Statistics. on Single Molecule Photon. Diffusion. Influence of Spectral

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1 Influence of Spectral Diffusion on Single Molecule Photon Statistics Eli Barkai Yong He Bar-Ilan Feb

2 Single Molecule Photon Statistics Sub-Poissonian Statistics, Mandel s Q parameter (198). Spectral Diffusion, Kubo-Anderson Line Shape Theory (195). Influence of Spectral Diffusion on Photon Statistics. Yong He Phys. Rev. Lett. 93, 6832 (24).

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5 Mandel s Q parameter N(t) is the Random Number of photons emitted in time t. Q = N 2 N 2 N 1 Q < sub Poissonian behavior. Q > super Poissonian behavior.

6 Sub-Poissonian Photon Statistics If time between successive emission events is a constant Q = 1. Such strong correlations are impossible due to uncertainty. Q < indicates correlations between emission events. The stream of photons emitted from many independent atoms is uncorrelated.

7 Resonance Fluorescence Photon Statistics Mandel showed that for a single atom in the process of resonance fluorescence Q <. The effect is related to anti-bunching, and Rabi oscillations which favors an emission process with some periodicity in time. Q = 6Ω2 Γ 2 (Γ 2 + 2Ω 2 ) 2 When Ω = Γ/ 2, Q attains its minimum, Q min = 3/4.

8 Single Molecule Two contributions to fluctuations: Spectral Diffusion, and Quantum behavior of spontaneous emission. Anti bunching and sub-poissonian statistics are measured in many single molecule systems (Orrit, Moerner, Wild, Dahan).

9 Spectral Trail (Experiment) Single molecule (terrylene) in an amorphous matrix (polyisobutylene) at T = 2K Vainer Kador. For a similar measurement on an ensemble of molecules, fluctuations would be averaged out.

10 Kubo Anderson Line Shape Theory 1954 The line Shape I (ω L ) = lim t t N(t) A stochastic approach where δ L (t) = ω L ω ω(t) is a stochastic time dependent detuning. Cor(t) = e i t dτδ(τ) Sto Use Wiener Khintchine formula (Fourier Transform) to obtain I (ω L ).

11 Questions Analyze transitions between sub and super Poissonian behavior, induced by the spectral diffusion time scale. What is the lower bound for Q? How to choose the Rabi frequency, in such a way that quantum sub-poissonian behavior becomes strongest.

12 Moment Generating Function P N (t) Prob. of N emission events. Moment Generating function: 2Y(s) N= s N P N (t)

13 Generating Function Formalism Generalized Bloch Eqs. (Cook, Brown, Mukamel) U (s) = Γ 2 U (s) + δ L(t)V (s) V (s) = δ L (t)u (s) Γ 2 V (s) ΩW (s) Ẇ (s) = ΩV (s) Γ 2 (1 + s) W (s) Γ 2 (1 + s) Y (s) Ẏ (s) = Γ 2 (1 s) W (s) Γ 2 (1 s) Y (s). Formal solution in terms of time ordering operator. These equation are exact within RWA and Bloch equation formalism. Γ radiative decay rate. Ω Rabi frequency.

14 Stochastic Process ω(t) = νh(t) where ν describe frequency shifts, and h(t) describes a random telegraph process: h(t) = 1 or h(t) = 1. R transition rate between state up (+) and state down (-). Bare SM Laser Freq.

15 ν frequency shifts R flipping rate. Parameters Describing The Problem ω L laser freq. ω molecule s absorption freq. Γ radiative rate of electronic transition. Ω Rabi s freq.

16 Exact Solution for Q Method of Marginal Averages yields: Q = Numerator[Q] Denominator[Q] Denominator[Q] = R (4ω 2 L (1 + 2 R) + 4 ν 2 (1 + 6 R + 8 R 2 ) + (1 + 4 R) (1 + 6 R + 8 R Ω 2 )) (16 ν 4 (1 + 2 R) + 8 ν 2 (1 4 ω 2 L (1 + 2 R) + 2 Ω R (3 + 4R)(1 + Ω 2 ))+ (1 + 4 ω 2 L + 2 Ω 2 ) (4ωL 2 (1 + 2 R)+ (1 + 4 R) (1 + 6 R + 8 R Ω 2 ))) 2 (1)

