Artificial electromagnetism and spin-orbit coupling for ... · Quantum Optics Group @ ITPA, Vilnius University Research activities: " Light-induced gauge potentials for cold atoms

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

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

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)

Quantum Optics Group @ 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

Vilnius (Lithuania) is in north of EUROPE

Vilnius University

Quantum Optics Group @ 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



This talk



This talk v  Slow light (multi-component) will be briefly

mentioned (at the end)

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

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

Trapped atoms - electrically neutral particles


n  A possible method to create an effective magnetic field (an artificial Lorentz force): Rotation à Coriolis force à

Trapped atoms - electrically neutral particles


n  A possible method to create an effective magnetic field: Rotation à Coriolis force (Mathematically equivalent to Lorentz force)

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:

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

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, 180401 (2002) q  D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003) q  E. Mueller, Phys. Rev. A 70, 041603 (R) (2004) q  A. S. Sørensen, E. Demler, and M. D. Lukin, Phys.

Rev. Lett. 94, 086803 (2005)

n  Beff 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 → Beff ≠ 0


n  Using (unconventional) optical lattices Initial proposals: q  J. Ruostekoski, G. V. Dunne, and J. Javanainen,

Phys. Rev. Lett. 88, 180401 (2002) q  D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003) q  E. Mueller, Phys. Rev. A 70, 041603 (R) (2004) q  A. S. Sørensen, E. Demler, and M. D. Lukin, Phys.

Rev. Lett. 94, 086803 (2005) n  A. R. Kolovsky, EPL, 93 20003 (2011)

n  Experimental implementation : q  M. Aidelsburger et al., PRL 111, 185301 (2013) q  H. Miyake et al., PRL 111, 185302 (2013)


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, 010403 (2005)


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, 010403 (2005)

(Spin-orbit coupling in optical lattices)

n  Distinctive features: q  No rotation is necessary q  No lattice is needed

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

Effective magnetic fields without rotation -- using Geometric Potentials

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 (Currently: a lot of activities)

n  Recent reviews: 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:1308.6533 (Submitted to Rep. Progr. Phys., 2013) .

Geometric potentials

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

.

Creation of Beff using geometric potentials

ë ( r-dependent “dressed” eigenstates)

ç (includes r-dependent atom-light coupling)

Atomic dynamics taking into account both internal (electronic) degrees of freedom and also center of mass motion.

(for c.m. motion)




Atomic dynamics taking into account both internal (electronic) degrees of freedom and also center of mass motion.

(for c.m. motion)




Atomic dynamics taking into account both internal degrees of freedom and also center of mass motion.

(for c.m. motion)




Atomic dynamics taking into account both internal degrees of freedom and also center of mass motion.

(for c.m. motion)

n  Adiabatic atomic energies

n  Full state vector:

– wave-function of the atomic centre of mass motion in the n-th atomic internal “dressed” state

n=1

n=2

n=3

r

Adiabatic approximation:

n=1

n=2

n=3

r

Non-degenerate state with n=1

Full state-vector


n=1

n=2

n=3

r


Full state-vector


n=1

n=2

n=3

r



Equation of the atomic c. m. motion in the internal state

n=1

n=2

n=3

r

€

ˆ H =p−A11( )2

2M+ V (r) + ε1(r)

Effective Vector potential appears (due to the position-dependence of the atomic internal “dressed” state )

€

A11 ≡A




n=1

n=2

n=3

r

€

ˆ H =p−A11( )2

2M+ V (r) + ε1(r)

Effective Vector potential appears (due to the position-dependence of the atomic internal “dressed” state )

€

A11 ≡A


R. Dum and M. Olshanii, Phys. Rev. Lett., 76,1788, 1996.



n=1

n=2

n=3

r

€

ˆ H =p−A11( )2

2M+ V (r) + ε1(r)

Effective Vector potential appears

€

A11 ≡A


- effective magnetic field



n=1

n=2

n=3

r

€

ˆ H =p−A11( )2

2M+ V (r) + ε1(r)

Effective Vector potential appears

€

A11 ≡A


- effective magnetic field (non-trivial situation if )

G.Juzeliūnas, and P. Öhberg, Phys. Rev. Lett. 93, 033602 (2004).

