Light-induced gauge potentials for ultracold atoms

Light-induced gauge potentials

for ultracold atomsGediminas Juzeliūnas

Institute of Theoretical Physics and Astronomy,Vilnius University, Vilnius, Lithuania

National Tsing-Hua University, Hsinchu, 14 May 2012

Collaboration (in the area of the artificial

magnetic field for cold atoms) P. Öhberg & group, Heriot-Watt University,

Edinburgh M. Fleischhauer & group, TU Kaiserslautern, L. Santos & group, Universität Hannover J. Dalibard, F. Gerbier & group, ENS, Paris I. Spielman, D. Campbel, C. Clark and J.

Vaishnav, NIST, USA M. Lewenstein, ICFO, ICREA, Barcelona

Quantum Optics Group @ ITPA, Vilnius University

V. Kudriasov, J. Ruseckas, G. J., A. Mekys, T. Andrijauskas Not in the picture: V. Pyragas and S. Grubinskas

Quantum Optics Group @ ITPA, Vilnius University Research activities: Light-induced gauge potentials for cold atoms

(both Abelian and non-Abelian) Ultra cold atoms in optical lattices Slow light (with OAM, multi-component, …) Graphene Metamaterials

Quantum Optics Group @ ITPA, Vilnius University Research activities: Light-induced gauge potentials for cold atoms Ultra cold atoms in optical lattices

This talk:A combination of first two topics

Quantum Optics Group @ ITPA, Vilnius University Research activities: Light-induced gauge potentials for cold atoms Cold atoms in optical (flux) lattices

This talk:A combination of first two topics

OUTLINE Background

Optical lattices Geometric gauge potentials

Optical flux lattices (OFL) Non-staggered artificial magnetic flux Ways of producing of OFL Non-Abelian gauge potentials Conclusions

Ultra-cold atomic gases 1995: Creation of the first Atomic Bose-

Einstein Condensate (BEC)

T<Tcrit~10-7K (2001 Nobel Price in Physics)

1999: Creation of the Degenerate Fermi gas of atoms: T<TF~10-7K

Currently: A great deal of interest in ultracold atomic gases

Cold atoms are trapped using:

1. (Parabolic) trapping potential produced by magnetic or optical means:

Cold atoms are trapped using:

1. (Parabolic) trapping potential produced by magnetic or optical means:

2. Optical lattice(periodic potential):

Optical lattices (ordinary): [Last 10 years] A set of counter-propagating light beams

(off resonance to the atomic transitions)

I. Bloch, Nature Phys. 1, 23 (2005)

Optical lattices (ordinary) A set of counter-propagating light beams

(off resonance to the atomic transitions) Atoms are trapped at intensity minima (or

intensity maxima) of the interference pattern (depending on the sign of atomic polarisability)

Vdip r dE r L E r 2

Optical lattices (ordinary) A set of counter-propagating light beams

(off resonance to the atomic transitions) Atoms are trapped at intensity minima (or

intensity maxima) of the interference pattern (depending on the sign of atomic polarisability)

2D square optical lattice:

3D cubic optical lattice:

Vdip r dE r L E r 2

Optical lattices (more sophisticated) Triangular or hexagonal optical lattices using

three light beams (propagagating at 1200)

three light beams (propagagating at 1200)Experiment:

Optical lattices (more sophisticated) Triangular or hexagonal optical lattice using

(a) Polarisations are perpendicular to the plane Triangular lattice

(b) Polarisations are rotating in the plane Hexagonal lattice: Analogies with electrons graphene

(b) Polarisations are rotating in the plane Hexagonal (spin-dependent) lattice

[traps differently atoms in different spin states]

Ultracold atoms Analogies with the solid state physics

Fermionic atoms ↔ Electrons in solids Atoms in optical lattices – Hubbard model Simulation of various many-body effects

Advantage : Freedom in changing experimental parameters

that are often inaccessible in standard solid state experiments

Ultracold atoms Analogies with the solid state physics

Fermionic atoms ↔ Electrons in solids Atoms in optical lattices – Hubbard model Simulation of various many-body effects

Advantage : Freedom in changing experimental parameters

that are often inaccessible in standard solid state experiments

e.g. number of atoms, atom-atom interaction, lattice potential

Trapped atoms - electrically neutral species No direct analogy with magnetic phenomena by

electrons in solids, such as the Quantum Hall Effect (no Lorentz force)

