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 !

More info : Phys. Rev. A 93, 022124 (2016)

F. Damanet, D. Braun, J. Martin Phys. Rev. A 93, 022124 (2016) 19 / 19