Counter-propagating beams with spatially shifted profiles

[G. Juzeliūnas, J. Ruseckas, P. Öhberg, and M. Fleischhauer, Phys. Rev. A 73, 025602 (2006).]

Artificial Lorentz force (due to photon recoil)

€

B =∇ ×A ≠ 0

[Interpretation: M. Cheneau et al., EPL 83, 60001 (2008).]

§

§

§

Position-dependent detuning δ

§

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

If one degenerate atomic internal dressed state

-- Abelian gauge potentials

If more than one degenerate atomic internal dressed state

--Non-Abelian (non-commuting) vector potential

(spin-orbit coupling)


n=3

r n=1, n=2

n=4

Degenerate states with n=1 and n=2

– wave-function of the atomic centre of mass motion in the n-th atomic internal “dressed” state (n=1,2)


n=3

r n=1, n=2

n=4


– wave-function of the atomic centre of mass motion in the n-th atomic internal “dressed” state (n=1,2)

- two-component atomic wave-function (spinor wave-function) à Quasi-spin 1/2

€

Ψ(r, t) =Ψ1(r,t)Ψ2(r,t)#

$ %

&

' (

Repeating the same procedure …


Equation of motion:

n=3

r n=1, n=2

n=4

€

ˆ H =p −A( )2

2M+V (r) +ε1(r)

appears due to position-dependence of

€

A


- effective vector potential

-  two-comp. atomic wave-function

2x2 matrix

Spin-orbit coupling is formed

€

Ψ(r, t) =Ψ1(r,t)Ψ2(r,t)#

$ %

&

' (


Equation of motion:

n=3

r n=1, n=2

n=4

€

ˆ H =p −A( )2

2M+V (r) +ε1(r)

appears due to position-dependence of

€

A




2x2 matrix

Spin-orbit coupling is formed

€

Ψ(r, t) =Ψ1(r,t)Ψ2(r,t)#

$ %

&

' (

H =p2 +p.A+A.p+A2

2M+V (r)+ε1(r)


Equation of motion:

n=3

r n=1, n=2

n=4

€

ˆ H =p −A( )2

2M+V (r) +ε1(r)



- effective magnetic field (curvature)


2x2 matrix

2x2 matrix €

Ψ(r, t) =Ψ1(r,t)Ψ2(r,t)#

$ %

&

' (

Non-trivial situation if


Equation of motion:

n=3

r n=1, n=2

n=4

€

ˆ H =p −A( )2

2M+V (r) +ε1(r)



- effective magnetic field (curvature)


2x2 matrix

2x2 matrix

If Ax, Ay, Az do not commute, B≠0 even if A is constant !!

à Non-Abelian vector potential is formed

€

Ψ(r, t) =Ψ1(r,t)Ψ2(r,t)#

$ %

&

' (

Non-trivial situation if

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 atom-

light coupling)

Tripod configuration

n  Two degenerate dark states n  (Superposition of atomic ground states immune to the atom-

light coupling) n  Dark states: destructive interference for transitions to the

excited state n  Lasers keep the atoms in these dark (dressed) states

(Non-Abelian) light-induced gauge potentials

for centre of mass motion of dark-state atoms: (Due to the spatial dependence of the dark states)

J. Ruseckas, G. Juzeliūnas and P.Öhberg, and M. Fleischhauer, Phys. Rev. Letters 95, 010404 (2005).

ë (two dark states)


for centre of mass motion of dark-state atoms: (Due to the spatial dependence of the dark states)

J. Ruseckas, G. Juzeliūnas and P.Öhberg, and M. Fleischhauer, Phys. Rev. Letters 95, 010404 (2005).

ë (two dark states)

A - effective vector potential à Spin-orbit coupling

An,m = i! Dn ∇Dm


Centre of mass motion of dark-state atoms:

A - effective vector potential à Spin-orbit coupling

- 2x2 matrix

ç Two component atomic wave-function

€

Ψ(r, t) =Ψ1(r,t)Ψ2(r,t)#

$ %

&

' ( An,m = i! Dn ∇Dm

Three plane wave setup

(centre of mass motion, dark-state atoms):

(acting on the subspace of atomic dark states)

à Spin-Orbit coupling of the Rashba-Dresselhaus type

Constant non-Abelian A with [Ax,Ay]~σz à B ~ ez

(à Rashba-type Hamilltonian)

– spin ½ operator

T. D. Stanescu,, C. Zhang, and V. Galitski, Phys. Rev. Lett 99, 110403 (2007). A. Jacob, P. Öhberg, G. Juzeliūnas and L. Santos, Appl. Phys. B. 89, 439 (2007).

