Synthetic gauge fields in Bose-Einstein Condensates 1. Alexander Fetter Stanford University. University of Hannover, May 2015

Transcription

1 Synthetic gauge fields in Bose-Einstein Condensates 1 Alexander Fetter Stanford University University of Hannover, May Two-component trapped spin-orbit coupled Bose-Einstein condensate (BEC) 2. Synthetic gauge fields in optical lattices a. shaken lattices b. laser-assisted tunneling in two-dimensional square lattices c. synthetic dimensions 1 For recent reviews, see Galitski and Spielman, Nature 494, 49 (2013), Goldman et al. arxiv: v2 and Dalibard arxiv: v1 1

2 1 Two-component spin-orbit coupled BEC early experiments used parabolic (harmonic) magnetic traps these traps can confine only certain hyperfine states for example, workhorse atom 87 Rb has nuclear spin I = 3 2 single 5S valence electron has spin s = 1 2 vector sum F = I + s yields two hyperfine manifolds F = 1 and 2 F = 1 has three magnetic sub-states m F = 1, 0, 1 of these, only F, m F = 1, 1 is confined in typical magnetic traps for spin-orbit effects, need external magnetic fields hence experiments instead rely on laser dipole trapping how does laser dipole trapping work? 2

3 For given electric dipole d in external electric field E, the energy is U = d E energy is minimum when d is parallel to E for a single neutral atom with polarizability α, dipole moment is d = αe turn on electric field adiabatically and get final energy U = 1 2 α E 2 for low frequency, polarizability is positive (α > 0) hence atoms go to regions of large E 2 in such a case, focused laser beam traps atoms at waist, where E 2 is largest 3

4 In general, polarizability depends on frequency ω of applied electric (laser) field for alkali metal atom with single valence electron in ns state, real part of frequency-dependent polarizability has the form α(ω) α(0) ω 2 SP ω 2 SP ω2 here ω SP is transition frequency ns np, where n = 2 for Li, n = 3 for Na, n = 4 for K, n = 5 for Rb, etc. for ω < ω SP, polarizability α(ω) is positive and laser field is attractive (called red-detuned ) for ω > ω SP, polarizability α(ω) is negative and laser field is repulsive (called blue-detuned ) 4

5 As an introduction to synthetic gauge fields, recall transformation to rotating frame H H Ω L rewrite last term as Ω L Ω r p = Ω r p combine with free-particle kinetic energy to obtain p 2 (p MΩ r)2 M Ω r 2 Ω r p = 2M 2M 2 where last term is a (negative) centrifugal potential that opposes an applied trapping potential we can now interpret the term MΩ r as an effective gauge potential A eff since it appears in the familiar combination (p A eff ) 2 /(2M) more generally, whenever the Hamiltonian contains a term linear in p, the coefficient can be taken to define an effective vector potential A eff 5

6 In classical physics, a particle with charge q experiences a Lorentz force F L = qv B in a magnetic field B quantum view is different, relying on vector potential A, where B = A when charged particle travels from r 1 to r 2 along path C, wave function acquires phase S = q 2 1 A(r ) dr if can create such phase, even a neutral atom can experience synthetic gauge field hence new perspective is on phase engineering of quantum wave function Spielman (NIST) proposed and then demonstrated synthetic magnetic effects for neutral atoms (2009) in optical traps 6

7 This field of synthetic gauge fields is growing rapidly, especially in connection with optical lattices. Below is partial list of important applications involving optical lattices creation of complex tunneling amplitudes by shaking an optical lattice (Sengstock 2012) uniform synthetic flux in 2D optical lattice with 1 2 (Bloch, Ketterle 2013) flux per plaquette creation of Haldane model with complex next-nearest neighbor tunneling in a deformed honeycomb lattice (Esslinger 2014) chiral edge currents in 1D lattices with finite transverse synthetic dimensions (Fallani, Spielman 2015) We ll come back to this topic later, but for now concentrate on synthetic gauge fields for neutral atoms in an optical trap 7