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

Trapped atoms - electrically neutral particles No direct analogy with magnetic phenomena by

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

Trapped atoms - electrically neutral particles No direct analogy with magnetic phenomena by

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

Trap rotationHamiltonian in the rotating frame [see e.g. A. Fetter, RMP 81, 647 (2009)] Trapping potentialor

Trap rotationHamiltonian in the rotating frame [see e.g. A. Fetter, RMP 81, 647 (2009)] Trapping potential

rotation vector

or rotation vector

Effective vector potential

or rotation vector

Effective vector potential(constant Beff~Ω)Coriolis force (equivalent to Lorentz force)

or rotation vector

Effective vector potential Centrifugal potential (constant Beff~Ω) (anti-trapping) Coriolis force (equivalent to Lorentz force)

Trap rotation: Summary of the main features

Constant Beff: Beff Trapping frequency: Landau problem

ROTATION Can be applied to utracold atoms both in usual

traps and also in optical lattices

(a) Ultracold atomic cloud (trapped):

(b) Optical lattice:

ROTATION 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 Using (unconventional) optical lattices

Initial proposals: J. Ruostekoski, G. V. Dunne, and J. Javanainen,

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

Rev. Lett. 94, 086803 (2005) Beff is produced by inducing an asymmetry

in atomic transitions between the lattice sites. Non-vanishing phase for atoms moving along

a closed path on the lattice (a plaquette) → Simulates non-zero magnetic flux → Beff ≠ 0

Effective magnetic fields without rotation Optical square lattices

D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003) J. Dalibard and F. Gerbier, NJP 12, 033007 (2010). -Ordinary tunneling along x direction (J). -Laser-assisted tunneling between atoms in different

internal states along y axis (with recoil along x).

J’exp(-ikx1) J’exp(ikx2)

J Atoms in different internal states (red or yellow)

are trapped at different lattice sites

J Atoms in different internal states (red or yellow)

are trapped at different lattice sites

J Non-vanishing phase for the atoms moving over a

plaquette: S=k(x2-x1)=ka

plaquette: S=k(x2-x1)=ka → Simulates non-zero magnetic flux (over plaquette)

Staggered flux!

plaquette: S=k(x2-x1)=ka non-zero magnetic flux: Experiment: M. Aidelsburger et al., PRL 107, 255301

(2011)

Effective magnetic fields without rotation Optical square lattices Experiment: M. Aidelsburger, M. Atala, S. Nascimbène,

S. Trotzky, Yu-Ao Chen and I Bloch, PRL 107, 255301 (2011)

-Ordinary tunneling along y direction. -Laser-assisted tunneling along x axis (with recoil).

Non-zero magnetic flux over a plaquette

Effective magnetic fields without rotation Optical lattices: The method can be extended to create Non-Abelian gauge potentials

(Laser assisted, state-sensitive tunneling)A proposal:

K. Osterloh, M. Baig, L. Santos, P. Zoller and M. Lewenstein, Phys. Rev. Lett. 95, 010403 (2005)

Distinctive features: No rotation is necessary No lattice is needed Yet a lattice can be an important ingredient in

creating Beff using geometric potentials Optical flux lattices

Effective magnetic fields without rotation-- using Geometric Potentials

Geometric potentials Emerge in various areas of physics (molecular,

condensed matter physics etc.) First considered by Mead, Berry, Wilczek and Zee

and others in the 80’s (initially in the context of molecular physics).

More recently – in the context of motion of cold atoms affected by laser fields (Currently: a lot of activities)

See, e.g.: J.Dalibard, F. Gerbier, G. Juzeliūnas and P. Öhberg, Rev. Mod. Phys. 83, 1523 (2011).

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

Creation of Beff using geometric potentials

( r-dependent “dressed” eigenstates)

(includes r-dependent atom-light coupling)

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

(for c.m. motion)

Adiabatic atomic energies

Full state vector:

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

Non-degenerate state with n=1 Adiabatic atomic energies

Full state vector:

Adiabatic approximation

(only the atomic internal state with n=1 is included)

Full state vector:

(only the atomic internal state with n=1 is included)

Full state vector:

What is the equation of motion for ?

