Slow Light and Superluminal Propagation M. A. Bouchene

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1 Slow Light and Superluminal Propagation M. A. Bouchene Laboratoire «Collisions, Agrégats, Réactivité», Université Paul Sabatier, Toulouse, France

2 Interest in Slow and Fast light Fundamental aspect in optical physics How can we stop (reversibly) the light? Can light travels faster than c? Applications: optical storage, optical information, telecom (buffering)

3 Outlook Group velocity, case of a resonant system (Ultra) Slow light: - Electromagnetic Induced Transparency - Coherent Population Oscillations - Zeeman Coherence Oscillations. Coherent control of the optical response Superluminal propagation: - Relativity implications - Different velocities for a light pulse - Backward propagation Conclusion

4 n( ω) = 1; αω ( ) = Etz E t z c ω k = c c (, ) ( / )e i k z ω = t Group Velocity c ( ) dn n( ω) 1 ( ω ) ω ω ( ) n + ( ) αω ( ) = cte dω ω Etz (, ) = GE( t z/v )e ω k = vϕ v ϕ = c/ n( ω): c v g = : dn n( ω) + ω dω Phase velocity ω g i( kz ω t) Group velocity PROPOGATION WITHOUT DISTORTION UNLESS RESHAPING EFFECTS (higher orders contributions) v g v ϕ z

5 Group Velocity n( ω) = 1; αω ( ) = Etz E t z c (, ) ( / )e i k z ω = t k ω = c c c ( ) dn n( ω) 1 ( ω ) ω ω ( ) n + ( ) αω ( ) = cte dω ω Etz (, ) = GE( t z/v )e k = ω v ϕ v g v ϕ g i( kz ω t) z

6 Group Velocity n( ω) = 1; αω ( ) = Etz E t z c (, ) ( / )e i k z ω = t k ω = c c c ( ) dn n( ω) 1 ( ω ) ω ω ( ) n + ( ) αω ( ) = cte dω ω Etz (, ) = GE( t z/v )e ω k = vϕ v = c/ n( ω ): Phase velocity ϕ v g v ϕ g i( kz ω t) z

7 Group Velocity n( ω) = 1; αω ( ) = Etz E t z c (, ) ( / )e i k z ω = t k ω = c c c ( ) dn n( ω) 1 ( ω ) ω ω ( ) n + ( ) αω ( ) = cte dω ω Etz (, ) = GE( t z/v )e ω k = vϕ v ϕ = c/ n( ω): c v g = : dn n( ω) + ω dω ω v g v ϕ Phase velocity g i( kz ω t) Group velocity z

8 n( ω) = 1; αω ( ) = Etz E t z c ω k = c c (, ) ( / )e i k z ω = t Group Velocity c ( ) dn n( ω) 1 ( ω ) ω ω ( ) n + ( ) αω ( ) = cte dω ω Etz (, ) = GE( t z/v )e ω k = vϕ v ϕ = c/ n( ω): c v g = : dn n( ω) + ω dω Phase velocity ω g i( kz ω t) Group velocity linear-dispersive medium with constant spectral gain/abs: PROPAGATION WITHOUT DISTORTION ( / ) High orders contributions: RESHAPING EFFECTS i ω c nl α L v g v ϕ e, z e

9 V c dn g = ; ng = n+ : n ω d ω g Group index

10 V c dn g = ; ng = n+ : n ω d ω g n 1 Group index Dilute medium but v g may be very different from c

11 V c dn g = ; ng = n+ : n ω d ω g n 1 Group index Dilute medium but v g may be very different from c v g V g =c -1 dn ω d ω

12 V c dn g = ; ng = n+ : n ω d ω g n 1 Group index Dilute medium but v g may be very different from c SUPERLUMINAL V g >c v g V g =c SLOW LIGHT V g <c -1 dn ω d ω V g < Backward propagation V g > Forward propagation ANOMALOUS DISPERSION NORMAL DISPERSION