8 For atomic system, rely on strong laser fields coupling two atomic states typical laser coupling yields set of dressed eigenstates ψ j (r) focus on ground state ψ 1 (r) (assumed nondegenerate) spatial dependence of ψ 1 along path 1 2 yields Berry s phase for wave function S B = A(r ) dr here, A = i ψ 1 ψ 1 is the synthetic gauge field note that S B requires explicit spatial dependence of state ψ 1 (r) here we deal with neutral atoms take effective charge as 1, so A has dimension of momentum 8

9 assume specific model with spatially dependent parameters χ(r) and η(r) ( ) cos(χ/2) ψ 1 (r) = e iη sin(χ/2) find A = (1 cos χ) η 2 hence state ψ 1 (r) must have spatially dependent phase η(r) correspondingly induced synthetic magnetic field is B = A detailed calculation gives B = 1 2 (cos χ) η basic conclusion is that synthetic magnetic field requires 1. spatial dependence of both phase η and polar angle χ 2. in addition, spatial gradients must be non-collinear to assure nonzero vector product in practice, Spielman (NIST) takes η x and χ y, leading to synthetic Landau gauge with A x y and nearly uniform B Bẑ 9

10 for original theory, see I. B. Spielman, PRA 79, (2009) NIST group used this technique to create vortices in nonrotating superfluid relevant angular momentum comes from synthetic electromagnetic field for experiment, see Y.-J. Lin et al., Nature 462, 628 (2009) vortex cores are dark spots remarkable demonstration of power of synthetic gauge field 10

11 How exactly does this NIST scheme work? trap a BEC in red-detuned (typically IR) dipole laser apply two counter-propagating Raman laser beams along ± ˆx with k 1, ω 1 and k 2, ω 2 take k 1 k 2 k 0 absorb photon from one beam and emit photon to other opposite beam momentum transfer is ±2 k 0 ˆx due to recoil frequency transfer is controlled by acoustic-optical modulators (AOM) that determine ω = ω 1 ω 2 11

12 In addition to Raman lasers along ˆx, apply Zeeman magnetic field B along ŷ splitting the three m F levels for the F = 1 manifold in practice, need correction to linear Zeeman theory quadratic Zeeman effect shifts states with m F state with m F = 0 = ±1 upward relative to can isolate two states 1 = 1, 0 and 2 = 1, 1 they serve as pseudo-spin in a two-component basis 12

13 In two-component basis, Raman beams act to couple the two pseudo-spin states can vary applied magnetic field to induce detuning δ/2 from Raman resonance associated with applied Raman field (see figure) 13

14 Most importantly, Raman lasers induce Rabi coupling between and relevant matrix element is from dipole coupling d E = ( Ω/2) e 2ik 0x where the phase factor reflects the spatial dependence of the total laser field obtained as the vector sum E 1 (x, t) + E 2 (x, t) here Ω is called the Rabi frequency (fixed by the applied laser strength) and k 0 k 1 k 2 is the Raman laser wave number in basis of two states and, find matrix single-particle Hamiltonian ( ) (p 2 /M) + δ Ω e 2ik 0x H 0 = 1 2 Ω e 2ik 0x (p 2 /M) δ here, k 0, δ and Ω are under experimental control 14

15 Spatial dependence of off-diagonal elements 1 2 Ωe±2ik 0x complicates the problem introduce unitary matrix U = ( e ik 0 x 0 ) 0 e ik 0x apply this unitary transformation to wave function induces a new single-particle Hamiltonian with spin-orbit structure H SO = U H 0 U ( ( 2 /M)( i + k 0 ˆx) 2 + δ Ω ) = 1 2 Ω ( 2 /M)( i k 0 ˆx) 2 δ ] [( 2 /M) ( i I + k 0 ˆxσ z ) 2 + δσ z + Ωσ x =

