Niklas Rehfeld. Universität Konstanz. Diploma Thesis Niklas Rehfeld p.1/21

Transcription

1 The Theory of the Manipulation of Molecules with Laser Beams Manipulation of the Spontaneous Emission Rate in Diatomic Molecules Diploma Thesis Niklas Rehfeld Universität Konstanz Diploma Thesis Niklas Rehfeld p.1/21

2 Overview Introduction Diploma Thesis Niklas Rehfeld p.2/21

3 Overview Introduction Diatomic molecules + radiation Diploma Thesis Niklas Rehfeld p.2/21

4 Overview Introduction Diatomic molecules + radiation The master equation Diploma Thesis Niklas Rehfeld p.2/21

5 Overview Introduction Diatomic molecules + radiation The master equation Maximization of the spontaneous emission rate Diploma Thesis Niklas Rehfeld p.2/21

6 Overview Introduction Diatomic molecules + radiation The master equation Maximization of the spontaneous emission rate Numerical examples Diploma Thesis Niklas Rehfeld p.2/21

7 Overview Introduction Diatomic molecules + radiation The master equation Maximization of the spontaneous emission rate Numerical examples Time evolution Diploma Thesis Niklas Rehfeld p.2/21

8 Overview Introduction Diatomic molecules + radiation The master equation Maximization of the spontaneous emission rate Numerical examples Time evolution Summary Diploma Thesis Niklas Rehfeld p.2/21

9 Principle µ µ 2 1 µ 0 } vibrational levels of the electronic upper state ν ν 0 1 } vibrational levels of the electronic ground state Suppose a system is in a superposition of excited states and couples to the vacuum. Diploma Thesis Niklas Rehfeld p.3/21

10 Principle µ µ 2 1 µ 0 } vibrational levels of the electronic upper state ν ν 0 1 } vibrational levels of the electronic ground state Suppose a system is in a superposition of excited states and couples to the vacuum. Assume that different pathways to the ground state interfere. Diploma Thesis Niklas Rehfeld p.3/21

11 Principle µ µ 2 1 µ 0 } vibrational levels of the electronic upper state ν ν 0 1 } vibrational levels of the electronic ground state Suppose a system is in a superposition of excited states and couples to the vacuum. Assume that different pathways to the ground state interfere. Effectively, this can result in a modification of the spontaneous emission rate. Diploma Thesis Niklas Rehfeld p.3/21

12 Why (diatomic) molecules? The pathways of orthogonal dipole moments do not interfere. Diploma Thesis Niklas Rehfeld p.4/21

13 Why (diatomic) molecules? The pathways of orthogonal dipole moments do not interfere. Electronic dipole moments of the same electronic transition are parallel for all vibrational quantum numbers. Diploma Thesis Niklas Rehfeld p.4/21

14 Why (diatomic) molecules? The pathways of orthogonal dipole moments do not interfere. Electronic dipole moments of the same electronic transition are parallel for all vibrational quantum numbers. Parallel dipole moments are not easy to obtain in atoms. Diploma Thesis Niklas Rehfeld p.4/21

15 Why (diatomic) molecules? The pathways of orthogonal dipole moments do not interfere. Electronic dipole moments of the same electronic transition are parallel for all vibrational quantum numbers. Parallel dipole moments are not easy to obtain in atoms. Advantage over atoms. Diploma Thesis Niklas Rehfeld p.4/21

16 Why (diatomic) molecules? The pathways of orthogonal dipole moments do not interfere. Electronic dipole moments of the same electronic transition are parallel for all vibrational quantum numbers. Parallel dipole moments are not easy to obtain in atoms. Advantage over atoms. Modification of the spontaneous emission rate could make cyclic processes like laser cooling of molecules feasable. Diploma Thesis Niklas Rehfeld p.4/21

17 Coupling to the electromagnetic field Born-Oppenheimer approximation in molecules Ψ = ψ e ( r; R)χ(R, ϑ, ϕ) = ψ e ( r; R)P (R)Φ(ϑ, ϕ) Diploma Thesis Niklas Rehfeld p.5/21

18 Coupling to the electromagnetic field Born-Oppenheimer approximation in molecules Ψ = ψ e ( r; R)χ(R, ϑ, ϕ) = ψ e ( r; R)P (R)Φ(ϑ, ϕ) Interaction Hamiltonian: H int = ( d + D) E Diploma Thesis Niklas Rehfeld p.5/21

19 Coupling to the electromagnetic field Born-Oppenheimer approximation in molecules Ψ = ψ e ( r; R)χ(R, ϑ, ϕ) = ψ e ( r; R)P (R)Φ(ϑ, ϕ) Interaction Hamiltonian: H int = ( d + D) E Dipole matrix element M fi = d 3n r d 3 R ψef ( r, R)χ f ( R)( d + D)ψ ei ( r, R)χ i ( R) = + dω d 3 R χ f dr Pf (R)Φ f (ϑ, ϕ) D χ i M e fi (R;ϑ,ϕ) {}}{ ( d 3n r ψ ef d ψ ei ) Φ i (ϑ, ϕ)p i (R) d 3 r ψ efψ ei }{{} =0 Diploma Thesis Niklas Rehfeld p.5/21

