Artificial electromagnetism and spin-orbit coupling for ultracold atoms

Transcription

1 Artificial electromagnetism and spin-orbit coupling for ultracold atoms Gediminas Juzeliūnas Institute of Theoretical Physics and Astronomy,Vilnius University, Vilnius, Lithuania ******************************************************************* Wednesday, May 28, 11:00am Institut für Quantenoptik Leibniz Universität Hannover

2 OUTLINE v Background v Geometric gauge potentials v Spin-orbit coupling (SOC) for ultracold atoms v NIST scheme for SOC v Magnetically generated SOC v Synthetic gauge fields using synthetic dimensions v Conclusions

3 Collaboration (in the area of artificial gauge potentials for ultracold atoms) n P. Öhberg & group, Heriot-Watt Univ., Edinburgh n M. Fleischhauer & group, TU Kaiserslautern, n L. Santos & group, Universität Hannover n J. Dalibard, F. Gerbier & group, ENS, Paris n I. Spielman, B. Anderson, V. Galitski, D. Campbel, C. Clark and J. Vaishnav, JQI, NIST/UMD, USA n M. Lewenstein & group, ICFO, ICREA, Barcelona n Shih-Chuan Gou & group, Taiwan n Wu-Ming Liu, Qing Sun, An-Chun Ji & groups (Beijing)

4 Quantum Optics ITPA, Vilnius University R. Juršėnas, J. Ruseckas, T. Andrijauskas, A. Mekys, G. J., E. Anisimovas, V. Novicenko, D. Efimov (St. Petersburg Univ.) and H. R. Hamedi (April 2014) Not present in the picture: A. Acus and V. Kudriašov

5 Vilnius (Lithuania) is in north of EUROPE

6 Vilnius University

7 Quantum Optics ITPA, Vilnius University Research activities: n Light-induced gauge potentials for cold atoms (includes Rashba-Dresselhaus SO coupling) n Slow light (with OAM, multi-component, ) n Cold atoms in optical lattices n Graphene n Metamaterials

8 Quantum Optics ITPA, Vilnius University n Light-induced gauge potentials for cold atoms (includes Rashba-Dresselhaus SO coupling) This talk

9 Quantum Optics ITPA, Vilnius University n Light-induced gauge potentials for cold atoms (includes Rashba-Dresselhaus SO coupling) This talk v Slow light (multi-component) will be briefly mentioned (at the end)

10 Ultracold atoms n Analogies with the solid state physics q Fermionic atoms Electrons in solids q Atoms in optical lattices Hubbard model q Simulation of various many-body effects n Advantage : q Freedom in changing experimental parameters that are often inaccessible in standard solid state experiments q e.g. number of atoms, atom-atom interaction, lattice potential

11 Trapped atoms - electrically neutral species n No direct analogy with magnetic phenomena by electrons in solids, such as the Quantum Hall Effect (no Lorentz force) n A possible method to create an effective magnetic field (an artificial Lorentz force): Rotation

12 Trapped atoms - electrically neutral particles n No direct analogy with magnetic phenomena by electrons in solids, such as the Quantum Hall Effect (no Lorentz force) n A possible method to create an effective magnetic field (an artificial Lorentz force): Rotation à Coriolis force à

13 Trapped atoms - electrically neutral particles n No direct analogy with magnetic phenomena by electrons in solids, such as the Quantum Hall Effect (no Lorentz force) n A possible method to create an effective magnetic field: Rotation à Coriolis force (Mathematically equivalent to Lorentz force)

14 ROTATION n Can be applied to utracold atoms both in usual traps and also in optical lattices (a) Ultracold atomic cloud (trapped): (b) Optical lattice:

15 ROTATION n Can be applied to utracold atoms both in usual traps and also in optical lattices (a) Ultracold atomic cloud (trapped): (b) Optical lattice: Not always convenient to rotate an atomic cloud Limited magnetic flux