Adiabatic approximation:

Non-degenerate state with n=1

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

ˆ H p A11 2

2MV (r)1(r)

Effective Vector potential appears

ˆ H p A11 2

2MV (r)1(r)

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

ˆ H p A11 2

2MV (r)1(r)

- effective magnetic field

ˆ H p A11 2

2MV (r)1(r)

- effective magnetic field (non-trivial situation if )

Large possibilities to control and shape the effective magnetic field B by changing the light beams

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

To summarise

Light induced effective magnetic field is due to Spatial dependence of atom-light coupling

Light induced effective magnetic field can be due to

1. Spatial dependence of laser amplitudes2. Spatial dependence of atom-light detuning3. Spatial dependence of both the laser

amplitudes and also atom-light detuning (e.g. optical flux lattices)

1. Spatial dependence of laser amplitudes

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

Artificial Lorentz force (due to photon recoil)

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

Total magnetic flux is proportional to the sample length L:

(one can not increase the total flux in the transverse direction)

Total magnetic flux is proportional to the sample length L:

(one can not increase the total flux in the transverse direction)

No lattice

No translational symmetry for shifted beams (in the transverse direction):

Light induced effective magnetic field due to Spatial dependence of laser amplitudes Spatial dependence of atom-light detuning

Position-dependent detuning δ

Position-dependent detuning δ=δ(y) B≠0

Light induced effective magnetic field due to Spatial dependence of atom-light detuning

Magnetic flux is again determined by the sample length (rather than the area)!

One can not create large magnetic flux

Detuning δ=δ(y)

1. Spatial dependence of laser amplitudes2. Spatial dependence of atom-light detuning3. Spatial dependence of both the laser

amplitudes and also atom-light detuning

Effective gauge potentials – due to position-dependence of both A) Detuning and

B) Laser amplitudes e.g. Optical flux lattices

N. R. Cooper, Phys. Rev. Lett. 106, 175301 (2011)

Magnetic flux is determined by the area (!!!) of atomic cloud

Related earlier work: A. M. Dudarev, R. B. Diener, I. Carusotto, and Q. Niu,

Phys. Rev. Lett. 92, 153005 (2004).

Magnetic flux is determined by the area (!!!) of atomic cloud

Two atomic internal states

Position-dependent detuning Δ(r) 2Ωz

Position-dependence of the (complex) Rabi frequencies of atom-light coupling Ω± (r) Ωx±iΩy

Position-dependent detuning Δ(r) 2Ωz Position-dependence of the (complex) Rabi frequencies of

atom-light coupling Ω± (r) Ωx±iΩy

Atom-light Hamiltonian:

ˆ H 0 r hz x iy

x iy z

(2×2 matrix)

Position-dependent detuning Δ(r) 2Ωz Position-dependence of the Rabi (complex) frequencies of

atom-light coupling Ω± (r) Ωx±iΩy

Atom-light Hamiltonian:

ˆ H 0 r hz x iy

x iy z

Ωx≠0, Ωy≠0, Coupling between the atomic states

ˆ H 0 r has position-dependent eigenstates , j=1,2

Effective vector potential for atomic motion in the lower dressed state :

ˆ H 0 r j r j r j r (j=1,2),

ˆ H 0 r hz x iy

x iy z

A r h2

See, e.g.: J.Dalibard, F. Gerbier, G. Juzeliūnas and P. Öhberg. Rev. Mod. Phys. 83, 1523 (2011).

Effective vector potential for atomic motion in the lower dressed state :

ˆ H 0 r j r j r j r (j=1,2),

ˆ H 0 r hz x iy

x iy z

A r h2

(AB singularity at the “South pole”)

Optical flux latticesTwo-level system:

Coupling and detun. - specially chosen periodic funct.

ˆ H 0 r hz x iy

x iy z

x cos(x /a)

y cos(y /a)

z II sin(x /a)sin(y /a)

x cos(x /a)

y cos(y /a)

i.e. is a periodic array of vortices & anti-vortices

ˆ H 0 r hz x iy

x iy z

x cos(x /a)

y cos(y /a)

ˆ H 0 r hz x iy

x iy z

oscillates

x cos(x /a)

y cos(y /a)

ˆ H 0 r hz x iy

x iy z

oscillates (changes the sign)

x cos(x /a)

y cos(y /a)

ˆ H 0 r hz x iy

x iy z

oscillates (changes the sign)

x cos(x /a)

y cos(y /a)

ˆ H 0 r hz x iy

x iy z

oscillates (changes the sign) Periodic vector potential with

Periodic coupling and periodic detuning

Periodic vector potential with

x cos(x /a)

y cos(y /a)