13 V c dn g = ; ng = n+ : n ω d ω g n 1 Group index Dilute medium but v g may be very different from c SUPERLUMINAL V g >c v g V g =c SLOW LIGHT V g <c -1 dn ω d ω V g < Backward propagation V g > Forward propagation ANOMALOUS DISPERSION NORMAL DISPERSION

14 Ultra slow light needs w dn/dw >>1 How to achieve?? v g > c : What about relativity? v g < : Propagation of energy?

15 What happens near a resonant transition? ω E ω i nl αl c E e e ω N

16 What happens near a resonant transition? ω E ω i nl αl c E e e ω N 2 γ α = α ω ω + γ ( ) 2 2 : absorptioncoefficient α = N 2 μω cε γ : absorption at resonance frequency, γ : relaxation rate

17 NORMAL DISPERSION ANOMALOUS DISPERSION Increasing n g is possible but absorption also increases!

18 What happens near a resonant transition? ω E ω i nl αl c E e e ω N n = + 2cαγ ω ω 1 : 2 2 ω ( ω ω) + γ refraction index α = N 2 μω cε γ : absorption at resonance frequency, γ : relaxation rate

19 NORMAL DISPERSION ANOMALOUS DISPERSION Increasing n g is possible but absorption also increases!

20 NORMAL DISPERSION ANOMALOUS DISPERSION Increasing n g is possible but absorption also increases! Difficult to observe experimentally

21 I- How to produce slow-fast light? Kramers-Kronigs relations V g = c : dn n + ω d ω c αω ( ') n( ω) 1 = dω ' 2 2 π ω' ω 2 4ω n( ω ') 1 αω ( ) = dω ' 2 2 πc ω' ω A narrow spectral hole produce a strong normal dispersion

22 Slow light Electromagnetic Induced Transparency Coherent Population Oscillations Zeeman Coherence Oscillations

23 II-a Electromagnetic Induced Transparency

24 II-a Electromagnetic Induced Transparency

25 II-a Electromagnetic Induced Transparency

26 Interpretation 1 Ω C Control field laser intensity 2 γ Propagating field

27 Interpretation ± = iωt 1 ± e 2 : adiabatic states 2 1 Ω C Control field laser intensity 2 γ Propagating field

28 Interpretation Ω C Control field laser intensity 2 1 ± = γ iωt 1 ± e 2 : adiabatic states 2 Ω C γ + Propagating field Propagating field

29 Interpretation Ω C Control field laser intensity 2 1 ± = γ iωt 1 ± e 2 : adiabatic states 2 Ω C Propagating field γ + } Autler-Townes Splitting Dynamical Stark effect Propagating field

30 Interpretation Ω C Control field laser intensity 2 1 ± = γ iωt 1 ± e 2 : adiabatic states 2 Ω C Propagating field γ + } Autler-Townes Splitting Dynamical Stark effect Propagating field Ω >> C γ : Propagating field is non-resonant

31 propag. 1 ω ω ABSORPTION COEFFICIENT

32 Interpretation ± = iωt 1 ± e 2 : adiabatic states 2 Ω γ C Ω C + Propagating field

33 Interpretation ± = iωt 1 ± e 2 : adiabatic states 2 Ω γ C Ω C + Propagating field Interference between quantum paths + and RESULT?