16 Effective magnetic fields without rotation n Using (unconventional) optical lattices Initial proposals: q J. Ruostekoski, G. V. Dunne, and J. Javanainen, Phys. Rev. Lett. 88, (2002) q D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003) q E. Mueller, Phys. Rev. A 70, (R) (2004) q A. S. Sørensen, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 94, (2005) n B eff is produced by inducing an asymmetry in atomic transitions between the lattice sites. n Non-vanishing phase for atoms moving along a closed path on the lattice (a plaquette) n Simulates non-zero magnetic flux B eff 0

17 Effective magnetic fields without rotation n Using (unconventional) optical lattices Initial proposals: q J. Ruostekoski, G. V. Dunne, and J. Javanainen, Phys. Rev. Lett. 88, (2002) q D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003) q E. Mueller, Phys. Rev. A 70, (R) (2004) q A. S. Sørensen, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 94, (2005) n A. R. Kolovsky, EPL, (2011) n Experimental implementation : q M. Aidelsburger et al., PRL 111, (2013) q H. Miyake et al., PRL 111, (2013)

18 Effective magnetic fields without rotation n Optical lattices: The method can be extended to create Non-Abelian gauge potentials (Laser assisted, state-sensitive tunneling) A proposal: q K. Osterloh, M. Baig, L. Santos, P. Zoller and M. Lewenstein, Phys. Rev. Lett. 95, (2005)

19 Effective magnetic fields without rotation n Optical lattices: The method can be extended to create Non-Abelian gauge potentials (Laser assisted, state-sensitive tunneling) A proposal: q K. Osterloh, M. Baig, L. Santos, P. Zoller and M. Lewenstein, Phys. Rev. Lett. 95, (2005) (Spin-orbit coupling in optical lattices)

20 Effective magnetic fields without rotation -- using Geometric Potentials n Distinctive features: q No rotation is necessary q No lattice is needed

21 Effective magnetic fields without rotation -- using Geometric Potentials q An artificial vector potential A is created q A can be a matrix à q à Spin-orbit coupling is formed

22 Geometric potentials n Emerge in various areas of physics (molecular, condensed matter physics etc.) n First considered by Mead, Berry, Wilczek and Zee and others in the 80 s (initially in the context of molecular physics). n More recently in the context of motion of cold atoms affected by laser fields n n n. (Currently: a lot of activities) Recent reviews: J.Dalibard, F. Gerbier, G. Juzeliūnas and P. Öhberg, Rev. Mod. Phys. 83, 1523 (2011). N. Goldman, G. Juzeliūnas, P. Öhberg and I.B. Spielman arxiv: (Submitted to Rep. Progr. Phys., 2013)

23 Geometric potentials n Advantage of such atomic systems: possibilities to control and shape gauge potentials by choosing proper laser fields..

24 Creation of B eff using geometric potentials Atomic dynamics taking into account both internal (electronic) degrees of freedom and also center of mass motion. ç (includes r-dependent atom-light coupling) (for c.m. motion) ë ( r-dependent dressed eigenstates)

25 Creation of B eff using geometric potentials Atomic dynamics taking into account both internal (electronic) degrees of freedom and also center of mass motion. ç (includes r-dependent atom-light coupling) (for c.m. motion) ë ( r-dependent dressed eigenstates)

26 Creation of B eff using geometric potentials Atomic dynamics taking into account both internal degrees of freedom and also center of mass motion. ç (includes r-dependent atom-light coupling) (for c.m. motion) ë ( r-dependent dressed eigenstates)

27 Creation of B eff using geometric potentials Atomic dynamics taking into account both internal degrees of freedom and also center of mass motion. ç (includes r-dependent atom-light coupling) (for c.m. motion) ë ( r-dependent dressed eigenstates)

28 n n Adiabatic atomic energies Full state vector: n=3 n=2 n=1 r wave-function of the atomic centre of mass motion in the n-th atomic internal dressed state

