adiabatic transfer processes TECHNISCHE UNIVERSITÄT KAISERSLAUTERN Lecture 7 Coherent light-matter interaction: Optically driven adiabatic transfer processes K. Bergmann Lecture course - Riga, fall 2013

summary of 6th lecture: spectral properties of 2 + 1system |a+> = c+,g |g> + c+,e |e> |a-> = c-,g |g> + c-,e |e> |f> e f g S P bare states weak strong J = 7 population may reach level f via two different paths interference structure Ref. 71

summary of 6th lecture: Rate equation (incoherent radiation) for 3-LS e.g. Gaussian pulse shape 2 1 3 P S 4 loss to other levels 50% 33% coincident pulses maximum transfer: 33% reached only without loss through spontaneous emission 50% 33% delayed pulses maximum transfer: 25 % reached only without loss through spontaneous emission

S P repetition: electromagnetically induced transparency Lorentzian profile amplitude of transition dipole moment: <1| |a>  <1||3>  <1||2> File: Lorentz0 …50_2xSUM needed bare P 1 3 S 2 adiabatic AT fluorescence two transition dipole moments: 180o out of phase Intensity ≠ 0 ? P – laser frequency

spectral profile probed on |1> -- |2> transition repetition: electromagnetically induced transparency spectral profile probed on |1> -- |2> transition File: EIT_Final-1 as coupling  of |2> -- |3> increases intensity start EIT/AT probe laser frequency (|1> -- |2>)  = decay rate of level |2>

repetition: electromagnetically induced transparency (EIT)  (much) smaller than : ( = natural line width) interference structure (narrow) observed, which is called: EIT also zero (EIT) between AT-components  (much) larger than : two separate features (of width ≈ ) observed, called AT-splitting

summary of 6th lecture: EIT bare states P S dressed states P S Ne* 2 1 3 P S Experiment transition dipole moments |1>  |a+> and |1>  |a-> are 180o out of phase transition amplitude zero when frequency tuned to the centre between the states Ref. 24 exactly zero

understanding the relevance of the “zero-energy” eigenvalue the goals for today understanding the structure of the 3-level RWA Hamiltonian in analogy to the 2-level case. understanding the structure of the 3-level eigenvalues und eigenfunctions understanding the relevance of the “zero-energy” eigenvalue understanding the basics of the STIRAP process for very efficient, lossless transfer op population between levels seeing examples for successful applications of the STIRAP method

3.3 The 3-level RWA Hamiltonian 3.3.1 Basics 3.3.2 Adiabatic energies and adiabatic states 3.3.3 The relevance of the zero-energy eigenstate 3.3.4 Variation of |<1|ao>|2 with laser overlap 3.3.5 Preview of the STIRAP process

3.3.1 3-level system: basics -- parameters reminder: some basic facts 2 Rabi frequency ik = ik EL / S  E2,3  E1,2 P 2,3 ik = <i|er|k> transition dipole moment 23 12 1,2 3 1 ik << E1,3  E1,3 each field couples one pair of levels only selectivity by frequency and/or polarization J = 0, M = 0 M = 0, J = 1 M = -1 M = +1  +  -  large pulse area:  T >  many Rabi cycles in two-level systems adiabatic evolution

3.3.1 3-level system: basics -- relevant detunings 2 1 3 P S P = S  0  E1,2 2 1 3 P S P = S = 0 one- and two-photon resonance only two-photon resonance P also quoted es 1, with index „1“ for one-photon detuning  E1,2 2 1 3 P S P P - S  E1,2 2 1 3 P S P S 1 = same thing differently drawn 2 = no resonance no resonance

 H ~ 3.3.1 3-level system: the RWA Hamiltonian 2 2  1  E1,2  E1,2 P S 1 2 unitary transformation  2-level system 3-level system  RWA Hamiltonian H ~ 

the adiabatic energies for X = 0 3.3.1 3-level system: the RWA Hamiltonian  E1,2 2 1 3 P S 1 2 choice of zero-point of energy no coupling between levels 1 and 3 „reading“ the 3-level RWA Hamiltonian coupling of level 1 to all other levels  coupling of level 2 to all other levels coupling of level 3 to all other levels the adiabatic energies for X = 0