16 in essence, Raman beams shift the minima of the two pseudo-spin dispersion relations to new and different local minima located at k 0 ˆx shifted minima represent gauge field: (p A) 2 with A = k 0 σ z ˆx, where σ z is usual Pauli matrix in addition, Rabi coupling induces off-diagonal coupling ( Ω/2) σ x here spin-orbit coupling involves non-commuting Pauli matrices σ z and σ x this spin-orbit Hamiltonian leads to nontrivial single-particle dynamics 16

17 To understand new physics, temporarily ignore trap potential and interaction energy in this approximation, seek 1D plane-wave solutions e ikx use k 1 0 as unit of length and recoil energy E R = 2 k 2 0/2M as unit of energy, find dimensionless spin-orbit coupled single-particle Hamiltonian ( ) (k + 1) 2 + δ/2 Ω/2 H SO = Ω/2 (k 1) 2 δ/2 readily find the eigenvalues E ± (k) = k ± 1 2 (4k + δ)2 + Ω 2 focus on lower (minus) band Two cases are of special interest: 17

18 Case 1 assume Ω >> 4 (namely strong Rabi coupling) expand E (k) in powers of Ω 1 and find E (k) Ω }{{} overall shift + Ω 4 Ω }{{} effective mass ) 2 ( k δ Ω 4 }{{} gauge potential shift first two terms are simply overall downward shift of energy factor in front of quadratic term is effective mass + note that minimum of parabolic dispersion relation is shifted away from origin this shift identifies the synthetic vector potential A x = δ/(ω 4) 18

19 basic result: detuning δ determines the synthetic gauge field A x δ in experiment, magnetic field gradient gives δ(y) δ y with constant δ hence find A x δ y with δ as control parameter this is like Landau gauge A x y A gives uniform synthetic magnetic field along ẑ remarkable result: real magnetic-field gradient δ along ŷ induces synthetic magnetic field along ẑ for neutral atoms, this is analogous to a rotation with MΩ B reflects close analogy between Lorentz force and Coriolis force in rotating frame we as observers are effectively in the rotating frame hence vortices are at rest in laboratory frame 19

20 Lin et al., Nature 462, 628 (2009) used this method to create stationary vortices in non-rotating condensate why is shape increasingly distorted from circular form? recall synthetic gauge field A ˆx y like Landau gauge A x y turn off A, which generates synthetic electric field E = A/ t pulsed electric field produces impulsive shear, as seen in above figure 20

21 Case 2 Return to H SO written in matrix notation H SO = 1 2 [M 1 (pi + k 0 ˆxσ z ) 2 + δσ z + Ωσ x ] displays presence of matrix synthetic gauge field A x = k 0 σ z cross term in kinetic energy is spin-orbit coupling p A = p x k 0 σ z here have only single component of A, so this is not non-abelian note that this spin-orbit coupling p x σ z is a little different from that familiar in atomic physics L σ = r p σ present form of spin-orbit coupling occurs in semiconductor physics it arises from semiconductor band structure depending on details, it is known as Rashba or Dresselhaus coupling 21

22 Focus on special case of small Rabi coupling Ω and zero detuning δ = 0 corresponding energy dispersion relation is E ± (k) = k ± k2 + Ω 2 if Rabi coupling vanishes (Ω = 0), get two shifted parabolas (k ± 1) 2 that intersect at k = 0 22

23 when Ω 0, avoided crossing splits dispersion curves into upper and lower band Lin et al., Nature 471, 83 (2011) mapped out this behavior in their study of spin-orbit coupling in cold 87 Rb atoms gray curves show two shifted parabolas for Ω = 0 colored curves show successive splitting of upper and lower bands with increasing Ω 23

24 For Ω < 4, find two local minima at k 2 = 1 (Ω/4) 2 In contrast, for Ω > 4, find single minimum at k = 0 experiments map out this locus of points note qualitatively different behavior for Ω < 4 (two minima at k = ±Ω/4) and for Ω > 4 (single minimum at k = 0) 24