29 Non-degenerate state with n=1 Adiabatic approximation: n=3 n=2 Full state-vector n=1 r

30 Non-degenerate state with n=1 Adiabatic approximation: n=3 n=2 Full state-vector n=1 r

31 Non-degenerate state with n=1 Adiabatic approximation: n=3 n=2 n=1 r

32 Non-degenerate state with n=1 Adiabatic approximation: Equation of the atomic c. m. motion in the internal state n=3 n=2 n=1 r ( ) 2 H ˆ = p A 11 2M + V (r) + ε 1(r) Effective Vector potential A 11 A appears (due to the positiondependence of the atomic internal dressed state )

33 Non-degenerate state with n=1 Adiabatic approximation: Equation of the atomic c. m. motion in the internal state n=3 n=2 n=1 r ( ) 2 H ˆ = p A 11 2M + V (r) + ε 1(r) Effective Vector potential A 11 A appears (due to the positiondependence of the atomic internal dressed state ) R. Dum and M. Olshanii, Phys. Rev. Lett., 76,1788, 1996.

34 Non-degenerate state with n=1 Adiabatic approximation: Equation of the atomic c. m. motion in the internal state n=3 n=2 n=1 r ( ) 2 H ˆ = p A 11 2M + V (r) + ε 1(r) Effective Vector potential A 11 A appears - effective magnetic field

35 Non-degenerate state with n=1 Adiabatic approximation: Equation of the atomic c. m. motion in the internal state n=3 n=2 n=1 r ( ) 2 H ˆ = p A 11 2M + V (r) + ε 1(r) Effective Vector potential A 11 A appears - effective magnetic field (non-trivial situation if ) G.Juzeliūnas, and P. Öhberg, Phys. Rev. Lett. 93, (2004).

36 Counter-propagating beams with spatially shifted profiles [G. Juzeliūnas, J. Ruseckas, P. Öhberg, and M. Fleischhauer, Phys. Rev. A 73, (2006).] B = A 0 Artificial Lorentz force (due to photon recoil) [Interpretation: M. Cheneau et al., EPL 83, (2008).]

37

38

39 Position-dependent detuning δ

40 Position-dependent detuning δ=δ(y) B=B(y)

41

42 If one degenerate atomic internal dressed state -- Abelian gauge potentials

43 If more than one degenerate atomic internal dressed state --Non-Abelian (noncommuting) vector potential (spin-orbit coupling)

44 Degenerate states with n=1 and n=2 Adiabatic approximation: n=4 n=3 n=1, n=2 r wave-function of the atomic centre of mass motion in the n-th atomic internal dressed state (n=1,2)

45 Degenerate states with n=1 and n=2 Adiabatic approximation: n=4 n=3 n=1, n=2 r wave-function of the atomic centre of mass motion in the n-th atomic internal dressed state (n=1,2) # Ψ(r,t) = Ψ(r,t) & 1 % ( $ Ψ 2 (r,t)' - two-component atomic wave-function (spinor wave-function) à Quasi-spin 1/2 Repeating the same procedure

46 Degenerate states with n=1 and n=2 Adiabatic approximation: Equation of motion: ( ) 2 H ˆ = p A 2M +V(r) +ε 1(r) # Ψ(r,t) = Ψ(r,t) & 1 % ( $ Ψ 2 (r,t)' 2x2 matrix n=4 n=3 n=1, n=2 - effective vector potential r - two-comp. atomic wave-function A appears due to position-dependence of Spin-orbit coupling is formed

47 Degenerate states with n=1 and n=2 Adiabatic approximation: Equation of motion: ( ) 2 H ˆ = p A 2M +V(r) +ε 1(r) # Ψ(r,t) = Ψ(r,t) & 1 % ( $ Ψ 2 (r,t)' 2x2 matrix n=4 n=3 n=1, n=2 - effective vector potential r - two-comp. atomic wave-function A appears due to position-dependence of Ĥ = p2 + p.a + A.p + A 2 2M +V(r)+ε 1 (r) Spin-orbit coupling is formed

