Unit ray matrix Resonance condition: kL 2n Optical design A single cavity 1D cavity array (a) ) )+1 !" ! "$% &' % &' ( #" &' % &' ( #"$% (b) )+2 ! "$( &' % &' ( #"$( •Bloch theory •Periodic & linear iK y a j ,l 1 e a j ,l ; a j 1,l eiK x a j ,l •Transfer matrix M BS i r it r it r i r •Tight binding approach j ,l ; j 'l ' dr ei 2x 0 (r ) 0 (r R j ',l ' ) E *j ,l E j ',l ' 2 Effective optical circuit Dispersion relation 0 | r |2 0 2 cos(K x -2x )+cos(K y -2 y ) , 2 Hamiltonian Abelian gauge field Ax , y 2x , y Gauge potential (field) y j0 x l0 •Non-Abelian gauge field •Polarization--spin •Wave plate e iAx , y Ax , y x , y 1, 2 2D physics Butterfly Edge state 0 E Quantum spin Hall effect Quantum Hall effect e2 xy C h Probing scheme Driving-damping dynamic Input-output diag{ 1, 2 ,, n ,} Photonic transmission System spectrum & edge state Butterfly spectrum Ttot j ',l ' 2 | T j ,l 0 | Cylindrical boundary No backscattering j ',l ', j Unidirectional Abelian 2.2 1 2.2 Edge state transport OAM distribution Connection ? Chern number OAM displacement le j ',l ' 2 l ' | T j ,0 | jedge j ',l ' Cylindrical boundary Quantum number k y In the gap—edge state Chern number le sgn( vm ) Ctot m Error bar: , , 0 1 / 20 Non-Abelian Abelian 0 1 / 6 Topological quantum phase transition Band structure 0 0,0.075,0.125 0 1.5 Topological quantum phase transition Polarized edge state Topological insulator 0 0 1.6 Normal insulator 0 0.125 Measurement of Chern number Chern number C 1 dk dk A (k x , k y ) x y k z 2 i A u k u u u0 ,..., uq1 umq q (kx , k y ) umq (kx , k y ) Gauge transformation e if ( k x , k y ) u C is invariant Global Stokes theorem ? 1 C dk A( k x , k y ) 2i C Always Zero Brillouin zone : no boundary However,the phase of Bloch wave is not well defined globally over the Brillouin zone – nontrivial topology. No global Stokes theorem! Measurement of Chern number e if ( k x , k y ) u 0 1 / 6 1 Regime B1, u0 is real. 1 Regime B2, u3 is real. Phase mismatch at edge: u i ( kx ,k y ) B1 e u B2 Separate Stokes therom u13 0 u10 0 Measurement of Chern number j , ql l T (k x , k y , lq ) F T0,0 q T (k x , k y ) i k x , k y E E i / 2 E 0,0 u T (k x , k y , 0),..., T (k x , k y , q 1) | T0,0j ,l |2 arg T (k x , k y ,3) arg T (k x , k y , 0) summary • 2D physics in 1D simulator – OAM • Connection between transmission coefficient and topological invariant – Chern number • Topological phase transition – QSH More detail : Nature Communications 6, 7704 (2015). Thanks for your attention ! 谢谢!