25 There are many proposals for more symmetric spin-orbit coupling typical case is pure Rashba 2D coupling H R = κ(p x σ y p y σ x )/M this is preferred theoretical model owing to high symmetry here κ is a coupling constant with dimension of wavenumber equivalently, H R = κẑ p σ/m now have two components of synthetic vector potential A x = κ σ y and A y = κ σ x note that [A x, A y ] = 2 κ 2 [σ y, σ x ] = 2i 2 κ 2 σ z 0 hence the two components of A do not commute in such a case, these are called non-abelian gauge fields 25

26 for pure Rashba coupling, minimum of dispersion relation is circle of radius p = κ in 2D momentum space like Mexican-hat potential this Hamiltonian is like massless Dirac equation in 2D with familiar Dirac cone (as in graphene) in practice, need to include other control terms in Hamiltonian detuning σ z now plays role of mass and lifts degeneracy at avoided crossing, splitting upper and lower bands various theory papers suggest using intense laser beams or rapid magnetic pulses around ˆx and ŷ to create symmetric Rashba coupling despite great efforts, such coupling has not yet been achieved 26

27 2 Synthetic gauge fields in optical lattices The NIST synthetic gauge fields occur in a BEC that is trapped with harmonic confinement. Optical lattices can provide a different and large class of synthetic gauge fields, and I here review a few of them. a. shaken lattices assume a one-dimensional optical lattice generated by two counter-propagating laser beams gives optical potential V lab (x) = 1 2 V 0 cos(2k 0 x) in laboratory frame strength V 0 fixed by laser intensity here k 0 is laser wave number let position of the lattice be modulated in predetermined way specified by X 0 (t) 27

28 hence lattice potential in laboratory becomes time-dependent, with V lab (x, t) = 1 2 V 0 cos{2k 0 [x X 0 (t)]} where X 0 (t) is a given time-dependent periodic function with period T in lab, now have time-dependent single-particle Hamiltonian H lab (t) = p 2 /(2M) + V lab (x, t) perform a series of unitary transformations to obtain time-dependent Hamiltonian in co-moving frame shift the position to local coordinate with time-dependent operator U 1 = exp[ix 0 (t)p/ ] as anticipated, yields U 1 xu 1 = x + X 0(t) and ( U 1 i ) U 1 t = i t Ẋ0(t) p 28

29 corresponding modified Hamiltonian in moving frame becomes H(t) = [p MẊ0(t)] 2 2M V 0 cos(2k 0 x) M 2 [Ẋ0(t)] 2 perform two additional unitary transformations (Arimondo et al., arxiv: ) to eliminate shift in momentum and time-dependent energy shift find new Hamiltonian H 0 + H 1 (t), where H 0 is original laboratory lattice Hamiltonian (now in the moving frame) and H 1 (t) is periodic with H 1 (t + T ) = H 1 (t) new oscillatory perturbation is H 1 (t) = F (t)x note the linear dependence on position through explicit factor x here, inertial force F (t) = MẌ0(t) provides the periodic perturbation 29

30 assume small tunneling between adjacent lattice sites interpret one-body Hamiltonian H 0 as tight-binding energy spectrum E(k) = 2J cos(kd), where d is the lattice spacing and J is an overlap integral involving two localized Wannier functions (hopping amplitude) assume that the driving force is fast compared to other low-energy scales (like width of lowest band), but small compared to gap to next lattice band periodic term in Hamiltonian allows use of Floquet theory (like Bloch theory but now in time domain) this theory is a bit intricate, and I here rely on simpler approach that yields the same result 30