20 The Franck Condon Integral When M e fi (R 0 + x; ϑ, ϕ) M e fi (R 0; ϑ, ϕ): M fi = f fi dω Φ f (ϑ, ϕ) M fi e(r 0; ϑ, ϕ)φ i (ϑ, ϕ) } {{ } selection rules of the rotational levels f fi := dr Pf (R)P i (R) Franck Condon Integral Diploma Thesis Niklas Rehfeld p.6/21

21 The Franck Condon Integral When M e fi (R 0 + x; ϑ, ϕ) M e fi (R 0; ϑ, ϕ): M fi = f fi dω Φ f (ϑ, ϕ) M fi e(r 0; ϑ, ϕ)φ i (ϑ, ϕ) } {{ } selection rules of the rotational levels f fi := dr Pf (R)P i (R) Franck Condon Integral Diploma Thesis Niklas Rehfeld p.6/21

22 Example µ µ 2 1 µ 0 } vibrational levels of the electronic upper state ν1 ν 0 } vibrational levels of the electronic ground state The adiabatic potential curves for the lowest electronic states of CN. The graph is taken from A. A. Radzig and B. M. Smirnov. Reference Data on Atoms, Molecules, and Ions. Springer- Verlag, Berlin, In general possible: Vibrational levels of the ground state that are energetically higher than levels of the first excited state Diploma Thesis Niklas Rehfeld p.7/21

23 The Master Equation Master equation in the Schrödinger picture: σ(t) = il mol σ(t) Tr R([V, t 0 dτe iτl 0 [V, 0 R 0 R σ(t τ)]]) 2 V :=V 0 ωk a ks ɛ ks dµν µ ν + H.C. k,s µ ν L mol := 1 [H mol, ] := [ µ µ µ µ + ν ν ν ν, ] L 0 := L mol + k,s [ω k a ks a ks, ] Diploma Thesis Niklas Rehfeld p.8/21

24 The Master Equation (2) σ µa µ B = i(µ A µ B )σ µa µ B d eg 2 { 6π 2 ɛ 0 c 3 ν,ν 1 (r ν 1µ B νµ A + r νµ A ν 1 µ B )σ νν1 } µ,ν σ νa ν B = i(ν A ν B )σ νa ν B d eg 2 { 6π 2 ɛ 0 c 3 µ,µ 1 (r µ 1ν B µν A + r µν A µ 1 ν B )σ µµ1 } ν,µ (r µν (r νµ µ A νσ µµb + r µ µν B νσ µa µ) ν A µ σ νν B + rν νµ B µ σ ν A ν) r µ 1ν 1 µν := Θ(µ 1 ν 1 )Θ(µ ν)f µνf µ1 ν 1 (µ 1 ν 1 ) 3 Diploma Thesis Niklas Rehfeld p.9/21

25 Maximization at a fixed time point σ νa ν B = i(ν A ν B )σ νa ν B + d eg 2 Assumption: pure state 6π 2 ɛ 0 c 3 ( µ,µ 1 (r µ 1ν B µν A + r µν A µ 1 ν B )σ µµ1 ) }{{} to be maximized Goal: maximization of the spontaneous emission rate into one special lower level Method: method of Langrangian multipliers Diploma Thesis Niklas Rehfeld p.10/21

26 Maximization 2 Def.: χ µ 1 µ := χ µ 1ν B µν A := r µ 1ν B µν A + r µν A µ 1 ν B Aussumption: σ µµ1 = c µ c µ 1 To be maximized: µµ 1 χ µ1 µ σ µµ 1 + λ( µ σ µµ 1) Result: µ χ α µc µ + λc α = 0 Diploma Thesis Niklas Rehfeld p.11/21

27 Results so far Maximization of the spontaneous emission rate into one special lower level is possible. Diploma Thesis Niklas Rehfeld p.12/21

28 Results so far Maximization of the spontaneous emission rate into one special lower level is possible. However, calculations are so far only for fixed time. Diploma Thesis Niklas Rehfeld p.12/21

29 Results so far Maximization of the spontaneous emission rate into one special lower level is possible. However, calculations are so far only for fixed time. The effect of maximization depends strongly on the involved Franck Condon factors. Diploma Thesis Niklas Rehfeld p.12/21

30 Numerical example: CN molecule Lennard-Jones potential fit. Relative transition rate: µ 0 ν 0 : 0.56 Σ(µ 0 µ 4 ) ν 0 : 0.94 Diploma Thesis Niklas Rehfeld p.13/21

31 Numerical example: N 2 -like molecule Relative transition rate: µ 0 ν 0 : Σ(µ 0 µ 9 ) ν 0 : Diploma Thesis Niklas Rehfeld p.14/21

32 Num. example: Harmonic potentials Joule Joule x x10 11 m ω e X m m 6x m 1x x10 11 m 1x x10 11 m 1x x10 11 m ω e X ω e X ω e X 6x m 6x m 6x m Diploma Thesis Niklas Rehfeld p.15/21