48 Degenerate states with n=1 and n=2 Adiabatic approximation: n=4 n=3 Equation of motion: ( ) 2 H ˆ = p A 2M +V(r) +ε 1(r) 2x2 matrix # Ψ(r,t) = Ψ(r,t) & 1 % ( $ Ψ 2 (r,t)' 2x2 matrix n=1, n=2 - effective vector potential - effective magnetic field (curvature) Non-trivial situation if r - two-comp. atomic wave-function

49 Degenerate states with n=1 and n=2 Adiabatic approximation: n=4 n=3 Equation of motion: ( ) 2 H ˆ = p A 2M +V(r) +ε 1(r) 2x2 matrix # Ψ(r,t) = Ψ(r,t) & 1 % ( $ Ψ 2 (r,t)' 2x2 matrix n=1, n=2 - effective vector potential - effective magnetic field (curvature) Non-trivial situation if r - two-comp. atomic wave-function If A x, A y, A z do not commute, B 0 even if A is constant!! à Non-Abelian vector potential is formed

50 Tripod configuration n M.A. Ol shanii, V.G. Minogin, Quant. Optics 3, 317 (1991) n R. G. Unanyan, M. Fleischhauer, B. W. Shore, and K. Bergmann, Opt. Commun. 155, 144 (1998) n Two degenerate dark states n (Superposition of atomic ground states immune to the atomlight coupling)

51 Tripod configuration n Two degenerate dark states n n n (Superposition of atomic ground states immune to the atomlight coupling) Dark states: destructive interference for transitions to the excited state Lasers keep the atoms in these dark (dressed) states

52 (Non-Abelian) light-induced gauge potentials for centre of mass motion of dark-state atoms: (Due to the spatial dependence of the dark states) ë (two dark states) J. Ruseckas, G. Juzeliūnas and P.Öhberg, and M. Fleischhauer, Phys. Rev. Letters 95, (2005).

53 (Non-Abelian) light-induced gauge potentials for centre of mass motion of dark-state atoms: (Due to the spatial dependence of the dark states) ë (two dark states) A n,m = i! D n D m J. Ruseckas, G. Juzeliūnas and P.Öhberg, and M. Fleischhauer, Phys. Rev. Letters 95, (2005). A - effective vector potential à Spin-orbit coupling

54 (Non-Abelian) light-induced gauge potentials Centre of mass motion of dark-state atoms: A n,m = i! D n D m - 2x2 matrix ç Two component atomic wave-function # Ψ(r,t) = Ψ(r,t) & 1 % ( $ Ψ 2 (r,t)' A - effective vector potential à Spin-orbit coupling

55 Three plane wave setup T. D. Stanescu,, C. Zhang, and V. Galitski, Phys. Rev. Lett 99, (2007). A. Jacob, P. Öhberg, G. Juzeliūnas and L. Santos, Appl. Phys. B. 89, 439 (2007). # Ψ(r,t) = Ψ(r,t) & 1 % ( $ Ψ 2 (r,t)' (centre of mass motion, dark-state atoms): (à Rashba-type Hamilltonian) σ = e x σ x + e y σ y spin ½ operator (acting on the subspace of atomic dark states) à Spin-Orbit coupling of the Rashba-Dresselhaus type Constant non-abelian A with [A x,a y ]~σ z à B ~ e z

56 Three plane wave setup Centre of mass motion of dark-state atoms: σ = e x σ x + e y σ y (à Rashba-type Hamilltonian) Plane-wave solutions: σ k g ± ± k = ±g k à two dispersion branches with positive or negative chirality

57 Dispersion of centre of mass motion for cold atoms in light fields !/v 0 " v g =dω/dk>0 v g =dω/dk< k/"

58 Dispersion of centre of mass motion for cold atoms in light fields E(k x,k y ) Picture by Patrik Öhberg Dirac regime k x Lower branch Mexican hat potential k y

59 Zittenbewegung of cold atoms Theory n J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100, (2008). n M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys. Lett. 83, (2008). n Q. Zhang, J. Gong and C. H. Oh, Phys. Rev. A. 81, (2010). n Y-C.i Zhang, S.-W. Song and C.-F. Liu and W.-M. Liu, Phys. Rev. A. 87, (2013). Experiment L. J. LeBlanc et al., New. J. Phys (2013). C. Qu et al. Phys. Rev. A A 88, (R) (2013).