31 use semiclassical dynamics for wave packet [Holthaus, PRL 69, 351 (1992)] ṗ = H x and where H( k, x, t) = 2J cos(kd) F (t)x hence k = F (t)/ and ẋ = (2Jd/ ) sin kd ẋ = H p integrate first equation to find k(t) = k t 0 dt F (t ) where k 0 = k(0) for definiteness, take simple sinusoidal oscillation: F (t) = F 0 sin ωt direct integration gives k(t) = k 0 + F 0 ω substitute into other equation and find ẋ(t) = 2Jd sin [ k 0 d + F 0d ω (1 cos ωt) ] (1 cos ωt) since the driving frequency is fast, average over one period T = 2π/ω with T = T 1 T 0 dt 31

32 standard mathematics shows that cos (A cos ωt) T = J 0 (A), where J 0 (A) is Bessel function of real argument A averaging ẋ(t) yields ẋ T = 1 T T 0 dt ẋ(t) = 2Jd sin ( k 0 d + F ) ( ) 0d F0 d J 0 ω ω here, F 0 d/ ω is determined by the strength of the shaking of the optical lattice can interpret this result as effective tunneling integral J eff = JJ 0 (F 0 d/ ω) modified by the shaking of the lattice for small argument, Bessel function J 0 (x) reduces to 1, but it vanishes at a finite value (x 2.40) and then becomes negative near zero of J 0, effective tunneling amplitude J eff vanishes, so particle is strongly localized simply by shaking the lattice with appropriate amplitude stronger shaking of optical lattice can change sign of J eff, which inverts the band structure here phase can only be 0 or π, since everything is real 32

33 This simple toy model shows how a one-dimensional shaken optical lattice can yield a renormalized hopping matrix element J eff /J = J 0 (F 0 d/ ω) that can change sign depending on amplitude F 0 of driving force but remains real With clever choice of driving force, it is also possible to induce effective hopping amplitudes J eff that have complex phases (Sengstock, Hamburg, 2012) for this discussion, it is preferable to use second-quantized bosonic operators b i and b i that create and destroy a particle at lattice site i they obey familiar commutation relations [b i, b j ] = δ ij 33

34 driven Hamiltonian now becomes [see, for example, Eckardt et al. EPL 89, (2010) and Struck et al., PRL 108, (2012)] H(t) = ij J ij b i b j + i v i (t) n i + H on site here ij denotes sum over nearest neighbors and J ij are real and positive last term includes time independent terms like interactions and trap potential assume lattice in one or more dimensions, with periodic force F (t) = M R 0 (t), directly generalizing previous 1D result F (t) = MẌ0(t) in this case, v i (t) = r i F (t) has dimension of energy and n i = b i b i is number operator for site i crucial assumption is that v i (t + T ) = v i (t) is periodic with period T also v i (t) has zero average over one period, with v i = T 1 T 0 dt v i(t) = 0 hence Floquet theory again applies, but prefer simpler approach 34

35 introduce unitary operator U(t) = exp i n j W j (t) j where W j (t) = t 0 dτ v j(τ) T 1 T 0 dt t 0 dτ v j(τ) and last term is a constant that ensures zero average over one cycle note that dw j (t)/dt = v j (t) by construction, this unitary operator eliminates time-dependent term H 1 (t) = i v i(t) n i in the transformed time-dependent Hamiltonian H(t) write state as ψ = U ψ, leading to new Hamiltonian H = U HU i U ( U/ t) it is easy to check that second term explicitly eliminates H 1 (t) from new Hamiltonian H now need to evaluate the first term in detail, which takes some careful analysis 35

36 need to transform typical hopping term: U b i b ju rewrite as U b i U U b j U for any site, note that [b, n] = b, so that bn k = (n + 1) k b hence b e λn = e (n+1)λ b and U b j U = e iw j/ b j similarly U b i U = eiw i/ b i finally U b i b j U = b i b j exp[i(w i W j )/ ] acquires an explicit phase factor final result for modified Hamiltonian is H (t) = ij J ij e [ i (W i (t) W j (t))] b i b j + H on site note that only remaining time dependence is in the phases 36