33 A simple picture Condition for maximum: µ χ α µc µ + λc α = 0 with σ αα1 = c α c α 1 χ µ 1 µ =r µ 1ν 0 µν 0 + r µν 0 µ 1 ν 0 Diploma Thesis Niklas Rehfeld p.16/21

34 A simple picture Condition for maximum: µ χ α µc µ + λc α = 0 with σ αα1 = c α c α 1 χ µ 1 µ =f µν 0 f µ1 ν 0 (µ 1 ν 0 ) 3 + f µ 1 ν 0 f µν0 (µ ν 0 ) 3 Diploma Thesis Niklas Rehfeld p.16/21

35 A simple picture Condition for maximum: µ χ α µc µ + λc α = 0 ω with σ αα1 = c α c α 1 ω eg χ µ 1 µ =f µν 0 f µ1 ν 0 (µ 1 ν 0 ) 3 + f µ 1 ν 0 f µν0 (µ ν 0 ) 3 ω eg ω : χ µ 1 µ 2ω 3 egf µ1 ν 0 f µν 0 = 2ω 3 eg f ν0 f ν0 This is true for real Franck Condon integrals. Diploma Thesis Niklas Rehfeld p.16/21

36 Discussion of the simple picture In this limit it is very easy to determine the optimal superposition: The coefficients of the superposition are simply the overlap (scalar product) of the upper nuclear radial wave function of concern (µ = 0... n) and the lower nuclear radial wave function (here ν 0 ). c α f αν0 Diploma Thesis Niklas Rehfeld p.17/21

37 Discussion of the simple picture In this limit it is very easy to determine the optimal superposition: The coefficients of the superposition are simply the overlap (scalar product) of the upper nuclear radial wave function of concern (µ = 0... n) and the lower nuclear radial wave function (here ν 0 ). c α f αν0 and the maximal spontaneous emission rate is Γ max = d eg 2 ( 3ɛ 0 π 2 c 3 ω3 eg f αν0 2) α Diploma Thesis Niklas Rehfeld p.17/21

38 Time evolution Problem: The different energies of the upper vibrational eigenstates cause the eigenfunctions to de-phase and do destroy the superposition. Diploma Thesis Niklas Rehfeld p.18/21

39 Time evolution Problem: The different energies of the upper vibrational eigenstates cause the eigenfunctions to de-phase and do destroy the superposition. Does this weaken the effect? The population of ν 0 of CN for t (0s, s). Superposition of 5 upper levels. Diploma Thesis Niklas Rehfeld p.18/21

40 Time evolution (2) Level 0 Level 1 Level 2 Level 3 Level 4 probability e-13 2e-13 3e-13 4e-13 5e-13 time [s] The population of level ν 0 of CN for t (0s, s). The straight lines stand for the population if the molecule initially is in state µ [0, 4]. In this case, there is no interference and the time evolution of the population is just like 1 e Γµ ν 0 t. Diploma Thesis Niklas Rehfeld p.19/21

41 Summary Modification of the spontaneous emission rate is possible because electronic dipole moments are parallel. Diploma Thesis Niklas Rehfeld p.20/21

42 Summary Modification of the spontaneous emission rate is possible because electronic dipole moments are parallel. A method to calculate the optimal superposition has been found. Diploma Thesis Niklas Rehfeld p.20/21

43 Summary Modification of the spontaneous emission rate is possible because electronic dipole moments are parallel. A method to calculate the optimal superposition has been found. The maximal spontaneous emission rate depends crucially on the involved Franck Condon factors. Diploma Thesis Niklas Rehfeld p.20/21

44 Summary Modification of the spontaneous emission rate is possible because electronic dipole moments are parallel. A method to calculate the optimal superposition has been found. The maximal spontaneous emission rate depends crucially on the involved Franck Condon factors. Unfortunately the energy separation of the levels of the excited electronic state leads to de-phasing and does destroy the superposition. Diploma Thesis Niklas Rehfeld p.20/21

45 Summary Modification of the spontaneous emission rate is possible because electronic dipole moments are parallel. A method to calculate the optimal superposition has been found. The maximal spontaneous emission rate depends crucially on the involved Franck Condon factors. Unfortunately the energy separation of the levels of the excited electronic state leads to de-phasing and does destroy the superposition. The effect is limited by a time τ vib π/ ω = π / E. After this time the spontaneous emission rate starts to rise again due to de-phasing. Diploma Thesis Niklas Rehfeld p.20/21

46 Improvements? Shallow adiabatic excited state potential could diminish the de-phasing. Diploma Thesis Niklas Rehfeld p.21/21

47 Improvements? Shallow adiabatic excited state potential could diminish the de-phasing. Coupling of strong lasers: Pulsed laser with very high repetition rate ( /sec) Strong continuous wave laser Diploma Thesis Niklas Rehfeld p.21/21

48 Improvements? Shallow adiabatic excited state potential could diminish the de-phasing. Coupling of strong lasers: Pulsed laser with very high repetition rate ( /sec) Strong continuous wave laser Only use two upper levels and make use of the periodic change of the spontaneous emission rate. Diploma Thesis Niklas Rehfeld p.21/21