60 Unconventional Bose-Einstein condensation!/v 0 " k/" Energy minimum is a ring (in the momentum space) T. D. Stanescu, B. Anderson and V. M. Galitski, PRA 78, (2008); J. Larson and E. Sjöqvist, Phys. Rev. A 79, (2009) C. Wang et al, PRL 105, (2010); T.-L. Ho and S. Zhang, PRL 107, (2011); Z. F. Xu, R. Lu, and L. You, PRA 83, (2011); S.-K. Yip, PRA 83, (2011); S. Sinha R. Nath, L. Santos, PRL 107, (2011); S.-W. Su, I.-K. Liu, Y.-C. Tsai, W. M. Liu, S.-C. Gou, Phys. Rev. A 86, (2012) H. Zhai, arxiv: (2014) [also covers fermions] etc. - Mexican hat potential

61 n Drawback of the tripod scheme: degenerate dark states are not the ground atomic dressed states à collision-induced loses n Closed loop setup overcomes this drawback: (a) Coupling scheme (with pi phase) (c) Uncoupled dispersion D. L. Campbell, G. Juzeliūnas and I. B. Spielman, Phys. Rev. A 84, (2011)

62 n Drawback of the tripod scheme: degenerate dark states are not the ground atomic dressed states à collision-induced loses n Closed loop setup overcomes this drawback: (a) Coupling scheme (with pi phase) D. L. Campbell, G. Juzeliūnas and I. B. Spielman, Phys. Rev. A 84, (2011) n n (c) Uncoupled dispersion Two degenerate internal ground states à non-abelian vector poten. for ground-state manifold

63 n Laser fields represent counter-propagating plane waves: BEC (a) Coupling scheme (d) Coupled dispersion (c) Uncoupled dispersion à Closed loop setup produce 2D Rashba- Dresselhaus SO coupling for cold atoms D. L. Campbell, G. Juzeliūnas and I. B. Spielman, Phys. Rev. A 84, (2011)

64 (a) Coupling scheme n Possible implementation of the closed loop setup using the Raman transitions: (with pi phase) (c) Uncoupled dispersion D. L. Campbell, G. Juzeliūnas and I. B. Spielman, Phys. Rev. A 84, (2011)

65 n Laser fields represent plane (b) Tetrehedral Geometry waves with wave-vectors forming a tetrahedron: (d) Spectrum à Modified closed loop setup produces a 3D Rashba-Dresselhaus SOC B. Anderson, G. Juzeliūnas, I. B. Spielman, and V. Galitski, Phys. Rev. Lett. 108, (2012) [Editor s Choice paper].

66 n Laser fields represent plane (b) Tetrehedral Geometry waves with wave-vectors forming a tetrahedron: (d) Spectrum à Closed loop setup produces a 3D Rashba-Dresselhaus SOC Minimum of atomic energy - the sphere (in the momentum space) B. Anderson, G. Juzeliūnas, I. B. Spielman, and V. Galitski, Phys. Rev. Lett. 108, (2012) [Editor s Choice paper].

67 The schemes for creating 2D and 3D SOC involve complex atom-light coupling and have not yet been implemented

68 Experimental implementation of the 1D SOC (using a simpler scheme) Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011).

69 Experimental implementation of the 1D SOC (using a simpler scheme) Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011). J.-Y. Zhang et al. Pan, Phys. Rev. Lett. 109, (2012). P. J. Wang et al, Phys. Rev. Lett. 109, (2012). L. W. Cheuk et a;, Phys. Rev. Lett. 109, (2012). C. Qu et al, Phys. Rev. A 88, (R) (2013).

