Artificial Gauge Fields for Neutral Atoms

Transcription

1 Artificial Gauge Fields for Neutral Atoms Simon Ristok University of Stuttgart 07/16/2013, Hauptseminar Physik der kalten Gase 1 / 29

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4 What are artificial gauge fields? Neutral cold atoms as toy model for various physical problems e.g. simulating electrons in a solid Condensed matter physics: easy application of magnetic fields Not all regimes are accessible 4 / 29

5 What are artificial gauge fields? Cold atoms can be controled with different methods Available tools: lasers, magnetic and electric fields, optical lattices,... Problem: atoms are neutral Real magnetic fields don t influence them How to simulate effects on charged particles? Artificial gauge fields: affect neutral atoms like a real magnetic field affects charged particles! 5 / 29

6 Example: Quantum Hall Effect 2D-System of electrons, e.g. AlGaAs-GaAs heterojunction Strong perpendicular magnetic field resistivity ρ xy is quantised ρ xy = 1 h n, n = 1,2,3,4,... e 2 Figure : [4] 6 / 29

7 Example: Quantum Hall Effect (red) (c) (green) (a) (b) Figure : [2] Figure : [1] Integer Quantum Hall Effect: explanation with Landau levels Fractional Quantum Hall Effect: more complicated Investigate this regime more closely with cold atoms 7 / 29

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9 From Rotation to Artificial Gauge Potential Charged particle in electromagnetic potentials: H = ( p q ) 2 A( r,t) 2m Rotating neutral atoms in harmonic trap: (2) can be transformed to (1) + qv( r,t) (1) H = p2 2m mω2 r 2 Ω L (2) What are q, A( r,t), and V( r,t)? 9 / 29

10 Transformed Hamiltonian H = p2 2m mω2 r 2 Ω L = Resulting parameters: q = 1 V( r) = 1 2 m ( ω 2 Ω 2) r 2 y A( r) = Ωm x 0 ( p ) 2 A( r) 2m x m ( ω 2 Ω 2) r 2 (3) z Ω = Ω ê z y Artificial magnetic field: B = A = 2Ωmê z 10 / 29

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12 Overview and principle BEC of 87 Rb, F = 1 Use lasers and magnetic field to change eigenstates Use dispersion relation for k x as effective Hamiltonian: E(k x ) h2 (k x k min ) 2 2m Artificial gauge potential A x = h k min Figure : [8] 12 / 29

13 Zeeman splitting and Raman coupling Constant magnetic field B 0 along y-axis splitting of m F states ω Z = g µ B B 0 / h Raman lasers along x-axis transitions between m F states Raman detuning δ = ω L ω Z hk x = ±2 hk L Coupling of: m F = 1,k x 2k L m F = 0,k x m F = 1,k x + 2k L Figure : [8] New eigenstates with interesting dispersion relations 13 / 29

14 Hamiltonian Full Hamiltonian: H = H 1 (k x ) + [ h 2 ( k 2 y + k 2 z) 2m + V( r) ] 1 (4) with h 2m (k x + 2k L ) 2 δ Ω R /2 0 H 1 (k x ) = h h Ω R /2 2m k2 x ε Ω R /2 0 Ω R /2 h 2m (k x 2k L ) 2 + δ Wavevector of Raman lasers k L = 2π λ Raman detuning δ = ω L ω Z Quadratic Zeeman shift for m F = 0 ε Raman Rabi frequency Ω R 14 / 29

15 Rabi frequency and dispersion relations Determine Ω R for given δ Black: m F = 1 Red: m F = 0 Blue: m F = 1 Recoil energy E r = h2 k 2 L 2m Figure : [8] Dispersion relations for δ = 0 (left) and δ < 0 (right) Energy [Er] Energy [Er] k x / k L Figure : [8] k x / k L Figure : [8] 15 / 29

16 Synthetic vector potential Expand dispersion relation around minimum: E(k x ) h2 (k x k min ) 2 2m (5) h k min resembles x-component of a vector potential Problem: k min = const. for δ = const. Resulting synthetic magnetic field B = A = 0 k min / k L ħδ/er Figure : [8] 16 / 29