17 Numerator[Q] = 2 Ω 2 (256 ν 8 R (1 + 2 R) 3 ( R 2 ) ( ω L 2 ) R (ω L 2 (4 + 8 R)+ (1 + 4 R) (1 + 6 R + 8 R Ω 2 )) ν 6 (1 + 2 R) 2 (4 ω 2 L (1 + 4 R + 4 R R R 4 ) 3 R (1 + 4 R) (24 R R 3 Ω R (1 + Ω 2 ))) + 32 ν 4 (1 + 2 R) (16 ω 4 L (1 + 2 R) 2 (2 + 5 R + 8 R 3 ) + 4 ω 2 L (1 + 6 R + 8 R 2 ) ( R R 5 4 Ω 2 + R 2 (76 8 Ω 2 ) R ( 4 + Ω 2 ) + 16 R 3 (1 + Ω 2 )) + 3 R (1 + 4 R) ( R R 5 + Ω 2 2 Ω R 3 ( Ω 2 ) + 4 R 2 ( Ω 2 ) R ( Ω Ω 4 ))) 8 ν 2 (64 ω 6 L (1 + 2 R) 4 R (1 + 4 R) (1 + 6 R + ( R R 3 Ω R (11 + Ω 2 )) 16 ω 4 L (1 + 2 R) 2 ( R R ( Ω 2 )) 4 ω 2 L (1 + 2 R) (768 R R R 5 (37 + Ω 2 ) (1 + 2 Ω 2 ) R 4 ( Ω 2 ) + 8R 3 ( Ω 2 8 Ω 4 )+ 4R 2 ( Ω 2 12 Ω 4 ) 2 R ( Ω Ω 4 )))). (2)

18 ω L ω ( MHz ).6.4 Q ( ω ω ) L (b).6 2 I ( ω ω ) L (a) 1 x 16 Γ = 4 MHz, Ω = Γ/ 2, ν = 2Γ fast R = 1Γ, intermediate R = Γ/2, slow R = Γ/1

19 Fast Modulation Limit Using natural units Γ = 1, ω = lim Q = 2Ω2 R ( 3 4ω 2 L ) ( 1 + 2Ω 2 + 4ω 2 L ) 2. A result obtained by Mandel. When R the laser field cannot respond to the fluctuations induced by spectral diffusion.

20 Let R and ν keep Γ SD lim ν,r ν 2 R In this case the line is Lorentzian and exhibits motional narrowing. [3 Qfast = 2Ω2 + 5Γ 2 SD + Γ3 SD 4ω L 2 ( + Γ SD 7 + 4ω L 2 )] ( [1 + Γ 2 SD + 2Ω 2 + 2ΓSD 1 + Ω 2) + 4ω L 2 ] 2. A sub-poissonian behavior.

21 Q ( ω L ω ) I ( ω L ω ) 1 x ( a ) ( b ) ωl ω ( MHz )

22 Q ( Ω ) I ( Ω ) 2 x ( a ) ( b ) Ω / Γ

23 Optimization of Rabi Freq. When are quantum fluctuations largest? How to choose the Rabi freq.? Global minimum at zero detuning Q min = 3/4 when ν = and Ω = 1/ 2. Minimum Qmin = Γ SD + 3 4(1 + Γ SD ) Ωmin = 1 + Γ SD 2.

24 Γ t 1 (ω L ω ) / Γ x counts (ω L ω ) / Γ total counts Slow Spectral Trail 1

25 Q ( ω L ω ) I ( ω L ω ) 6 x 16 5 ( a ) x ( b ) ωl ω ( MHz )

26 Qslow = ( I + I ) 2 2R(I + +I ), I± = Ω 2 1+2Ω 2 +4(ω L ω ν) 2 The intensity is jumping between two states, simple random walk picture. I± are steady state solutions of time independent Bloch equations.

27 A E SM - Glass System J TLS SM 2 1 Phonon

28 Single Molecules in Glass-Slow Dynamics Single molecule (terrylene) in an amorphous matrix (polyisobutylene) at T = 2K Vainer Kador, Orrit.

29 Q(t) in Single Molecule Experiments Single molecules in low temperature glass, exhibit behavior compatible with the slow modulation limit. Single molecules in low temperature ordered matrix, reveal: anti-bunching, Rabi-Oscillations. Q(t) similar to Mandel s theory.

30 Summary The line shape is describes well by Kubo-Anderson classical model. The fluctuations of the number of photon counts exhibit quantum behavior. Mandel s Q parameter exhibits a transition between sub-poissonian behavior, in the fast modulation limit, to super-poissonian behavior, in the slow modulation limit. An optimal Rabi frequency exists, which maximizes quantum fluctuations. Our results are in agreement with experiments investigating slow dynamics (Orrit).

31 Γ t 1 (ω L ω ) / Γ x counts (ω L ω ) / Γ total counts Fast Spectral Trail 1

32 Q ( ω L ω ) I ( ω L ω ) 3 x ( a ) ( b ) ωl ω ( MHz )

33 Q ( ω L ω ) I ( ω L ω ) 5 x ( a ) ( b ) ωl ω ( MHz )