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Theory of Spin-Orbit-Coupled Cold Atomic Systems Victor Galitski Joint Quantum Institute and Physics Department University of Maryland Acknowledgements: Work supported by NSF-PFC & ARO-MURI Collaborators !"#$%&$' ($%)"*&$' +,-./012345' ($%")B' E&G)"@*&$' +,-./01EH5' ;<#"=)*';=#"6' +892/12345' K)%F7F$#*'' 8?L)=F?$#*' +CF=$F?*5' 8?"#D'E#%FA' ''''+,-.5' 3#$6#"'.#*'' 3#"7#'' +892:,-.5' '''4?%&"'3@#$)*A?' '''+,-./0,BC#5' 2#$'3MF)=7#$' +892:12345' ."I'8#>'C#F*<$#J' ''''+12345' 8#6)'4#>=&"' +892:12345' Outline • Synthetic spin-orbit coupling schemes, old and new and many more • A zoo of spin-orbit-coupled BECs: (mostly) theory • How to create vortices in spin-orbit-coupled BECs? • Topological optical lattices & lattice quantum Hall states • Practical applications: Quantum gravimetry and interferometry !"#$%&'()!*+#,-./+$)012*3+#45) +#)0136,7$18)!"5$&85) Atom in a r-dependent laser field. Tripod scheme E)N*O''E?*)A6#*P'8?L)=F?$#*P'Q)<G)"RP'S'T=)F*A<<#?)"P'U<>*I'E)JI'H)VI'9:P'WXWYWY'+ZWW[5' '''''''''''3@#$)*A?P'\<#$RP'S'K#=F@*6FP'U<>*I'E)JI'H)VI'99P'XXWYW]'+ZWW^5'' Two equivalent descriptions To get the effective Hamiltonian... 1. Diagonalize Ĥa−l via a unitary rotation, Ĥa−l → Û †(r)Ĥa−lÛ (r) 2. Project the result onto the dark subspace � � − �2 ∇2 † Ĥeff = P̂darkÛ (r) Û (r)P̂dark 2m • Picture I: Particle in a non-Abelian “gauge field” �2 1 � i Ĥeff = −i�∇ − A (r)σ̂i 2m • Picture II: Spin-orbit coupled particle p2 Ĥeff (p) = + b(p) · σ̂ 2m � � For the tripod scheme in 2D: b(p) = 0, vpx, v �py SU(3) “spin”-orbit coupling A different choice of phases in the tetragonal scheme leads to a Hamiltonian that can NOT be spanned by Pauli matrices alone, but may be spanned by 3 × 3 Gell-Mann matrices � � k λ̂ λ̂i, λ̂j = ifij k Gell-mann matrices are generators of su(3). su(3) spin-orbit coupled system: 8 � p2 Ĥsu(3) = + bi(p)λ̂i 2m i=1 No analogue in condensed matter (or any other matter)... G. Boyd, B. Anderson, & V. Galitski, tbp ;&<)$"*&)1=)>15&,?+#5$&+#)(1#6@$&) • 4F7)/&N/_FR<@'N&"'@<)'?*?#='!`;' • 4&T'N&"'#'*MF$/&"GF@'A&?M=)%'!`;' A1.'(&5)+#)!*+#,-./+$) >15&,?+#5$&+#)01#6@$&5) How to create vortices in a BEC? How to create vortices in a BEC? How to create vortices in a BEC? • Rotation: If an ordinary BEC is stirred by a laser “spoon” or the anisotropic trap is rotated: •– Rotation: If an ordinary is stirred by a laser “spoon” or the The Hamiltonian in theBEC rotating anisotropic trap is rotated: frame is time-independent: – The Hamiltonian in the rotating frame is time-independent: a)V)"=)P'-24'+ZWWX5' HRF = H0 − Ω · L H =H −Ω·L – Equilibrium stat. mechanics RF apply 0 Equilibrium stat. mechanics apply – – Vortex lattice appears – Vortexmagnetic lattice appears • Synthetic field for neutral • Synthetic field formagnetic neutral atoms leads to an effective magatoms leads magnetic to an effective 3MF)=7#$P'892/1234'+ZWWb5' netic for states dressed states field for field dressed Rotating the trap generally does NOT work for spin-orbit BECs, as there is no rotating frame where the Hamiltonian is time-independent (nonequilibrium physics, heating). However, to combine synthetic spin-orbit coupling with synthetic magnetic field should work. ing scheme creates an effective gauge field Heff p2 = + vpxσ̂z + Ωσ̂x + δ(y)σ̂z 2m Spin-orbit coupling + spatially-dependent detuning troducing spatially-dependent detuning δ(r) δ( inr)I.inSpielman’s exist• Introducing spatially-dependent detuning I. Spielman’s existg scheme an effective gaugegauge field field ing creates scheme creates an effective Heff 2 p2 p = vpxσ̂ +z vp σ̂z + Ωσ̂δ(y)σ̂ x + δ(y)σ̂ = Heff + +xΩσ̂ x+ z z 2m 2m • The combination of the SOC and synthetic gauge field yields two main effects: 1. Spatial separation of the left- and right-movers 2. Synthetic mag. field for each component and vortex nucleation • The combination of the SOC and synthetic gauge field yields two main effects: he combination of the SOC and synthetic gauge field yields two 1. Spatial separation of the left- and right-movers ain effects: 2. Synthetic mag. field for each component and vortex nucleation Spatial separation of the left- and right-movers Parity effect in vortex nucleation Due to symmetry of the effective gauge field with respect to reflection about the y = 0 axis and almost spin-independent interactions, an interesting parity effect is observed (in GPE simulations): the number of vortices is the same in both components. B1*1314+(@3)) -*'(@3)C@D(&5) delpuoC-tibrOsmetsyS c ikstila tnemtrapeD scisyhP dn dnalyraM f IRUM-ORA & CFP-FSN yb Synthetic gauge fields Lattice formed by 3 standing waves Trap ?E@($)5+#43&,*@.'(3&)5*&($.28)@#6)6+$")1=)5$@$&5) F&18&$."O'('c$F@)/*FL)'%F*6'*#7M=)' ?E@($)5+#43&,*@.'(3&)5*&($.28)@#6)6+$")1=)5$@$&5) ?E@($))G<@H&,=2#('1#G))*.1I3&) J.@('(@3)7**3+(@'1#5)1=)$%&) !"#$%&'()F@24&)K+&365)=1.) J.&(+5+1#)L2@#$28)M#$&.=&.18&$.")) Steven Chu From Steven Chu group web-site at Stanford: <VMO::BBBI*@#$N&"%I)%?:R"&?M:A<?R"&?M:' Existing gravimeters do not probe time-dependent gravity/acceleration, ȧ(t). But it is possible with synthetic gauge fields [B. Anderson, J.Taylor, & VG, Phys. Rev. A 83, 031602(R) (2011)]. “Vector potential:” An(p) = i �un(p)|∂pun(p)� Dual “magnetic field:” Bdual (p) = ∇p × An(p) n Trajectories for “spin-up” Semiclassical Eqs. of motion (Luttinger, 50’s; and Niu, Haldane, 2000’s): “spin-down” differ depending ∂En(p) ṙon =time-dependent + Bdual (p) × ṗ n gravity! ∂p ṗ = −eE + eB × ṙ Summary of the recent progress • Dressed states with cold atoms potentially host a much richer variety of spin-orbit structures than that available in solids (Rashba, Dresselhaus, “Weyl,” su(3)-SOC, ...) • Synthetic spin-structures may be practically useful for quantum interferometry • Abelian spin-orbit BECs have already been observed. Theory predicts macroscopically-entangled states in non-Abelian SO-BECs (staying tuned for new experiments...) • Synthetic SOC + synthetic magnetic field = new vortex structures • SO coupling on a lattice may lead to lattice quantum Hall effect • Future work: Non-equilibrium physics + synthetic fields. Cold atoms may be ideal candidates to realize Floquet topological insulators.