ˆ H 0 r hz x iy

x iy z

Periodic vector potential with (B – non-staggered)!!!

x cos(x /a)

y cos(y /a)

ˆ H 0 r hz x iy

x iy z

Periodic vector potential with (B – non-staggered)!!!

x cos(x /a)

y cos(y /a)

ˆ H 0 r hz x iy

x iy z

ˆ H 0 r hz x iy

x iy z

x cos(x /a)

y cos(y /a)

( i.e. is a periodic array of vortices & anti-vortices) Periodic vector potential with B – non-staggered !!! (A has the AB singularities if and )

x iy 0

ˆ H 0 r hz x iy

x iy z

x cos(x /a)

y cos(y /a)

( i.e. is a periodic array of vortices & anti-vortices) Periodic vector potential with B – non-staggered !!! (A has the AB singularities if and ) Additionally - Periodic trapping potential

Optical flux lattice!!!

x iy 0

Optical flux lattice (square)

Non-zero background magnetic flux over an elementary cell& Periodic trapping potential (the lattice)

AB fluxes compensatethe non-staggered background flux

OFL can be produced Using Raman transitions between the

hyperfine states of alkali atoms (and specially shaped laser fields)Triangular optical flux latticeN. R. Cooper and J. Dalibard, EPL, 95 (2011) 66004.

Square optical flux lattice:G. Juzeliunas and I.B. Spielman, in preparation

Raman transitions between the hyperfine states of alkali atoms (and specially shaped laser fields)

Square optical flux lattice:G. Juzeliūnas and I.B. Spielman, in preparation

Time-delayed polarisation-dependent standing waves:

Raman coupling:

• Position-dependent Raman coupling• Position-dependent detuning (light shifts)

Non-staggered effective magnetic field

Non-staggered effective magnetic field (& lattice):

Optical flux lattice is produced !!!

Non-staggered effective magnetic field (& lattice):

Characteristic features of light-induced gauge potentials No rotation of atomic gas Effective magnetic field can be shaped by

choosing proper laser beams The magnetic flux can be made proportional

to the area using the optical flux lattices Extension to the non-Abelian case

If one degenerate atomic internal dressed state

-- Abelian gauge potentials

Equation of the atomic motion in the internal state

ˆ H p A11 2

2MV (r)1(r)

If more than one degenerate atomic

internal dressed state--Non-Abelian gauge

potentials

Adiabatic approximation: n=3

rn=1, n=2

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)

Adiabatic approximation: n=3

rn=1, n=2

– 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) Quasi-spin 1/2

(r, t) 1(r,t)2(r, t)

Repeating the same procedure …

Equation of motion:

rn=1, n=2

ˆ H p A 2

2MV (r)1(r)

appears due to position-dependence of

- effective vector potential

- two-comp. atomic wave-function

2x2 matrix

(r, t) 1(r, t)2(r, t)

Equation of motion:

rn=1, n=2

ˆ H p A 2

2MV (r)1(r)

2x2 matrix

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

(r, t) 1(r, t)2(r, t)

Equation of motion:

rn=1, n=2

ˆ H p A 2

2MV (r)1(r)

2x2 matrix

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

Non-Abelian gauge potentials are formed

(r, t) 1(r, t)2(r, t)

Non-Abelian gauge potentials

More than one degenerate dressed state

n=1, n=2

Tripod configuration

M.A. Ol’shanii, V.G. Minogin, Quant. Optics 3, 317 (1991) R. G. Unanyan, M. Fleischhauer, B. W. Shore, and K.

Bergmann, Opt. Commun. 155, 144 (1998) Two degenerate dark states (Superposition of atomic ground states immune of the atom-

light coupling)

Tripod configuration

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

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

excited state 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).

(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).

(two dark states)

Centre of mass motion of dark-state atoms:

A - effective vector potential (Mead-Berry connection)

- 2x2 matrix

Two component atomic wave-function

Tripod scheme

A is 2×2 matrix

Non-Abelian case if Ax, Ay, Az do not commute

B – curvature

Two degenerate dark states

Tripod scheme

A is 2×2 matrix

Non-Abelian case if Ax, Ay, Az do not commute

B – curvature

Can be achieved using a plane-wave setup

Two degenerate dark states

Three plane wave setup

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

(acting on the subspace of atomic dark states) Spin-Orbit coupling of the Rashba-Dresselhaus type

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

( 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).