70 Spin-orbit coupling using a simpler scheme Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011). Raman coupling between the F=1 ground state manifold of 87 Rb: (transitions between m F =-1 and m F =0 are in resonance)

71 Spin-orbit coupling using a simpler scheme Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011). Raman coupling between the F=1 ground state manifold of 87 Rb: (transitions between m F =-1 and m F =0 are in resonance) Large quadratic Zeeman shift à m F =1 state can be neglected

72 Spin-orbit coupling using a simpler scheme Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011). Raman coupling between the F=1 ground state manifold of 87 Rb: (transitions between m F =-1 and m F =0 are in resonance) Large quadratic Zeeman shift à m F =1 state can be neglected à An effective two level system (coupled by Raman transitions with a recoil)

73 Spin-orbit coupling using a simpler scheme Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011). Raman coupling between the F=1 ground state manifold of 87 Rb: (transitions between m F =-1 and m F =0 are in resonance) Large quadratic Zeeman shift à m F =1 state can be neglected à An effective two level system (coupled by Raman transitions with the recoil) Two states [to be called spin up and spin down states]

74 Spin-orbit coupling using a 2 level scheme Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011). Raman coupling between the the m F =-1 and m F =0 states of the F=1 ground state manifold of 87 Rb Two-level system Dispersion for δ=0 [Two parabolas shifted by the recoil momentum]

75 Spin-orbit coupling using a 2 level scheme Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011). Dispersion for δ=0 ( H = p χσ y) 2 2m + Ω 2 σ z Abelian (1D) vector potential (but non-trivial situation if Ω 0) (1D SOC)

76 Spin-orbit coupling using a 2 level scheme Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011). Dispersion for δ=0 ( H = p χσ y) 2 2m + Ω 2 σ z Abelian (1D) vector potential (but non-trivial situation if Ω 0) H = p2 2m χ pσ y m + Ω 2 σ z + const (1D SOC)

77 A relatively simple scheme to create 2D or 3D SOC?

78 Magnetically-induced SOC B.M. Anderson, I.B. Spielman and G. Juzeliūnas, Phys. Rev. Lett. 111, (2013). Z.F. Xu, L. You and M. Ueda, Phys. Rev. A (2013). A relatively simple scheme to create 2D or 3D SOC (without involving any light beams, any complex atom-light coupling)

79 Magnetically-induced SOC Atom in inhomogeneous -me- dependent magne-c field B=B(r,t):

80 Magnetically-induced SOC Atom in inhomogeneous -me- dependent magne-c field B=B(r,t): F spin operator For spin ½, F =σ/2

81 Magnetically-induced SOC Atom in inhomogeneous -me- dependent magne-c field B=B(r,t): Generation of 2D SOC Magnetic pulses:

82 Short Magnetic pulses: Ω y Ω x

83 Short Magnetic pulses: Ω y Ω x & Ω y induce spin-dep. force along y & x: Ω x à spin-dependent mom. kick along y & x:

84 Short Magnetic pulses: Ω y Ω x Ω x induces spin-dep. momentum kick along y: à Evolution operator:

85 Short Magnetic pulses: Ω y Ω x Ω x induces spin-dep. momentum kick along y: à Evolution operator: à Formation of 1D SOC (in y direction)

86 Short Magnetic pulses: Ω y U y : x y Ω x Ω x induces spin-dep. momentum kick along y: à Evolution operator: à Formation of 1D SOC (in y or x direction)

87 Total evolution operator after two pulses: H 2D - 2D Rashba-type Hamiltonian (small τ ):

88 Total evolution operator after two pulses: H 2D - 2D Rashba-type Hamiltonian (small τ ): à Formation of 2D SOC (Zero- order Floquet Hamiltonian)

89 Total evolution operator after two pulses: H 2D - 2D Rashba-type Hamiltonian (small τ ): 2D spin-orbit coupling Additional quadratic Zeman term

90 Total evolution operator after two pulses: H 2D - 2D Rashba-type Hamiltonian (small τ ): Can be extended to non-delta pulses:

