Electromagnetically induced transparencyin cold Rydberg Gases

S. Sevincli1,2, N. Henkel1, C. Ates3, T. Pohl1

1 Max Planck Institute for the Physics of Complex Systems, Dresden, Germany2 Aarhus University, Denmark

3 University of Nottingham, United Kingdom

Introduction

Electromagnetically induced transparency continues to play an important role in quantum and nonlinear optics. In particular, cold Rydberg gases areopening up new avenues for studying nonlinearities even at low intensities. In most contexts, nonlinear effects play a role only at high intensities. Butthe extreme interactions between Rydberg atoms allow to expand the study of nonlinear effects to the few-photon regime.We consider a gas of three-level atoms, with one of the states a strongly interacting and long-lived Rydberg state. Transitions between the states aredriven by a probe laser with Rabi frequency Ω2 and a control laser with Ω1 ≪ Ω2. We calculate the fully correlated steady-state populations. Fromthose, we derive the nonlinear optical response to classical light fields in the presence of arbitrarily strong atomic interactions. Upon proper scaling ourresults reveal an interesting universal behavior of the nonlinear absorption [1, 2, 3].The nontrivial optical susceptibility also leads to interesting propagation dynamics [4]. We find that the interaction as well as the resulting propagationdynamics of the probe field bear strong similarity to the behaviour found in Rydberg-dressed Bose-Einstein condensates, leading to roton instabilities [5]and self-trapped states [6]. |1〉

|2〉

|3〉

γ γ

|1〉

|2〉

|3〉

C6

r6

ij

atom i atom jrij

∆1 ∆1

Ω1Ω1

Ω2 Ω2

∆2 ∆2

Calculation of optical response

The gas can be described by the the von Neumann equation

iρ(N) = [H0 + V, ρ]− iL[ρ(N)], (1)

for the N -body density matrix ρ(N). L is the Lindblad operator that accounts for spontaneous decay.Adiabatically eliminating coherences from the single-particle OBE yields

d

dt

ρ11ρ22ρ33

=

−a11 a12 a13a21 −a22 a23a31 a32 −a33

ρ11ρ22ρ33

, (2)

where the interaction is included in the local detunings ∆(i)2 = ∆2 −

∑′j 6=i Vij.

The susceptibility of the probe transition is proportional to the single-particle coherence ρ12

χ12 =2µ212ρ0ε0~Ω1

ρ12 , (3)

In the steady-state the absorption is given by Im(ρ12) =γΩ1ρ22

Probe Susceptibility

Symbols: reduced density matrix,Lines: MC simulations of themany-body rate equation

Parameters: ∆2 = 0 and (Ω1,Ω2) =(1, 2), (Γ, γ12, γ23) = (6.1, 0.1, 0.1) in

MHz.for Rb55S state

-3 -2 -1 0 1 2 30.03

0.06

0.09

0.12

0.15

∆1 [MHz]

ρ0 = 3 · 108cm

−3

ρ0 = 109cm

−3

ρ0 = 5 · 109cm

−3

Im[χ

12][

µ12ρ0

ǫ0Ω

1

]

Position and width of EIT-window changes only slightly with increasingdensity/interaction

Complex susceptibility as a function ofdensity ρ0 in a Rb gas under EIT condi-tions, ∆1 = ∆2 = 0.

107

108

109

1010

1011

1012

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

ρ0 [cm−3]

χ12

[

µ2 12ρ0

ǫ0Ω

1

]

Im[χ12]

Re[χ12]

Im[χ(∞)12 ]

Strong non-linear dependence on density/interaction strength

Universal scaling of the signalabsorption on resonance.(a)Unscaled for various combina-tions of the principal quantumnumber n and probe Rabi fre-quency Ω1.squares( n = 50,Ω1 = 1.0MHz), circles(50, 1.0),triangles (70, 1.0) anddiamonds (70, 0.5)(b) Scaled with the absorption inthe high density limit and plottedversus the fraction fbl of excita-tion blocked atoms.