34 ABSORPTION COEFFICIENT propag. 1 ω ω

35 ABSORPTION COEFFICIENT propag. 1 ω ω TOTAL DESTRUCTIF INTERFERENCE

36 Coherent Population Trapping γ 1 B = sinϕ + cosϕ 2 : Bright state Ω C 2 Ω P D = sinϕ 2 + cosϕ :Dark state Ω tan ϕ= Ω p c

37 Coherent Population Trapping ω p =ω1 1 No coupling D B

38 Coherent Population Trapping ω p =ω1 1 No coupling Ω= Ω +Ω 2 2 P C D B

39 Coherent Population Trapping ω p =ω1 1 D B

40 Coherent Population Trapping 1 D B

41 Coherent Population Trapping 1 D B

42 Coherent Population Trapping 1 D B

43 Coherent Population Trapping 1 D B ALL THE ATOMS ARE IMMUNE TO INTERACTION: perfect transparency

44 Autler-Townes splitting CPT

45 NATURE, 397, 594, (1999) v g 7 c = 1!!

46 V g decreases with Ω C

47 V g decreases with Ω C Is it possible to stop the light?

48 V g decreases with Ω C Is it possible to stop the light? YES

49 V g decreases with Ω C Is it possible to stop the light? YES What means a stopped light??

50 V g decreases with Ω C Is it possible to stop the light? YES What means a stopped light??

51 Ω C E(Z = ) E(Z = L)

52 Ω C E(Z = ) E(Z = L) STORAGE

53 Ω C E(Z = ) E(Z = L) What kind of storage? STORAGE

54 Ω(t) cosθ= Ω 2 2 (t) + g N ψ (z,t) = cos θ(t) E(z,t) sin θ(t)g N σ bc Polariton (mixture field-atom) t + θ ψ = z 2 ccos (t) (z,t) Shape-preserved propagation with 2 v(t) g = ccos θ(t)

55 II-b Coherent Population Oscillations Bichromatic field : b δ ω 1 ω 2 a

56 II-b Coherent Population Oscillations Bichromatic field : = + + ( ω ω ) I I I 2 I I cos t b δ BEATING ω 1 ω 2 a

57 II-b Coherent Population Oscillations Bichromatic field : = + + ( ω ω ) I I I 2 I I cos t b δ BEATING ω 2 ω 1 n = n n = n + n e + n e () ( + ) iδt ( ) iδt ba b a ba ba ba a TEMPORAL GRATING

58 II-b Coherent Population Oscillations ω2 ω 2 + δ =ω1 Reinforcement of the emission at ω1 δ ω 2 ω2 δ

59 II-b Coherent Population Oscillations ω 2 ω + δ =ω 2 1 δ Absorption δ

60 II-b Coherent Population Oscillations ω 2 ω + δ =ω 2 1 δ Absorption 1 T 1 1 T 2 δ Δ t = T Δ t =δ POPULATION 1 OSCILLATION 1

62 II-c Coherent Zeeman Oscillations A double two level system F=½ c Δ d z π polarized control field y Control field π polarized Probe field σ polarized π x σ polarized probe field prop. axes F=½ a m F =½ b m F =-½

63 σ π II-c Coherent Zeeman Oscillations Bichromatic field, with strong resonant pulse and polarization produces a time /space grating e π π σ Φ r, t = Δ t k. r ( ) δ detuned pulse Eσ, et = eπ + eσ e E π Φ( i r t) e σ π 2 π 3π 2 Linear ellipt. L linear ellip. R linear 2π Φ ( r, t)

64 grating imprints on Zeeman coherences 1 2 n= 1 ρ = ρ + ρ = Z z z ρ n= 1 ( n) e in t k r ( Δ δ. ) ρ z2 ρ z1

65 grating imprints on Zeeman coherences 1 2 n= 1 ρ = ρ + ρ = Z z z ρ n= 1 ( n) e in t k r ( Δ δ. ) ρ z2 ρ z1 ( ) ρ ( ) σ ρσ θπ ρ θ Φ i r, t = Im Γ Z i σe ne ng 2 θ π deπ = ; θσ = Γ de σ Γ

66 grating imprints on Zeeman coherences n e 1 2 n= 1 ρ = ρ + ρ = Z z z ρ n= 1 ( n) e in t k r ( Δ δ. ) n g ( ) ρ ( ) σ ρσ θπ ρ θ Φ i r, t = Im Γ Z i σe ne ng 2 θ π deπ = ; θσ = Γ de σ Γ Absorption by population