91 Non-delta pulses:

92 Proposed experimental setup:

93 Finite pulse durations:

94 3D SOC:

95 For more details see B.M. Anderson, I.B. Spielman and G. Juzeliūnas, Phys. Rev. Lett. 111, (2013).

96 Spin-orbit Coupling using bilayer atoms, Q. Sun et at, arxiv: v 1D SOC in each layer (in 1 or 2 direct.) + Interlayer coupling (a) (b) (c) q y Q 2 Q1 Q 1 Q 2 q x

97 Spin-orbit Coupling using bilayer atoms, Q. Sun et at, arxiv: v 1D SOC in each layer (in 1 or 2 direct.) + Interlayer coupling v Effectively 2D SOC is generated (a) (b) (c) q y Q 2 Q1 Q 1 Q 2 q x

98 Spin-orbit Coupling using bilayer atoms, Q. Sun et at, arxiv: v 1D SOC in each layer (in 1 or 2 direct.) + Interlayer coupling v Effectively 2D SOC is generated v Interesing quantum phases for such a SOC BEC (a) (b) PW SP I SP II (c) q y Q 2 q x α HSL SP II Q1 Q 1 Q g / g g

99 Optical lattices in extra dimensions

100 Optical lattices in extra dimensions F=1, m = -1, 0, 1 Raman transitions between magnetic sublevels m (extra dimension)

101 Optical lattices in extra dimensions F=1, m = -1, 0, 1 Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency (recoil in x direction)

102 Optical lattices in extra dimensions F=1, m = -1, 0, 1 Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency (recoil in x direction) 1D chain of atoms (real dimension) x

103 Optical lattices in extra dimensions F=1, m = -1, 0, 1 Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency γ=ka 1D chain of atoms (real dimension) Tunneling in real dimension and Raman transitions in the extra dimensions yield a 2D lattice involving real and extra dimensions x

104 Optical lattices in extra dimensions F=1, m = -1, 0, 1 Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency γ=ka 1D chain of atoms (real dimension) Combination of real and extra dimensions yields strong & non-staggered magnetic flux γ=ka per 2D plaquette x

105 Optical lattices in extra dimensions F=1, m = -1, 0, 1 Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency γ=ka 1D chain of atoms (real dimension) Combination of real and extra dimensions yields strong & non-staggered magnetic flux γ=ka per 2D plaquette x

106 Optical lattices in extra dimensions F=1, m = -1, 0, 1 Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency γ=ka 1D chain of atoms (real dimension) Combination of real and extra dimensions yields strong & non-staggered magnetic flux γ=ka per 2D plaquette x

107 Optical lattices in extra dimensions Raman transitions between magnetic sublevels m (extra dimension) F=1, m = -1, 0, 1 Ω0e ikx - Raman Rabi frequency γ=ka 1D chain of atoms (real dimension) Sharp boundaries in extra dimension: Conducting edge states in extra dimension x

108 Optical lattices in extra dimensions Dispersion branches: Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency γ=ka x γ=ka=1.0 1D chain of atoms (real dimension) Ω0/t=0.1 Sharp boundaries in extra dimension: Conducting edge states in extra dimension

109 Optical lattices in extra dimensions Dispersion branches: Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency γ=ka x γ=ka=1.0 Edge states 1D chain of atoms (real dimension) Ω0/t=0.1 Sharp boundaries in extra dimension: Conducting edge states in extra dimension Atoms with opposite spins move in opposite directions

110 Optical lattices in extra dimensions Dispersion branches: Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency γ=ka Landau Level x γ=ka=1.0 Edge states 1D chain of atoms (real dimension) Ω0/t=0.1 Sharp boundaries in extra dimension: Conducting edge states in extra dimension Atoms with opposite spins move in opposite directions

111 Optical lattices in extra dimensions Dispersion branches: Raman transitions between magnetic sublevels m (extra dimension) Ω0e ikx - Raman Rabi frequency γ=ka x γ=ka=1.0 Edge states 1D chain of atoms (real dimension) Ω0/t=0.1 Atoms with opposite spins move in opposite directions Scattering of edge states by a short-range potential (???)