€

Ψ(r, t) =Ψ1(r,t)Ψ2(r,t)#

$ %

&

' (

σ ⊥ = exσ x + eyσ y

Three plane wave setup

Centre of mass motion of dark-state atoms:

Plane-wave solutions:

à two dispersion branches with positive or negative chirality

(à Rashba-type Hamilltonian)

€

σkgk± = ±gk

±

σ ⊥ = exσ x + eyσ y

Dispersion of centre of mass motion for cold atoms in light fields

vg=dω/dk>0

vg=dω/dk<0

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

!/v

0 "

k/"

Dispersion of centre of mass motion for cold atoms in light fields

Lower branch – Mexican hat potential

Picture by Patrik Öhberg

kx

ky

E(kx,ky)

Dirac regime

Strongly correlatedregime

Zittenbewegung of cold atoms

Theory n  J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100, 153002

(2008). n  M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys.

Lett. 83, 54002 (2008). n  Q. Zhang, J. Gong and C. H. Oh, Phys. Rev. A. 81, 023608 (2010). n  Y-C.i Zhang, S.-W. Song and C.-F. Liu and W.-M. Liu, Phys. Rev.

A. 87, 023612 (2013).

Experiment L. J. LeBlanc et al., New. J. Phys. 15 073011 (2013). C. Qu et al. Phys. Rev. A A 88, 021604(R) (2013).

- Mexican hat potential

Unconventional Bose-Einstein condensation

Energy minimum is a ring (in the momentum space)

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

!/v

0 "

k/"

T. D. Stanescu, B. Anderson and V. M. Galitski, PRA 78, 023616 (2008); J. Larson and E. Sjöqvist, Phys. Rev. A 79, 043627 (2009) C. Wang et al, PRL 105, 160403 (2010); T.-L. Ho and S. Zhang, PRL 107, 150403 (2011); Z. F. Xu, R. Lu, and L. You, PRA 83, 053602 (2011); S.-K. Yip, PRA 83, 043616 (2011); S. Sinha R. Nath, L. Santos, PRL 107, 270401 (2011); S.-W. Su, I.-K. Liu, Y.-C. Tsai, W. M. Liu, S.-C. Gou, Phys. Rev. A 86, 023601 (2012) H. Zhai, arXiv:1403.8021 (2014) [also covers fermions] etc.

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:

D. L. Campbell, G. Juzeliūnas and I. B. Spielman, Phys. Rev. A 84, 025602 (2011)

(a) Coupling scheme (b) Geometry

(c) Uncoupled dispersion (d) Coupled dispersion

BEC

(with pi phase)

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:

n  Two degenerate internal ground states n  à non-Abelian vector poten. for ground-state manifold




BEC

(with pi phase)

n  Laser fields represent

counter-propagating plane waves:

à Closed loop setup produce 2D Rashba-Dresselhaus SO coupling for cold atoms



BEC



BEC


n  Possible implementation of the closed loop setup using the Raman transitions:




BEC

(with pi phase)

n  Laser fields represent plane

waves with wave-vectors forming a tetrahedron:

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, 235301 (2012) [Editor’s Choice paper].

(b) Tetrehedral Geometry

(d) Spectrum

n  Laser fields represent plane

waves with wave-vectors forming a tetrahedron:

à Closed loop setup produces a 3D Rashba-Dresselhaus SOC

(b) Tetrehedral Geometry

(d) Spectrum

Minimum of atomic energy - the sphere (in the momentum space)

B. Anderson, G. Juzeliūnas, I. B. Spielman, and V. Galitski, Phys. Rev. Lett. 108, 235301 (2012) [Editor’s Choice paper].