17 δ = ω L ω Z Figure : [7] Creating a non-zero synthetic magnetic field New setup geometry k L = 2π λ 2 Varying real magnetic field B(y) = ( B0 b y ) ê y ω Z = g µ B B(y)/ h Detuning gradient δ = δ y = g µ Bb / h Synthetic vector potential A x = A x(δ) Synthetic magnetic field B = A x y = δ A x δ Figure : [7] 17 / 29

18 Experimental proof: formation of vortices Absorption-imaging after time-of-flight Figure : [7] Stern-Gerlach effect: spin components are separated along y 18 / 29

19 Experimental proof: formation of vortices Increasing detuning gradient δ more vortices Figure : [7] 19 / 29

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21 Bloch electrons in strong magnetic fields Electrons in a periodic potential Bloch waves: ψ n k ( r) = u n k ( r) e i k r 2D-square-lattice (x-y-plane) with lattice spacing a Bloch ( ) energy function: W k = 2E 0 (cos(k x a) + cos(k y a)) α ε 0 Figure : [5] Harper s equation: g(m + 1) + g(m 1) + 2cos(2π mα ν)g(m) = ε g(m) Douglas R. Hofstadter: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B, 14, Published 3 September / 29

22 Hofstadter butterfly and sceleton 1 1 α 0.5 α ε 0 Figure : [5] ε 0 Figure : [5] α = a2 B 2π( hc e ) = 1 Φ B 2π Φ 0 = 1 2π ϕ ε = E E 0 ϕ = phase gained when moving around one lattice cell 22 / 29

23 Proposal for experimental realisation Cold atoms in 2D optical lattice Description with Bose-Hubbard-model Trap different internal states in different columns of lattice Induce hopping amplitudes of same magnitude along x- and y-axis Non-vanishing phase for atoms moving around a lattice cell Figure : [6] 23 / 29

24 Proposal for experimental realisation Kinetic energy induced hopping only along y-axis Add linear potential in x-direction Accelerating the lattice: H acc = Ma acc x Electric field E = E x: H acc = µ E x Generate spacially varying Rabi frequencies Ω 1,2 Figure : [6] 24 / 29

25 Proposal for experimental realisation Lasers create y-dependent phase ϕ m = qm λ 2 q: y-component of wavevector (of coupling lasers) Phase gained when moving around a lattice cell: ϕ = ϕ m+1 ϕ m = q λ 2 = 2π α α = qλ 4π Figure : [6] 25 / 29

26 Proposal for experimental realisation Small filling factors n << 1 neglect interaction terms Hopping amplitudes J x J y Hamiltonian: ( H(α) = JΣ m,n e 2πiα m c n,m c n+1,m + c n,m c n,m+1 + h.c. ) Equivalent to H for electron moving on lattice with magnetic field B = 2πα A e with A = a x a y (area of one lattice cell) Figure : [6] 26 / 29

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28 Why do we need artificial magnetic fields for neutral atoms? Quantum-Hall effect Engineering Hamiltonians for neutral atoms artificial vector potential Light-induced vector potential Theory of the Hofstadter butterfly Possible experimental realisation 28 / 29

29 Thank you for your attention! 29 / 29

30 Advanced Quantum Mechanics II. Quantum Hall Effect. Abo-Shaeer, J. R. ; Raman, C. ; Vogels, J. M. ; Ketterle, W.: Observation of Vortex Lattices in Bose-Einstein Condensates. In: Science 292 (2001), Nr. 5516, S Goerbig, M. O.: Quantum Hall Effects. In: arxiv: v2 (1998) Hofstadter, D. R.: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. In: Phys. Rev. B 14 (1976), Nr. 6, S Jaksch, D. ; Zoller, P.: Creation of effective magnetic fields in optical lattices: the. In: New J. Phys. 5 (2003), Nr. 56 Lin, Y.-J. ; Compton, R. L. ; Jiménez-García, K. ; Porto, J. V. ; Spielman, I. B.: Synthetic magnetic fields for ultracold neutral atoms. In: nature 462 (2009), S Lin, Y.-J. ; Compton, R. L. ; Perry, A. R. ; Phillips, W. D. ; Porto, J. V. ; Spielman, I. B.: Bose-Einstein Condensate in a Uniform Light-Induced Vector Potential. In: Phys. Rev. Lett. 102 (2009), S / 29