Master equation for collective spontaneous emissionwith quantized atomic motion
Phys. Rev. A 93, 022124 (2016)
Francois Damanet1, Daniel Braun2, John Martin1
1 Institut de Physique Nucleaire, Atomique et de Spectroscopie. Universite de Liege,4000 Liege Belgium.
2 Institut fur theoretische Physik, Universitat Tubingen,72076 Tubingen, Germany.
DPG Spring Meeting - 2 March 2016
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 1 / 19
Motivation
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 2 / 19
System
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 3 / 19
System
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 3 / 19
The Lamb-Dicke parameter
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 4 / 19
Description of internal dynamicsPrevious works
Most of theoretical methods• are restricted to the Lamb-Dicke regime
See e. g. PRL 104, 043003 (2010); PRA 84, 043825 (2011)• account either for recoil or indistinguishability, but not for both
See e. g. PRA 51, 3959 (1995); PRA 53, 390 (1996); PRA 59, 3797 (1999)
Our contributionMaster equation for internal degrees of freedom beyond theLamb-Dicke regime
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 5 / 19
Description of internal dynamicsPrevious works
Most of theoretical methods• are restricted to the Lamb-Dicke regime
See e. g. PRL 104, 043003 (2010); PRA 84, 043825 (2011)• account either for recoil or indistinguishability, but not for both
See e. g. PRA 51, 3959 (1995); PRA 53, 390 (1996); PRA 59, 3797 (1999)
Our contributionMaster equation for internal degrees of freedom beyond theLamb-Dicke regime
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 5 / 19
Derivation of a master equationSystem-bath decomposition
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 6 / 19
Derivation of a master equationTimescales and approximations
τB︸︷︷︸S-B correlation
. τS︸︷︷︸internal dynamics
� τR︸︷︷︸relaxation
� τM︸︷︷︸motional dynamics
• τB � τR : Born-Markov condition (for field degrees of freedom)⇒ no memory effects (no photon absorption)
• τS � τR : RWA condition⇒ only energy conserving transitions
• τR � τM : Born condition (for motional degrees of freedom)⇒ freezing of the intrinsic motional dynamics (rj (t) −→ rj )
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 7 / 19
Master equation
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 8 / 19
Master equation
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 8 / 19
Master equation
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 8 / 19
Internal dynamicsLamb-Dicke regime : η → 0 (classical fixed atomic positions)
∆ij −→ ∆cl(rij) = 3γ04
[− pij
cos ξijξij
+ qij
(sin ξij
ξ2ij
+ cos ξij
ξ3ij
)]
γij −→ γcl(rij) = 3γ02
[pij
sin ξijξij
+ qij
(cos ξij
ξ2ij− sin ξij
ξ3ij
)]
ξij = 2π rijλ
pij , qij angular factors which depend on the dipolar transition (π or σ±)γ0 : single-atom spontaneous emission rate
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 9 / 19
Internal dynamics in Lamb-Dicke regime (η → 0)
Regime Relevant Phenomena
1� ξij γij ' γ0δij ⇒ independant spontaneous emissions
ξij � 1 γij ' γ0 ⇒ cooperative effects
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 10 / 19
Internal dynamics beyond the Lamb-Dicke regime
Regime Relevant Phenomena
1 . η � ξij recoil
η � 1� ξij independent spontaneous emissions
η . ξij � 1 cooperative effects
1� ξij . η indistinguishability, recoil
ξij � 1 . η cooperative effects, recoil, indistinguishability
ξij . η � 1 cooperative effects, indistinguishability
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 11 / 19
Internal dynamics beyond the Lamb-Dicke regimeRESULTS :
γii = γ0 ∀i = 1, . . . ,N|γij | ≤ γ0 ∀i , j = 1, . . . ,N
}for any motional state
γij and ∆ij depend on the motional state ρexA through Cex
ij (k) = 〈e ik·rij 〉ρexA
Analytical expressions of γij found forI Gaussian statesI Fock statesI thermal states
Numerical computations of ∆ij for Gaussian states
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 12 / 19
Motional Gaussian states - decay rates γijDistinguishable atoms
Analytical expression valid for any ξij and η
γsepij (ξij , η) =
3γ0
16η5
(√π
6e
−ξ2
ij4η2[
16η4 − qij(
4η4 + 3ξ2ij − 6η2
) ]Re{
erf(η +
iξij
2η
)}− qijηe−η2
0[
2η2 cos ξij − ξij sin ξij])
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 13 / 19
Motional Gaussian states - dipole-dipole shifts ∆ijDistinguishable atoms
∆ij =[∫ −ε−∞
+∫ ∞ε
]∆cl(x)F−1
x[Cex
ij (kx )]
dx
∆ij diverges when F−1x=0[Cex
ij (kx )]6= 0
A cutoff ε must be introduced (ε ' a0 = Bohr radius)
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 14 / 19
Motional Gaussian states - γij and ∆ijDistinguishable and indistinguishable atoms
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 15 / 19
Motional Fock states - decay rates γijDistinguishable atoms
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 16 / 19
ApplicationSpatial Pauli-blocking of spontaneous emission : N = 2 indistinguishable atoms
2 4 6 8 10
0.2
0.4
0.6
0.8
1.0
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 17 / 19
Conclusion and applications
ConclusionDerivation of a general master equation for internal dynamics includingquantized atomic motion
General expressions of decay rates and dipole-dipole shifts valid for anymotional stateAnalytical expressions obtained for relevant motional states
Applications and outlookQuantum programmed internal dynamics through motional state engineering
spatial Pauli-blocked spontaneous emissioncontrol of cooperative processes (super and subradiance)tuning of the dipole-dipole couplings between atoms (Rydberg atoms)
F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 18 / 19
Thanks for your attention !
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