light-induced gauge potentials for ultracold atoms
DESCRIPTION
Light-induced gauge potentials for ultracold atoms. G ediminas 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 ). - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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
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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
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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
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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
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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
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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
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Cold atoms are trapped using:
1. (Parabolic) trapping potential produced by magnetic or optical means:
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Cold atoms are trapped using:
1. (Parabolic) trapping potential produced by magnetic or optical means:
2. Optical lattice(periodic potential):
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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)
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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)
I. Bloch, Nature Phys. 1, 23 (2005)
Vdip r dE r L E r 2
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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:
I. Bloch, Nature Phys. 1, 23 (2005)
Vdip r dE r L E r 2
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Optical lattices (more sophisticated) Triangular or hexagonal optical lattices using
three light beams (propagagating at 1200)
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Optical lattices (more sophisticated) Triangular or hexagonal optical lattices using
three light beams (propagagating at 1200)Experiment:
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Optical lattices (more sophisticated) Triangular or hexagonal optical lattices using
three light beams (propagagating at 1200)
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Optical lattices (more sophisticated) Triangular or hexagonal optical lattice using
three light beams (propagagating at 1200)
(a) Polarisations are perpendicular to the plane Triangular lattice
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Optical lattices (more sophisticated) Triangular or hexagonal optical lattice using
three light beams (propagagating at 1200)
(a) Polarisations are perpendicular to the plane Triangular lattice
(b) Polarisations are rotating in the plane Hexagonal lattice: Analogies with electrons graphene
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Optical lattices (more sophisticated) Triangular or hexagonal optical lattice using
three light beams (propagagating at 1200)
(a) Polarisations are perpendicular to the plane Triangular lattice
(b) Polarisations are rotating in the plane Hexagonal (spin-dependent) lattice
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Optical lattices (more sophisticated) Triangular or hexagonal optical lattice using
three light beams (propagagating at 1200)
(a) Polarisations are perpendicular to the plane Triangular lattice
(b) Polarisations are rotating in the plane Hexagonal (spin-dependent) lattice
[traps differently atoms in different spin states]
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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
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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
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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
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Trapped atoms - electrically neutral particles 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 Coriolis force
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Trapped atoms - electrically neutral particles 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:Rotation Coriolis force (Mathematically equivalent to Lorentz force)
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Trap rotationHamiltonian in the rotating frame [see e.g. A. Fetter, RMP 81, 647 (2009)] Trapping potentialor
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Trap rotationHamiltonian in the rotating frame [see e.g. A. Fetter, RMP 81, 647 (2009)] Trapping potential
rotation vector
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Trap rotationHamiltonian in the rotating frame [see e.g. A. Fetter, RMP 81, 647 (2009)] Trapping potential
or rotation vector
Effective vector potential
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Trap rotationHamiltonian in the rotating frame [see e.g. A. Fetter, RMP 81, 647 (2009)] Trapping potential
or rotation vector
Effective vector potential(constant Beff~Ω)Coriolis force (equivalent to Lorentz force)
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Trap rotationHamiltonian in the rotating frame [see e.g. A. Fetter, RMP 81, 647 (2009)] Trapping potential
or rotation vector
Effective vector potential Centrifugal potential (constant Beff~Ω) (anti-trapping) Coriolis force (equivalent to Lorentz force)
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Trap rotation: Summary of the main features
Constant Beff: Beff Trapping frequency: Landau problem
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ROTATION Can be applied to utracold atoms both in usual
traps and also in optical lattices
(a) Ultracold atomic cloud (trapped):
(b) Optical lattice:
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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
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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
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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
J’exp(-ikx1) J’exp(ikx2)
J Atoms in different internal states (red or yellow)
are trapped at different lattice sites
x
y
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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
J’exp(-ikx1) J’exp(ikx2)
J
x
y
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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
J’exp(-ikx1) J’exp(ikx2)
J Atoms in different internal states (red or yellow)
are trapped at different lattice sites
x
y
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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
J’exp(-ikx1) J’exp(ikx2)
J Non-vanishing phase for the atoms moving over a
plaquette: S=k(x2-x1)=ka
x
y
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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
J’exp(-ikx1) J’exp(ikx2)
J Non-vanishing phase for the atoms moving over a
plaquette: S=k(x2-x1)=ka → Simulates non-zero magnetic flux (over plaquette)
x
y
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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
J’exp(-ikx1) J’exp(ikx2)
J
Staggered flux!
x
y
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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
J’exp(-ikx1) J’exp(ikx2)
J Non-vanishing phase for the atoms moving over a
plaquette: S=k(x2-x1)=ka non-zero magnetic flux: Experiment: M. Aidelsburger et al., PRL 107, 255301
(2011)
x
y
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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
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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)
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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
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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.