67 grating imprints on Zeeman coherences 1 2 n= 1 ρ = ρ + ρ = Z z z ρ n= 1 ( n) e in t k r ( Δ δ. ) ρ z2 n =1 π σ ρ z1 ( ) ρ ( ) σ ρσ θπ ρ θ Φ i r, t = Im Γ Z i σe ne ng 2 Diffraction of strong field by Zeeman coherence

68 grating imprints on Zeeman coherences 1 2 n= 1 ρ = ρ + ρ = Z z z ρ n= 1 ( n) e in t k r ( Δ δ. ) ρ z2 n =1 π σ ρ z1 ( ) ρ ( ) σ ρσ θπ ρ θ Φ i r, t = Im Γ Z i σe ne ng 2 Diffraction of strong field by Zeeman coherence Cancel exactly at Δ = Absorption by population

69 II-c Coherent Zeeman Oscillations Width χ ( / Γ) 2 π Susceptibility (in units of 2α k -1 ) 1. Ω π = Im(χ).8 Ω π =.1 Γ.6 Ω π =.2 Γ Re(χ) Δ (in units of Γ) F. A. Hashmi and M. A. Bouchene Phy. Rev. A 77, 5183(R) 28

70 II-c Coherent Zeeman Oscillations Pure Non-linear effect No dark state Hybrid properties between CPO and EIT

71 IId- Coherent Control of the Susceptibility Giant Non-linearity: lin (2) 2 (3) 3 P= χ E+ χ E + χ E +... Increase NL effects increase E or χ If E is strong : ionisation If resonance : absorption Slow light techniques: reduction of absorption AND resonance Effects of giant non linearities observable

72 Δ = : Susceptibility (in units of 2α k -1 ) Re(χ) Im(χ) Δ (in units of Γ) Ω π = Ω π =.1 Γ Ω π =.2 Γ Optical response canceled in the k σ direction Optical response determined by other emitted waves k σ δ k k π 2k k π σ

73 II-d Coherent Control of Susceptibility Δ =, δ k = F=½ c Δ d z π polarized control field k, k σ π y π σ x σ polarized probe field prop. axes F=½ a b Φ ( rt, ) =Δt δ kr. + ϕ = ϕ ρ = ρ + ρ = ρ ( ϕ):coherent Control σ da cb σ

74 II-d Coherent Control of Susceptibility ρ = ρ + ρ σ da cb c d ρ da = a b Absorption path k σ

75 II-d Coherent Control of Susceptibility ρ = ρ + ρ σ da cb c d c d ρ da = + π π a b a b Absorption path k σ Cross- Kerr type path k + k k = k ( ) π σ π σ

76 II-d Coherent Control of Susceptibility ρ = ρ + ρ σ da cb c d c d ρ da = + π π a b a b Absorption path k σ Cross- Kerr type path k + k k = k ( ) π σ π σ Cancel each other Δ =

77 II-d Coherent Control of Susceptibility ρ = ρ + ρ σ da cb c d c d c d ρ da = + + π π π π a b a b a b Absorption path k σ Cancel each other Δ = Cross- Kerr type path k + k k = k ( ) π σ π σ Phase Conjugate type path k ( k k ) = 2k k = k π σ π π σ Define the optical response σ

78 if Ω <<Γ and α L << ( thin sample ) : π II-d Coherent Control of Susceptibility 1 c d π π a b

79 if Ω π <<Γ and α L << ( thin sample ) : iϕ 1 Problem equivalent to linear propagation of field Ω σ e II-d Coherent Control of Susceptibility in an assembly of two level atoms : c d a b

80 if Ω < < Γand α L < ( thin sample ) : < 1 Problem equivalent to linear propagation of field Ω P π II-d Coherent Control of Susceptibility iϕ σ e in an assembly of two level atoms : ( ) ( ) ( 2 ρ χ ) lin Ω σ e = χline iϕ iϕ iϕ σ σ σ Ω e c a d b

81 if Ω < < Γand α L < ( thin sample ) : < 1 Problem equivalent to linear propagation of field Ω P π II-d Coherent Control of Susceptibility iϕ σ e in an assembly of two level atoms : ( ) ( ) ( 2 ρ χ ) lin Ω σ e = χline iϕ iϕ iϕ σ σ σ χ eff Ω e 2i lin e ϕ χ! c a d b Independent from pump field characteristics!