37 Now need to choose explicit form for the period forcing function and then evaluate the phase factors W i (t) already saw that forcing F (t) = sin ωt (pure sine wave forcing) yields real renormalized hopping amplitude Struck et al., PRL 108, (2012) use more complicated form: sin ω 1 t (single sine wave) of length T 1 = 2π/ω 1 followed by zero forcing for time T 2, with a total period T = T 1 + T 2 this form breaks time-reversal symmetry and leads to complex phase for renormalized hopping amplitude 37

38 Final result assumes that forcing frequency 2π/T is large compared to other natural frequencies. It gives complex effective hopping amplitude J ij e iθ ij = J ij e i (W i W j )/ T where T denotes an average of the full period T = T 1 + T 2 for one-dimensional system, detailed integration gives { W i (t) = z if 0 cos ω 1 t T 2 /T for 0 < t < T 1 ω 1 T 1 /T for T 1 < t < T 1 + T 2 note z i z j = ±d for nearest-neighbor hopping in one dimension 38

39 use of previous identity for averaging gives J eff J bare = T 2 T eik(t 1/T ) + T 1 T J 0(K) e ik(t 2/T ) = J eff J bare e iθ where K = F 0 d/ ω 1 is the forcing amplitude note that right and left hopping induce opposite phase this phase also shifts tight-binding spectrum E(k) = 2 J eff cos(kd θ) as expected for usual coupling based on (p A) 2 39

40 Peierls (1933) noted that this phase can be interpreted as a line integral of an actual (or synthetic) vector potential in going from site i to adjacent site j quoted in seminal paper on square two-dimensional lattice in uniform magnetic field [Hofstadter, Phys. Rev. B 14, 2239 (1976)] along a straight line note that θ ji = θ ij θ ij = j i dr A(r) this discrete phase is defined only on links joining lattice sites conceptually θ ij is quite different from the continuous phase of the condensate wave function in a trapped condensate, discussed in the context of the NIST experiments 40

41 b. laser-assisted tunneling in two-dimensional square lattices We have seen how to generate a complex tunneling amplitude in a one-dimensional lattice this system can only generate a synthetic magnetic field B = A if the effective vector potential has nontrivial spatial dependence and extends in two or more dimensions seminal work from Munich [Aidelsburger et al., PRL 107, (2011)] created a staggered flux in two-dimensional optical square lattice, with adjacent vertical columns having alternating flux per plaquette along y impose standard periodic optical lattice along x, use two lasers with short wavelength λ s and long wavelength 2λ s to make standing-wave superlattice with alternating local minima (high and low) separated by an energy that inhibits tunneling along the x direction (see figure part a below) 41

42 two Raman laser beams with k 1 (k 2 ) and frequency difference ω 1 ω 2 = / induce resonant hopping along x with complex amplitude K this quantity is the hopping matrix element of the laser electric field between adjacent sites with phase factor ±δk R mn = ± 1 2π(m+n) for hopping along x from low site to high site (+ to right and to left) around a closed square plaquette, acquire net phase φ = ±π/2, alternating in sign (see figure part b above) altering angles of Raman beams would alter value of phase per plaquette 42

43 generates effective hopping Hamiltonian H = ) (Ke ±iδk R a R a R+d x + Ja R a R+d y R + H.c. turn off trapping lasers and see different bosonic matter-wave interference patterns depending on presence of Raman laser beams and ratio J/K measures momentum distribution after time of flight, with good agreement between theory and experiment for two values of the ratio J/K 43

44 How can one engineer a system with uniform (not staggered/alternating) flux per plaquette? This was achieved in separate independent studies from Munich [Aidelsburger et al., Phys. Rev. Lett. 111, (2013)] and MIT [Miyake et al., Phys. Rev. Lett. 111, (2013)] again use Raman laser-assisted tunneling in x direction, but now use lattice with uniform tilt, either from gravity or from magnetic field gradient tune laser-frequency difference close to energy difference between levels in adjacent tilted wells note that all hopping processes now move to right 44