.
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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)
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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)
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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)
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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)
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Adiabatic atomic energies
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
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Non-degenerate state with n=1 Adiabatic atomic energies
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
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Non-degenerate state with n=1 Adiabatic atomic energies
Full state vector:
Adiabatic approximation
(only the atomic internal state with n=1 is included)
– wave-function of the atomic centre of mass motion in the n-th atomic internal “dressed” state
n=1
n=2
n=3
r
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Non-degenerate state with n=1 Adiabatic atomic energies
Full state vector:
Adiabatic approximation
(only the atomic internal state with n=1 is included)
– wave-function of the atomic centre of mass motion in the n-th atomic internal “dressed” state
n=1
n=2
n=3
r
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Non-degenerate state with n=1 Adiabatic atomic energies
Full state vector:
Adiabatic approximation
What is the equation of motion for ?
– wave-function of the atomic centre of mass motion in the n-th atomic internal “dressed” state
n=1
n=2
n=3
r
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Adiabatic approximation:
n=1
n=2
n=3
r
Non-degenerate state with n=1
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Adiabatic approximation:
Equation of the atomic c. m. motion in the internal state
n=1
n=2
n=3
r
ˆ H p A11 2
2MV (r)1(r)
Effective Vector potential appears
A11 A
Non-degenerate state with n=1
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Adiabatic approximation:
Equation of the atomic c. m. motion in the internal state
n=1
n=2
n=3
r
ˆ H p A11 2
2MV (r)1(r)
Effective Vector potential appears (due to the position-dependence of the atomic internal “dressed” state )
A11 A
Non-degenerate state with n=1
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Adiabatic approximation:
Equation of the atomic c. m. motion in the internal state
n=1
n=2
n=3
r
ˆ H p A11 2
2MV (r)1(r)
Effective Vector potential appears
A11 A
Non-degenerate state with n=1
- effective magnetic field
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Adiabatic approximation:
Equation of the atomic c. m. motion in the internal state
n=1
n=2
n=3
r
ˆ H p A11 2
2MV (r)1(r)
Effective Vector potential appears
A11 A
Non-degenerate state with n=1
- effective magnetic field (non-trivial situation if )
![Page 60: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/60.jpg)
Large possibilities to control and shape the effective magnetic field B by changing the light beams
- effective magnetic field (non-trivial situation if )
Effective Vector potential appears (due to the position-dependence of the atomic internal “dressed” state )
A11 A
To summarise
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Light induced effective magnetic field is due to Spatial dependence of atom-light coupling
![Page 62: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/62.jpg)
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)
![Page 63: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/63.jpg)
Light induced effective magnetic field can be due to
1. Spatial dependence of laser amplitudes
![Page 64: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/64.jpg)
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
L
B A 0
![Page 65: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/65.jpg)
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)
L
B A 0
![Page 66: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/66.jpg)
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)
L
B A 0
[Interpretation: M. Cheneau et al., EPL 83, 60001 (2008).]
![Page 67: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/67.jpg)
Counter-propagating beams with spatially shifted profiles
kL
[G. Juzeliūnas, J. Ruseckas, P. Öhberg, and M. Fleischhauer, Phys. Rev. A 73, 025602 (2006).]
Total magnetic flux is proportional to the sample length L:
L
(one can not increase the total flux in the transverse direction)
![Page 68: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/68.jpg)
Counter-propagating beams with spatially shifted profiles
kL
[G. Juzeliūnas, J. Ruseckas, P. Öhberg, and M. Fleischhauer, Phys. Rev. A 73, 025602 (2006).]