(r, t) 1(r,t)2(r, t)

Three plane wave setup

Centre of mass motion of dark-state atoms:

Plane-wave solutions:

two dispersion branches with positive or negative chirality

( Rashba-type Hamilltonian)

For small k: Similarities to graphene: Dirac-type Hamiltonian:Two dispersion cones: Quasirelativistic behaviour or cold atoms

G. Juzeliūnas, J. Ruseckas, L. Santos, M. Lindberg and P. Öhberg, Phys. Rev. A 77, 011802(R) (2008)

vo≈1cm/s

Similar to electrons in graphene

Graphene – hexagonal 2D crystalof carbon atoms Electron energy spectrum near EF

Near EF: Linear dispersion, (Two cones with positive and negative ) Electrons behave like relativistic massless particles (Dirac type effective Hamiltonian)

vo≈106m/s

For small k: Similarities to graphene: Dirac-type Hamiltonian:Two dispersion cones: Quasirelativistic behaviour or cold atoms

vo≈1cm/s

Zittenbewegung of cold atoms J. Y. Vaishnav and C. W. Clark, Phys. Rev.

Lett. 100, 153002 (2008). M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P.

Öhberg, Europhys. Lett. 83, 54002 (2008). Q. Zhang, J. Gong and C. H. Oh, Phys. Rev.

A. 81, 023608 (2010).

Dispersion of centre of mass motion for cold atoms

in light fields vg=dω/dk>0

vg=dω/dk<0

- sharp Mexican hat

Unconventional Bose-Einstein condensation T. D. Stanescu, B. Anderson and V. M. Galitski,

PRA 78, 023616 (2008);C. Wang et al, PRL 105, 160403 (2010);T.-L. Ho and S. Zhang, PRL 107, 150403 ;Z. F. Xu, R. Lu, and L. You, PRA 83, 053602 (2011);S.-K. Yip, PRA 83, 043616 (2011);S. Sinha R. Nath, and L. Santos, PRL 107, 270401 (2011); etc.

Negative refraction of cold atoms at a potential barrier

Veselago-type lenses for cold atoms

vg=dω/dk>0

vg=dω/dk<0

Veselago-type lenses for ultra-cold atoms

Double and negative reflection of atoms

G, Juzeliunas, J. Ruseckas, A. Jacob, L. Santos, P. Ohberg, Phys. Rev. Lett. 100, 200405 (2008).

- sharp Mexican hat

(a) k2 – closer to the normal (k2>k)

(b) Negatively reflected wave – closer to the surface (k2<k)

Resembles Andreev reflection Electron is converted into a hole with a

negative effective mass upon reflection

Double and negative reflection of atoms:Wave-packet simulations

Drawback of the tripod scheme: degenerate dark states are not the ground atomic dressed states collision-induced loses

Closed loop setup overcomes this drawback:

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

(with pi phase)

Drawback of the tripod scheme: degenerate dark states are not the ground atomic dressed states collision-induced loses

Closed loop setup overcomes this drawback:

Two degenerate internal ground states non-Abelian gauge fields for ground-state manifold

(with pi phase)

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

(with pi phase)

Laser fields represent counter-propagatingplane waves:

Closed loop setup produce 2D Rashba-Dresselhaus SO coupling for cold atomsD. L. Campbell, G. Juzeliūnas and I. B. Spielman, Phys. Rev. A 84, 025602 (2011)

Laser fields represent plane waves with wave-vectors forming a tetrahedron:

Closed loop setup produces a 3D Rashba-Dresselhaus SOC

B. Anderson, G. Juzeliūnas, I. B. Spielman, and V. Galitski, arXiv 1112.6022To appear in Phys. Rev. Letters.

Conclusions Abelian gauge potentials appear if there is

non-trivial spatial dependence of amplitudes or phases of laser fields (or spatial variation of atomic levels).

Non-Abelian fields can be formed using even the plane-wave setups. They can simulate the spin 1/2 Rashba-type Hamiltonian for cold atoms (even in 3D!).

Spin 1 Rashba coupling can also be generated [G.J., J. Ruseckas and J. Dalibard, PRA 81, 053403 (2010)].

For more see: J.Dalibard, F. Gerbier, G. Juzeliūnas and P. Öhberg. . Colloquium: Artificial gauge potentials for neutral atoms, Rev. Mod. Phys. 83, 1523 (2011).

Thank you!

Light-induced gauge potentials for ultracold atoms

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