45 for general phase φ mn = δk R mn = mφ x + nφ y with phases φ x and φ y, find following picture of flux per plaquette flux in each plaquette comes from sum of phases around the square boundary in positive (anticlockwise) direction net flux arises only from y dependent term φ y since φ x cancels out these experiments realize Harper-Hofstadter model Hamiltonian for twodimensional square lattice in uniform magnetic field 45

46 spatially dependent phase imprinted by Raman lasers gives a perturbative hopping amplitude K = 1 2 Ω d 2 r w(r R m,n ) e iδk r w(r R m,n d ˆx) Ke iδk R m,n where w(r) denotes a localized Wannier function (related to familiar Bloch function by essentially a Fourier transform) applied Raman beams determine the associated synthetic vector potential A = [ (k x x + k y y)/d] ˆx here, it is not exactly Landau gauge because of first term, which is a total derivative 46

47 very recently, MIT group has achieved BEC in this system (arxiv: ), as verified by matter-wave interference seen after turning off trap potential for various values of band filling note good agreement between experimental images (lower images) and theoretical predictions (upper images) 47

48 Separately, Esslinger group (ETH Zurich) has realized Haldane Hamiltonian for deformed but topologically equivalent honeycomb lattice with complex next-nearest neighbor hopping [Haldane, Phys. Rev. Lett. 61, 2015 (1988); Jotzu et al., Nature 515, 237 (2014)] no time to discuss this system in detail, but it is equally fascinating 48

49 c. synthetic dimensions Two very recent preprints have achieved long-strip geometry (Hall ribbon) using one-dimensional optical lattice and three (can in principle be more) internal states to create transverse synthetic dimension [Florence: arxiv: v1 Mancini et al. using fermionic 173 Yb and NIST: arxiv: v1 Stuhl et al. using bosonic 87 Rb, based in part on theoretical proposal Celi et al., PRL 112, (2014)] here concentrate on NIST experiment since I focus on bosons create effective flux with Aharonov-Bohm phase φ AB and synthetic flux φ AB /2π 4/3 per plaquette, which is equivalent to 1/3 because physics is 2π periodic in applied flux use hyperfine manifold F = 1 with Zeeman states m F = 1, 0, 1 serving as transverse synthetic dimension 49

50 geometry is long one-d optical lattice in x direction and three sites in transverse (synthetic) dimension using the three F = 1 hyperfine states m F = 1, 0, 1 either use RF fields to induce transitions with real amplitude t s between hyperfine states with no induced Aharonov-Bohm flux φ AB or use Raman beams to induce transitions with finite momentum transfer and hence induced Aharonov-Bohm flux φ AB for hopping amplitude t x e ±iφ AB in x direction (see figure) 50

51 schematic geometry of experiment use both φ AB = 0 and φ AB > 0 to compare outcomes load atoms adiabatically in various m values without or with synthetic flux φ AB 51

52 top (A,B) shows ground state with no flux with both absorption image and fractional population bottom three panels show atoms loaded into upper edge, bulk, and lower edge for φ AB /2π = 4/3, with localization along synthetic dimension 52

53 in presence of synthetic flux, atoms prepared in m = 0 site tunnel and execute skipping orbits at top and bottom of the three-state ribbon system prepared at m = ±1 site develops weakly damped periodic skipping orbits analogous to chiral edge magnetoplasmons in quantum Hall physics arxiv: v1 contains other convincing experimental demonstrations 53

54 Conclusions spin-orbit coupled trapped BECs have been created with Raman beams and applied real magnetic fields for large Rabi frequency Ω, use magnetic field gradient to make synthetic gauge field in Landau gauge hence mimic rotation and create vortices in a non-rotating condensate for zero detuning and small Rabi coupling, theory has simple spin-orbit structure that can be verified experimentally optical lattices have served to create synthetic gauge fields using many approaches here focus on shaken lattices, laser-assisted tunneling, and synthetic dimensions using internal atomic states other new techniques are developing rapidly 54