strongly correlated phenomena in cavity qed fernando g.s.l. brandão 1,2 michael j. hartmann 1,2...
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Strongly correlated phenomena in cavity QED
Fernando G.S.L. Brandão1,2
Michael J. Hartmann1,2 Martin B. Plenio1,2
1Institute for Mathematical Sciences, Imperial College London 2QOLS, Blackett Laboratory, Imperial College London
London, 04/05/2007
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Cavity QED systems
Strong Coupling:
,g
Non-trivial joint dynamics for atoms and photons
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Array of coupled cavities
Atoms in different cavities can “talk” to each other mediated by the photons
Photons in the same cavity can “talk” to each other mediated by the atoms
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Summary Photon nonlinearities EIT-based schemes
Stark-shift based scheme
Bose-Hubbard models Polaritons in coupled array of cavities
The photonic limit
Spin Chains Heisenberg model (XYZ)
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Summary Photon nonlinearities EIT-based schemes
Stark-shift based scheme
Bose-Hubbard models Polaritons in coupled array of cavities The photonic limit
Spin Chains Heisenberg model (XYZ)
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Photon-Photon interactions
Kerr-type nonlinear interaction:
Several applications
Photon blockade Imamoğlu et at, PRL 79, 1467 (1997)
nonlinear optics Boyd, Nonlinear Optics, (1992) Quantum nondemolition measurents Imoto et al , PRA 32, 2287 (1985)
Optical quantum computing Turchette et al, PRL 75, 4710 (1995) etc…
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Photon-Photon interactions
Kerr-type nonlinear interaction:
Natural Kerr interactions are far too small…
,
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Electromagnetically Induced Transparency nonlinearities
12
3
4
Imamoğlu et at, PRL 79, 1467 (1997)
gh
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Electromagnetically Induced Transparency nonlinearities
12
3
gN x
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Electromagnetically Induced Transparency nonlinearities
12
3
gN x
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Electromagnetically Induced Transparency nonlinearities
12
3
gN x
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Electromagnetically Induced Transparency nonlinearities
h
12
3
g
4
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Electromagnetically Induced Transparency nonlinearities
h
12
3
g
4
Only dark state polaritons p 0 couple to level 4!
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Electromagnetically Induced Transparency nonlinearities
h
12
3
g
4
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Electromagnetically Induced Transparency nonlinearities
h
2
3
g
4
1
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Electromagnetically Induced Transparency nonlinearities
h
2
3
g
4
1
We didn’t assume:
h
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Electromagnetically Induced Transparency nonlinearities
Example: Toroidal Microcavities
Spillane et al, PRA 71, 013817 (2005)
Aoki et al, Nature 443 671 (2006)
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Electromagnetically Induced Transparency nonlinearities
Example: Toroidal Microcavities
Spillane et al, PRA 71, 013817 (2005)
Aoki et al, Nature 443 671 (2006)
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Could we find a simpler set-up producing a nonlinearity
comparable with the EIT one?
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A.C. Stark shift nonlinearity
12
3
g
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A.C. Stark shift nonlinearity
,21
2
1
g
g
12
3
g
Dispersive regime:
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A.C. Stark shift nonlinearity
2
21,
2
21
2
2
2
2
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A.C. Stark shift nonlinearity
1
2
2
22:
g
Dispersive regime:
2
2
2
2
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A.C. Stark shift nonlinearity
1
2
2
22:
g
Dispersive regime:
2
2
2
2
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A.C. Stark shift nonlinearity
- Same strength as EIT scheme
- One level less
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Summary Photon nonlinearities EIT-based schemes
Stark-shift based scheme
Bose-Hubbard model Polaritons in coupled array of cavities
The photonic limit
Spin Chains Heisenberg model (XYZ)
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Bose Hubbard Model
Fisher et al, PRB 40, 546 (1989)
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Cold atoms in Optical Lattices
Jaksch et al, PRL 81, 3108 (1998) Greiner et al, Nature 415, 39 (2002)
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Cold atoms in Optical Lattices
Jaksch et al, PRL 81, 3108 (1998) Greiner et al, Nature 415, 39 (2002)
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The set-up
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The set-up
Photons can hope from one cavity to a neighbouring one
Yariv et al, Optics Lett. 24, 711 (1999)
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The polaritonic case
h
1
2
3
g
4
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The polaritonic case
h
12
3
g
4
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The polaritonic case
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The polaritonic case
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The polaritonic case
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real pred.
Fabry-Perot: 160 5 x 103
Photonic bgc: 10 5.5 x 105
MCs @ Imperial: 40 ?
Micro-toroid: 53 5 x 106
Spillane et al, PRA 2005 Soda et al, Nature Materials 2005
2 /g 2 /g
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The polaritonic case
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The photonic case
a.c. Stark shift nonlinearity EIT nonlinearity
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The photonic case
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real pred.
Fabry-Perot: 2.6 10
Photonic bgc: 0.1 4 x 103
MCs @ Imperial: 0.8 ?
Micro-toroid: 2.6 1.25 x 105
Spillane et al, PRA 2005 Soda et al, Nature Materials 2005
/g /g
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Summary Photon nonlinearities EIT-based schemes
Stark-shift based scheme
Photonic Bose-Hubbard models Polaritons in coupled array of cavities
The photonic limit
Spin Chains Heisenberg model (XYZ)
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Spins Lattices
Open questions in condensed-matter physics: high Tc
superconductivity, frustration, etc…
Applications in quantum information science: entanglement propagation, measurement-based quantum computation, etc…
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Spins Lattices: Heisenberg (XYZ) model
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XX and YY interactions:
Spins Lattices: Heisenberg (XYZ) model
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XX and YY interactions:
Spins Lattices: Heisenberg (XYZ) model
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XX and YY interactions:
Spins Lattices: Heisenberg (XYZ) model
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ZZ interactions + magnetic field:
Spins Lattices: Heisenberg (XYZ) model
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+
Suzuki-Trotter Decomposition:
Spins Lattices: Heisenberg (XYZ) model
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Cluster state generation
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Spins Lattices: XYZ model
h g
2g
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real pred. real pred
Fabry-Perot: 160 5 x 103 60 420
Photonic bc: 10 5.5 x 105 100 105
MCs @ Imperial: 40 ? 50 ?
Micro-toroid: 53 5 x 106 20 400
Spillane et al, PRA 2005 Soda et al, Nature Materials 2005
2 /g /g
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References Nonlinearities:
EIT scheme: Imamoğlu et at, PRL 79, 1467 (1997)
Hartmann, Plenio, arXiv:0704.2575
Light shift scheme: Brandão, Hartmann, Plenio, arXiv:0705.xxxx
Bose Hubbard model: Hartmann, Brandão, Plenio,
Nature Physics 2, 849 (2006), quant-ph/0606097
Subsequent proposals:
Angelakis, Santos, Bose, quant-ph/0606159
Greentree, Tahan, Cole, Hollenberg, Nature Physics 2, 856 (2006), quant-ph/0609050
Na, Utsonumiya, Tian, Yamamoto, quant-ph/0703219
Rossini, Fazio, Phase diagram of strongly correlated polaritons in a 1D array of coupled cavities, in preparation
Spin Hamiltonians: Hartmann, Brandão, Plenio, arXiv:0704.3056
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Thank you!
Michael J. Hartmann
Martin B. Plenio