The schemes for creating 2D and 3D SOC involve

complex atom-light coupling and have not yet been

implemented

Experimental implementation of the 1D SOC (using a simpler

scheme) Y.-J. Lin, K. Jimenez-Garcıa and I. B. Spielman, Nature 471, 83 (2011).

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, 115301 (2012). P. J. Wang et al, Phys. Rev. Lett. 109, 095301 (2012). L. W. Cheuk et a;, Phys. Rev. Lett. 109, 095302 (2012). C. Qu et al, Phys. Rev. A 88, 021604(R) (2013).

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 87Rb: (transitions between mF=-1 and mF=0 are in resonance)


Large quadratic Zeeman shift à mF=1 state can be neglected



Large quadratic Zeeman shift à mF=1 state can be neglected àAn effective two level system (coupled by Raman transitions with a recoil)



Large quadratic Zeeman shift à mF=1 state can be neglected àAn effective two level system (coupled by Raman transitions with the recoil)


Two states [to be called “spin up” and “spin down” states]

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 mF=-1 and mF=0 states of the F=1 ground state manifold of 87Rb

Dispersion for δ=0 Two-level system

[Two parabolas shifted by the recoil momentum]


H =p− χσ y( )

2

2m+Ω2σ z

Dispersion for δ=0

Abelian (1D) vector potential (but non-trivial situation if Ω≠0)

(1D SOC)


H =p− χσ y( )

2

2m+Ω2σ z

Dispersion for δ=0

H =p2

2m−χ pσ ym

+Ω2σ z + const

Abelian (1D) vector potential (but non-trivial situation if Ω≠0)

(1D SOC)

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

Magnetically-induced SOC

B.M. Anderson, I.B. Spielman and G. Juzeliūnas, Phys. Rev. Lett. 111, 125301 (2013). Z.F. Xu, L. You and M. Ueda, Phys. Rev. A 87 063634 (2013).

A relatively simple scheme to create 2D or 3D SOC (without involving any light beams, any complex atom-light coupling)

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


F – spin operator For spin ½, F =σ/2


Generation of 2D SOC Magnetic pulses:

Short Magnetic pulses:

Ωx

Ωy


Ωx

Ωy Ωx & Ωy induce spin-dep. force along y & x:

à spin-dependent mom. kick along y & x:


Ωx induces spin-dep. momentum kick along y:

à Evolution operator:

Ωx

Ωy




Ωx

Ωy

à Formation of 1D SOC (in y direction)




Ωx

Ωy

à Formation of 1D SOC (in y or x direction)

Uy : x↔ y

Total evolution operator after two pulses:

H2D - 2D Rashba-type Hamiltonian (small τ ):



à Formation of 2D SOC (Zero-‐order Floquet Hamiltonian)



Additional quadratic Zeman term 2D spin-orbit coupling



Can be extended to non-delta pulses:

Non-delta pulses:

Proposed experimental setup:

Finite pulse durations:

3D SOC:

For more details see

B.M. Anderson, I.B. Spielman and G. Juzeliūnas, Phys. Rev. Lett. 111, 125301 (2013).

Spin-orbit Coupling using bilayer atoms, Q. Sun et at, arXiv:1403.4338

v 1D SOC in each layer (in 1 or 2 direct.) + Interlayer coupling

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−Q1 −Q2


v 1D SOC in each layer (in 1 or 2 direct.) + Interlayer coupling v Effectively 2D SOC is generated

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՛ܬ ՝ܬ

(a)

(c)

ȳଵభڄԦ

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ȁ ՛ଵ

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ଵെଵ െଶଶ

qxqy

Q1Q2

−Q1 −Q2 0.8 0.9 1 1.1 1.2 1.3 1.40

0.2

0.4

0.6

0.8

1

1.2

g↑↓/√g↑g↓

α

HSLSPïII

SPïII

PW SPïI

Optical lattices in extra dimensions

F=1, m = -1, 0, 1

Raman transitions between magnetic sublevels m (extra dimension)