107

108

109

1010

1011

0.000

0.005

0.010

0.015

0.020

0.025

10-210-1

100101102103

0

0.2

0.4

0.6

0.8

1

ρ0 [cm−3] fbl

χ12

[

µ2 12ρ0

ǫ0Ω

1

]

χ12

(a) (b)

χ12 =fbl

1+fbl

Universal behavior of probe absorption in interacting Rydberg gas

Optical susceptibility

• In the limit of low probe intensities (Ω1(ri) ≪ Ω2)

d

dtσ(i)13 = −Ω2

Ω1(ri) + Ω2σ(i)13

2Γ−γ132σ(i)13 − i

∑

j 6=i

Vijσ(j)33 σ

(i)13 , Γ = γ + γ12 − 2i∆1 (4)

• Assuming that γ12 ≪ γ, γ13 ≪ δEIT = Ω22/2∆1, the optical susceptibility is

χ(r) =2µ212ε0~Ω

22

ρ2∫

dr′2Ω1(r

′)2V (r− r′)

Ω22 − (γ + 2i∆1)V (r− r′)

. (5)

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.80

1

2

3

Ωp0/Ωc

−ln

(T)

Ωp0/Ωc

ρ = 3.5× 109cm−3

Ωc

2π= 3.8MHz

ρ = 1.2× 1010

cm−3

Ωc

2π= 4.8MHz

• Shown is the transmission T ,which follows from the suscepti-bility.

• solid line: prediction from (5);dashed line: simple prediction ne-glecting attenuation and the spa-tial beam profile.

• Good agreement with experi-ments [2]

Description of the beam propagation

• For far-detuned media ∆1 ≫ γ and assuming that γ12 ≪ γ, γ13 ≪ δEIT = Ω22/2∆1,

χ(r) = −6πγρ2

k3∆1Ω2c2

∫

dr′|Ω1(r

′⊥, z)|

2

1 +|r′−r|6

R6c

− iγ

2∆1

|Ω1(r′⊥, z)|

2

[

1 +|r′−r|6

R6c

]2, (6)

• The light wave can be described by the Gross-Pitaevskii (non-linear Schrodinger) equation

(

−1

2∇2 + α

∫

dr′ |ψ(r′, t)|2U(|r− r′|)

)

ψ(r, t) = i∂tψ(r, t) (7)

with the effective potential (including dissipation effects)

U(r) =

∫ ∞

−∞dz

(

1

1 + r6− i

γ

2∆1

1

(1 + r6)2

)

and α =36π2ρ2γ2Pp

~k4cR3cΩ

42

C6, Rc = (C6/2~δEIT)1/6

For sufficiently large α > 0 (αcrit ≈ 50), the system is roton unstable, which leads to the formationof intensity modulations in the light field. For α < 0 self-focussing soliton states are formed.

Propagation dynamics

•Output beam profile |Ω(r⊥)|2 for Rb(70S1/2) (repulsive interactions)

Ω(0)1 /2π = 0.35MHz,

Ω2/2π = 80MHz and∆1/2π = 0.6GHz at den-sities (a) 4 × 1013cm−3,(b) 5.5 × 1013cm−3 and(c) 8× 1013cm−3.

• Intensity profiles |Ω(r⊥)|2 for Sr(501S0) (attractive interactions)

Ω(0)1 /2π = 0.3MHz,

Ω2/2π = 15MHz,∆1/2π = 1.6GHz atdifferent densities of8 × 1011cm−3 (a,c,e),1.2× 1012cm−3 (b,d,f).Panels (a,b) show thestable soliton solutions,panels (c-f) show thecompressed outputintensity profile.

References

[1] C. Ates, S. Sevincli and T. Pohl, PRA 83 041802(R) (2011)

[2] J.D. Pritchard et al., PRL 105, 193603 (2010).

[3] D. Petrosyan, J. Otterbach, and M. Fleischhauer, PRL 107, 213601 (2011).

[4] S. Sevincli, N. Henkel, C. Ates and T. Pohl, PRL 107 153001 (2011)

[5] N. Henkel, R. Nath, T. Pohl, PRL 104, 195302 (2010).

[6] F. Maucher et al., PRL 106 170401 (2011)