Total magnetic flux is proportional to the sample length L:
L
(one can not increase the total flux in the transverse direction)
No lattice
No translational symmetry for shifted beams (in the transverse direction):
![Page 69: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/69.jpg)
Light induced effective magnetic field due to Spatial dependence of laser amplitudes Spatial dependence of atom-light detuning
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§
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§
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§
Position-dependent detuning δ
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§
Position-dependent detuning δ=δ(y) B≠0
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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)
![Page 75: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/75.jpg)
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
![Page 76: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/76.jpg)
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)
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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
![Page 78: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/78.jpg)
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)
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
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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
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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
Atom-light Hamiltonian:
ˆ H 0 r hz x iy
x iy z
(2×2 matrix)
![Page 81: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/81.jpg)
Two atomic internal states
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
j r
ˆ H 0 r has position-dependent eigenstates , j=1,2
![Page 82: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/82.jpg)
Effective vector potential for atomic motion in the lower dressed state :
n=1
n=3
n=2
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
cos 1
Ω
Ωy
Ωx
Ωz
See, e.g.: J.Dalibard, F. Gerbier, G. Juzeliūnas and P. Öhberg. Rev. Mod. Phys. 83, 1523 (2011).
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Effective vector potential for atomic motion in the lower dressed state :
n=1
n=3
n=2
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
cos 1
Ω
Ωy
Ωx
Ωz
A r h
for
(AB singularity at the “South pole”)
![Page 84: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/84.jpg)
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)
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Optical flux latticesTwo-level system:
Coupling and detun. - specially chosen periodic funct.
x cos(x /a)
y cos(y /a)
z II sin(x /a)sin(y /a)
i.e. is a periodic array of vortices & anti-vortices
ˆ H 0 r hz x iy
x iy z
![Page 86: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/86.jpg)
Optical flux latticesTwo-level system:
Coupling and detun. - specially chosen periodic funct.
x cos(x /a)
y cos(y /a)
z II sin(x /a)sin(y /a)
ˆ H 0 r hz x iy
x iy z
i.e. is a periodic array of vortices & anti-vortices
oscillates
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Optical flux latticesTwo-level system:
Coupling and detun. - specially chosen periodic funct.
x cos(x /a)
y cos(y /a)
z II sin(x /a)sin(y /a)
ˆ H 0 r hz x iy
x iy z
i.e. is a periodic array of vortices & anti-vortices
oscillates (changes the sign)
![Page 88: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/88.jpg)
Optical flux latticesTwo-level system:
Coupling and detun. - specially chosen periodic funct.
x cos(x /a)
y cos(y /a)
z II sin(x /a)sin(y /a)
ˆ H 0 r hz x iy
x iy z
i.e. is a periodic array of vortices & anti-vortices
oscillates (changes the sign)
![Page 89: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/89.jpg)
Optical flux latticesTwo-level system:
Coupling and detun. - specially chosen periodic funct.
x cos(x /a)
y cos(y /a)
z II sin(x /a)sin(y /a)
ˆ H 0 r hz x iy
x iy z
i.e. is a periodic array of vortices & anti-vortices
oscillates (changes the sign) Periodic vector potential with
B A 0
![Page 90: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/90.jpg)
Optical flux latticesTwo-level system:
Periodic coupling and periodic detuning
Periodic vector potential with
x cos(x /a)
y cos(y /a)
z II sin(x /a)sin(y /a)
B A 0
ˆ H 0 r hz x iy
x iy z
![Page 91: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/91.jpg)
Optical flux latticesTwo-level system:
Periodic coupling and periodic detuning
Periodic vector potential with (B – non-staggered)!!!
x cos(x /a)
y cos(y /a)
z II sin(x /a)sin(y /a)
B A 0
ˆ H 0 r hz x iy
x iy z
![Page 92: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/92.jpg)
Optical flux latticesTwo-level system:
Periodic coupling and periodic detuning
Periodic vector potential with (B – non-staggered)!!!
WHY?
x cos(x /a)
y cos(y /a)
z II sin(x /a)sin(y /a)
B A 0
ˆ H 0 r hz x iy
x iy z
![Page 93: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/93.jpg)
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)
( i.e. is a periodic array of vortices & anti-vortices) Periodic vector potential with B – non-staggered !!! (A has the AB singularities if and )
B A 0
x iy 0
z 1
![Page 94: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/94.jpg)
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)
( 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!!!
B A 0
x iy 0
z 1
![Page 95: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/95.jpg)
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
![Page 96: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/96.jpg)
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
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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)
![Page 98: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/98.jpg)
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:
Non-staggered effective magnetic field
![Page 99: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/99.jpg)
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:
Non-staggered effective magnetic field (& lattice):
![Page 100: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/100.jpg)
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:
Optical flux lattice is produced !!!