82 if Ω < < Γand α L < ( thin sample ) : < 1 Problem equivalent to linear propagation of field Ω P π II-d Coherent Control of Susceptibility iϕ σ e in an assembly of two level atoms : ( ) ( ) ( 2 ρ χ ) lin Ω σ e = χline iϕ iϕ iϕ σ σ σ χ eff Ω e 2i lin e ϕ χ! c a d b Independent from pump field characteristics! -The system behaves as a linear medium with a modified (linear) suceptibility controlled by the phase shift -Giant non-linearity F. A. Hashmi and M. A. Bouchene Phys. Rev. Letters 11, 2136 (28)

83 II-d Coherent Control of Susceptibility χ 2iϕ eff χline : gain dispersion coupling Absorber.5 Anomalous dispersion 1. ϕ = Ω π =1 Ω σ =.1Γ; Γ ze =2Γ d =Γ; α L=.2.5. ϕ =π/4 Anomalous «dispersion-like» gain «absorption-like» dispersion χ eff in units of 2α k Amplifier Normal dispersion -.5 ϕ =π/ ϕ =3π/4 Re χ eff Im χ eff Normal «dispersion-like» gain «gain-like» dispersion Δ = ω ω Δ in units of Γ F. A. Hashmi and M. A. Bouchene, Phys. Rev. Letters 11, 2136 (28)

84 Superluminal propagation

85 CAUSALITY AND RELATIVITY (R) (1) ( x,t) (2) ( x+ Δ x,t+δt) (R ) v v Δ t Δx 2 c Δt uv Δx Δ t' = = 1 ; u = 1 v/c 1 v/c c Δt ( ) ( ) RELATIVITY (implicit) v < c CAUSALITY Δt' Δt u c v< c u: signal velocity

86 WHAT IS A SIGNAL?

87 WHAT IS A SIGNAL? In Optics : Wave packet with sharp front (Sommerfeld 191)

88 WHAT IS A SIGNAL? In Optics : Wave packet with sharp front (Sommerfeld 191) z

89 WHAT IS A SIGNAL? In Optics : Wave packet with sharp front (Sommerfeld 191) C exactly! z

90 WHAT IS A SIGNAL? In Optics : Wave packet with sharp front (Sommerfeld 191) >c or <c C exactly! z - Envelope faster than c! (Vg > c), Vg still meaningful -Pulse distortion occurs if the back part of the pulse catch up the sharp front Extension: non analyticity (Chiao et al. 2)

91 C

92 Five velocities for a light pulse: Phase velocity v c/ n ( cor c) ϕ = > <

93 Five velocities for a light pulse: Group velocity v = c/ n ( > cor < c) g g Meaningful in a linear dispersive medium with constant spectral gain

94 Five velocities for a light pulse: Signal velocity v s : speed of non analytical point = c v s

95 Five velocities for a light pulse: Signal velocity v s : speed of non analytical point = c «detectable Information» v s

96 Five velocities for a light pulse: centroid of field energy r cor c u G 3 ru 2 2 Fd r ε E B ( ), 3 F udr 2 2 F = > < = + μ G

97 Five velocities for a light pulse: centroid of field energy v ( ), ε E B μ 2 2 Field En. = rg > cor < c uf = + t 2 2 G