112 Optical lattices in extra dimensions Dispersion branches: Spin-independent potential (road block) γ=ka x γ=ka=1.0 Edge states 1D chain of atoms (real dimension) Ω0/t=0.1 Atoms with opposite spins move in opposite directions Scattering of edge states by a short-range potential (???)

113 Optical lattices in extra dimensions Dispersion branches: Spin-dependent potential (perturbation for m=1) γ=ka x γ=ka=1.0 Edge states 1D chain of atoms (real dimension) Ω0/t=0.1 Atoms with opposite spins move in opposite directions Scattering of edge states by a short-range potential (???)

114 Optical lattices in extra dimensions Dispersion branches: Spin-dependent potential (perturbation for m=0) γ=ka x γ=ka=1.0 Edge states 1D chain of atoms (real dimension) Ω0/t=0.1 Atoms with opposite spins move in opposite directions Scattering of edge states by a short-range potential (???)

115 Optical lattices in extra dimensions Dispersion branches: Spin-dependent potential (perturbation for m= -1) γ=ka x γ=ka=1.0 Edge states 1D chain of atoms (real dimension) Ω0/t=0.1 Atoms with opposite spins move in opposite directions Scattering of edge states by a short-range potential (???)

116 Optical lattices in extra dimensions Dispersion branches: Spin-independent potential (road block) γ=ka x γ=ka=1.0 Edge states 1D chain of atoms (real dimension) Ω0/t=0.1 Atoms with opposite spins move in opposite directions Scattering of edge states by a short-range potential (???)

117 Optical lattices in extra dimensions Scattering of edge state atoms by a short-range potential: Dispersion branches: γ=ka=1.0 Ω0/t=0.1 Blue line - spin-independent perturbation (road block), red - perturbation for m=±1, green - perturbation for m=0 Edge (spin-up & spin-down) states

118 Optical lattices in extra dimensions Scattering of edge state atoms by a short-range potential: Dispersion branches: γ=ka=1.0 Ω0/t=0.1 Blue line - spin-independent perturbation (road block), red - perturbation for m=±1, green - perturbation for m=0 Edge (spin-up & spin-down) states Two scattering peaks for higher energies Immune to scattering

119 Closed boundary condition along the extra dimension Various possibilities: - Combination of Raman and two-photon IR transitions: mf = Connecting different F manifolds via rf fields: rf rf

120 Energy spectrum of spin-1 atoms, closed boundary Formation of Hofstadter butterfly using artificial dimensions

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125 n n n n Conclusions The light induced non-abelian vector potential can simulate the spin-orbit coupling (SOC) of the Rashba-Dresselhaus-type for ultracold atoms (2D SOC or even 3D SOC!). Up to now experimentally only a simpler 1D SOC has been implemented (Abelian yet non-trivial). Possible solutions: Magnetically induced SOC: [B.M. Anderson, I.B. Spielman and G. Juzeliūnas, PRL 111, (2013). Z.F. Xu, L. You and M. Ueda, PRA (2013)] or using bilayer atoms: Q. Sun et at, arxiv: Non-staggered magnetic flux in optical lattices using synthetic dimensions [A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, and M. Lewenstein, Phys. Rev. Lett. 112, (2014)].

126 Recent reviews: Artificial gauge potentials n J. Dalibard, F. Gerbier, G. Juzeliūnas and P. Öhberg, Rev. Mod. Phys. 83, 1523 (2011). n N. Goldman, G. Juzeliūnas, P. Öhberg and I.B. Spielman, arxiv: (Submitted to Rep. Progr. Phys., 2013). Spin-orbit coupling n V. Galitski and I.B. Spielman I B Nature 494, 49 (2013) H. Zhai, arxiv: (2014) (Submitted to Rep. Progr. Phys., 2013)

127 Thank you!