Non-staggered effective magnetic field (& lattice):
![Page 101: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/101.jpg)
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
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If one degenerate atomic internal dressed state
-- Abelian gauge potentials
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Adiabatic approximation:
Equation of the atomic motion in the internal state
n=1
n=2
n=3
r
ˆ H p A11 2
2MV (r)1(r)
Effective Vector potential appears
A11 A
Non-degenerate state with n=1
- effective magnetic field (non-trivial situation if )
![Page 104: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/104.jpg)
If more than one degenerate atomic
internal dressed state--Non-Abelian gauge
potentials
![Page 105: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/105.jpg)
Adiabatic approximation: n=3
rn=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)
![Page 106: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/106.jpg)
Adiabatic approximation: n=3
rn=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)
- two-component atomic wave-function(spinor wave-function) Quasi-spin 1/2
(r, t) 1(r,t)2(r, t)
Repeating the same procedure …
![Page 107: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/107.jpg)
Adiabatic approximation:
Equation of motion:
n=3
rn=1, n=2
n=4
ˆ H p A 2
2MV (r)1(r)
appears due to position-dependence of
A
Degenerate states with n=1 and n=2
- effective vector potential
- effective magnetic field (non-trivial situation if )
- two-comp. atomic wave-function
2x2 matrix
2x2 matrix
(r, t) 1(r, t)2(r, t)
![Page 108: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/108.jpg)
Adiabatic approximation:
Equation of motion:
n=3
rn=1, n=2
n=4
ˆ H p A 2
2MV (r)1(r)
appears due to position-dependence of
A
Degenerate states with n=1 and n=2
- effective vector potential
- effective magnetic field (non-trivial situation if )
- two-comp. atomic wave-function
2x2 matrix
2x2 matrix
If Ax, Ay, Az do not commute, B≠0 even if A is constant !!!
(r, t) 1(r, t)2(r, t)
![Page 109: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/109.jpg)
Adiabatic approximation:
Equation of motion:
n=3
rn=1, n=2
n=4
ˆ H p A 2
2MV (r)1(r)
appears due to position-dependence of
A
Degenerate states with n=1 and n=2
- effective vector potential
- effective magnetic field (non-trivial situation if )
- two-comp. atomic wave-function
2x2 matrix
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)
![Page 110: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/110.jpg)
Non-Abelian gauge potentials
More than one degenerate dressed state
n=1, n=2
n=4
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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)
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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
![Page 113: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/113.jpg)
(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)
![Page 114: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/114.jpg)
(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)
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(Non-Abelian) light-induced gauge potentials
Centre of mass motion of dark-state atoms:
A - effective vector potential (Mead-Berry connection)
- 2x2 matrix
Two component atomic wave-function
![Page 116: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/116.jpg)
Tripod scheme
A is 2×2 matrix
Non-Abelian case if Ax, Ay, Az do not commute
B – curvature
Two degenerate dark states
![Page 117: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/117.jpg)
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
![Page 118: Light-induced gauge potentials for ultracold atoms](https://reader036.vdocuments.mx/reader036/viewer/2022062814/56816865550346895ddebc81/html5/thumbnails/118.jpg)
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)
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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)
kgkgk
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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
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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
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For small k: Similarities to graphene: Dirac-type Hamiltonian:Two dispersion cones: Quasirelativistic behaviour or cold atoms
vo≈1cm/s
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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).
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Dispersion of centre of mass motion for cold atoms
in light fields vg=dω/dk>0
vg=dω/dk<0
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- sharp Mexican hat
k1
k2
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.
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Negative refraction of cold atoms at a potential barrier
Veselago-type lenses for cold atoms
vg=dω/dk>0
vg=dω/dk<0
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Veselago-type lenses for ultra-cold atoms
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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
k1
k2
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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
k1
k2
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Double and negative reflection of atoms
(a) k2 – closer to the normal (k2>k)
(b) Negatively reflected wave – closer to the surface (k2<k)
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Resembles Andreev reflection Electron is converted into a hole with a
negative effective mass upon reflection
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Double and negative reflection of atoms:Wave-packet simulations
G, Juzeliunas, J. Ruseckas, A. Jacob, L. Santos, P. Ohberg, Phys. Rev. Lett. 100, 200405 (2008).
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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)
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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
D. L. Campbell, G. Juzeliūnas and I. B. Spielman, Phys. Rev. A 84, 025602 (2011)
(with pi phase)
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Possible implementation of the closed loop setup using the Raman transitions:
D. L. Campbell, G. Juzeliūnas and I. B. Spielman, Phys. Rev. A 84, 025602 (2011)
(with pi phase)
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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)
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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.
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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).
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Thank you!