98 Five velocities for a light pulse: centroid of field energy v ( ), ε E B μ 2 2 Field En. = rg > cor < c uf = + t 2 2 G v Field En. :" watched " velocity in pulse propagation experiment v = v + v( α / ω) = v if dispersive medium with cte gain Field En. g g

99 Five velocities for a light pulse: centroid of field energy v ( ), ε E B μ 2 2 Field En. = rg > cor < c uf = + t 2 2 G v Field En. :" watched " velocity in pulse propagation experiment v = v + v( α / ω) = v if dispersive medium with cte gain Field En. g g

100 Five velocities for a light pulse: centroid of field energy v ( ), ε E B μ 2 2 Field En. = rg > cor < c uf = + t 2 2 G v Field. En. 3 Sd r 3 uf d r v Field En. :" watched " velocity in pulse propagation experiment v = v + v( α / ω) = v if dispersive medium with cte gain Field En. g g

101 Can the energy go faster than c??? relativity: energy matter and v < c!! How to solve the paradox??

102 Energy has to be defined as the total energy

103 Five velocities for a light pulse: centroid of field energy v ( ), rufd r ε E B Field En. = > cor< c u 3 F = + t udr 2 2 F μ «Open»energy G v Field. En. 3 S dr 3 uf d r vfield En = v + v( α / ω) = v if dispersive medium with cte gain. g g v :" watched " velocity in pulse propagation experiment Field En.

104 Five velocities for a light pulse: centroid of total energy v =, u = + + E dt' + u( t ) t rud r ε E B P Tot. En. 3 t ud r 2 2μ t «Closed»energy G

105 Five velocities for a light pulse: centroid of total energy v =, u = + + E dt' + u( t ) t rud r ε E B P Tot. En. 3 t ud r 2 2μ t G v c Subluminal transport of the total energy Tot. En. < :

106 Five velocities for a light pulse: centroid of total energy v =, u = + + E dt' + u( t ) t rud r ε E B P Tot. En. 3 t ud r 2 2μ t G v = 3 Sd r Tot. En. 3 ud r v c Subluminal transport of the total energy Tot. En. < : J. Peatross et al. PRL 84, 237 (2) S. Glasgow et al. PRE 64, 4661 (21)

107

108 NATURE, 46 (2)

109 Supraluminal propagation: Backward propagation. PROPAGATION THROUGH NEGATIVE MEDIUM (n g =-2)

110 SCIENCE, 312 (26)

111 SCIENCE, 312 (26) Energy flow in the forward direction B E S k Paradox: v g and vtot. En. not only differ by their magnitude but also by their Pulse envelope displacement: spatial pulse shaping sign!!

112 Alternative description amplified absorbed Any saturable absorber may re-shape (distort) the pulse Here (constant spectral gain), the re-shaping is symmetric!, Vg meaningful

113 Alternative description amplified absorbed Construc. interf Destruc. interf Any saturable absorber may re-shape (distort) the pulse Here (constant spectral gain), the re-shaping is symmetric!, Vg meaningful

114 Conclusion Fast light: better understanding of relativity implications and light pulse velocities. Relativity implications on signal and information propagation: *Signal: non analyticity *transmission of (detectable) Information faster than c possible! Relativity implications on energy propagation: *Group velocity characterizes the displacement of the envelope but is not representative of the (total) energy flux! *Group velocity is significative in a constant spectral gain medium: displacement of the EM energy, watched velocity. *Superluminal pulse envelope propagation is the result of both temporal and spatial amplitude pulse shaping by the medium. *Superluminal propagation appears when truncated balance of energy Challenge in slow light: extension to short pulses (higher bandwidth): SRS, SBS techniques. Development of quantum optical memories

9 Atomic Coherence in Three-Level Atoms

9 Atomic Coherence in Three-Level Atoms 9.1 Coherent trapping - dark states In multi-level systems coherent superpositions between different states (atomic coherence) may lead to dramatic changes of light