F=1, m = -1, 0, 1


Ω0eikx - Raman Rabi frequency (recoil in x direction)


F=1, m = -1, 0, 1


1D chain of atoms (real dimension)

Ω0eikx - Raman Rabi frequency Ω0eikx - Raman Rabi frequency (recoil in x direction)

x



Tunneling in real dimension and Raman transitions in the extra dimensions yield a 2D lattice involving real and extra dimensions

F=1, m = -1, 0, 1


Ω0eikx - Raman Rabi frequency

x

γ=ka



Combination of real and extra dimensions yields strong & non-staggered magnetic flux γ=ka per 2D plaquette

F=1, m = -1, 0, 1



x

γ=ka




F=1, m = -1, 0, 1



x

γ=ka




F=1, m = -1, 0, 1



x

γ=ka


1D chain of atoms (real dimension) Sharp boundaries in extra dimension: ⟹Conducting edge states in extra dimension

F=1, m = -1, 0, 1 Ω0eikx - Raman Rabi frequency

x

γ=ka



1D chain of atoms (real dimension) Sharp boundaries in extra dimension: ⟹Conducting edge states in extra dimension


x

γ=ka

γ=ka=1.0 Ω0/t=0.1

Dispersion branches: Raman transitions between magnetic sublevels m (extra dimension)


1D chain of atoms (real dimension) Sharp boundaries in extra dimension: ⟹Conducting edge states in extra dimension ⟹Atoms with opposite spins move in opposite directions


x

γ=ka

γ=ka=1.0 Ω0/t=0.1

Dispersion branches:

Edge states



1D chain of atoms (real dimension) Sharp boundaries in extra dimension: ⟹Conducting edge states in extra dimension ⟹Atoms with opposite spins move in opposite directions



x

γ=ka

γ=ka=1.0 Ω0/t=0.1


Edge states

Landau Level



Atoms with opposite spins move in opposite directions Scattering of edge states by a short-range potential (???)



x

γ=ka

γ=ka=1.0 Ω0/t=0.1


Edge states




Spin-independent potential (road block)

x

γ=ka

γ=ka=1.0 Ω0/t=0.1


Edge states




Spin-dependent potential (perturbation for m=1)

x

γ=ka

γ=ka=1.0 Ω0/t=0.1


Edge states




Spin-dependent potential (perturbation for m=0)

x

γ=ka

γ=ka=1.0 Ω0/t=0.1


Edge states




Spin-dependent potential (perturbation for m= -1)

x

γ=ka

γ=ka=1.0 Ω0/t=0.1


Edge states




Spin-independent potential (road block)

x

γ=ka

γ=ka=1.0 Ω0/t=0.1

Edge states



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


Optical lattices in extra dimensions Scattering of edge state atoms by a short-range potential:

Scattering of edge state atoms by a short-range potential:

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


rf rf

Various possibilities: - Combination of Raman and two-photon IR transitions:

- Connecting different F manifolds via rf fields:

mF = -1 0 1

Closed boundary condition along the extra dimension

Energy spectrum of spin-1 atoms, closed boundary

Formation of Hofstadter butterfly using artificial dimensions

Conclusions n  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!).

n  Up to now experimentally only a simpler 1D SOC has been implemented (Abelian yet non-trivial).

n  Possible solutions: Magnetically induced SOC: [B.M. Anderson, I.B. Spielman and G. Juzeliūnas, PRL 111, 125301 (2013). Z.F. Xu, L. You and M. Ueda, PRA 87 063634 (2013)] or

using bilayer atoms: Q. Sun et at, arXiv:1403.4338. n  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, 043001 (2014)].

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:1308.6533 (Submitted to Rep. Progr. Phys., 2013).

n  V. Galitski and I.B. Spielman I B Nature 494, 49 (2013)

H. Zhai, arXiv:1403.8021 (2014) (Submitted to Rep. Progr. Phys., 2013)

Recent reviews:

Artificial gauge potentials

Spin-orbit coupling

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Artificial electromagnetism and spin-orbit coupling for ... · Quantum Optics Group @ ITPA, Vilnius University Research activities: " Light-induced gauge potentials for cold atoms