P S eff = (2P + 2S) 1/2 1 = 0, eff = 0  1 = 0, eff  0  3.3.1 3-level system: basics 2 1 3 P S E eff = (2P + 2S) 1/2 Interaction Hamiltonian: two photon resonance P = S preview: energy of adiabatic states: 1 = 0, eff = 0  1 = 0, eff  0  eff

S P eff = (2P + 2S) 1/2 eff  1 = 0,  = 0 ε Ω t 3.3.1 3-level system: basics 2 1 3 P S E eff = (2P + 2S) 1/2 Interaction Hamiltonian: two photon resonance P = S preview: Ω ΩS ΩP t ε ε+ ε0 ε- eff energy of adiabatic states: 1 = 0,  = 0 

3.3.2 3-level system: adiabatic energies and adiabatic states time-independent Schroedinger equation: (TISE) |a> = (a1, a2, a3) is a vector in a three-dimensional Hilbert space TISE in matrix form (and RWA): three linear independent (mutually orthogonal) solutions |ao,> with energies wanted rewritten and with indices for solutions added:

WP WS 3.3.2 3-level system: adiabatic energies and adiabatic states special case: index omitted 1 3 2 WP WS 4 solution, when determinant = 0, yields eigenvalues, for one- and two-photon resonance: ! two-photon resonance one-photon resonance one solution, two more solution: ½

WP WS 3.3.2 3-level system: adiabatic energies and adiabatic states ½ the three-state adiabatic energies (for one- and two-photon resonance) are: 1 3 2 WP WS 4 ½ new feature for 3-level system eff P = S = 0 P and/or S  0

3.3.2 3-level system: adiabatic energies and adiabatic states detuning i  0 allowed 2 1 WS 2 = 0 WP 3 solution when determinant = 0 yields eigenvalues 4 1 ! leads, in general, to a 3rd power polynomial which may not have a (physically meaningful) solution two-photon resonance X X is again a solution, IF 2 = 0 2 = 0 ½

WS WP 3.3.2 3-level system: adiabatic energies and adiabatic states 1 adiabatic energies (eigen values) with 1 0 allowed WS ½ 2 = 0 WP  exists, whenever 2 = 0 3 4 1 eff = 0 1 ½ for 1 = 0: o, - and + degenerate when eff = 0 for 1 > 0: o and - degenerate when eff = 0 for 1 < 0: o and + degenerate when eff = 0

o = 0 WP WS P = 0  = 0 P = S  = 45o S = 0  = 90o 3.3.2 3-level system: adiabatic energies and adiabatic states 1 3 2 WP WS 4 the RWA Hamiltonian, 1 = 0: o = 0 to be shown: verify for yourself: the adiabatic state eigenfunctions: special ! P = 0  = 0 P = S  = 45o S = 0  = 90o independent of S independent of P

3.3.2 3-level system: adiabatic energies and adiabatic states verification that is the eigenfunction with eigenvalue  = 0 for ! = 0 ! o = 0 ! verified

o = 0 WP WS 1 = 0   = 45o  cos  = sin  = 3.3.2 3-level system: adiabatic energies and adiabatic states 1 3 2 WP WS 4 the RWA Hamiltonian, 1  0: reported only: o = 0 the adiabatic state eigenfunctions, 1≠ 0 allowed: special solution remains ! 1 = 0   = 45o  cos  = sin  =

WP WS 3.3.2 3-level system: adiabatic energies and adiabatic states 1 3 2 WP WS 4 |1| >> eff then tan 2 ≈ sin 2 ≈ 2  the adiabatic state eigenfunctions, 1≠ 0 allowed: decay rate leak through „leak“ (state 2) during the transfer process: proportional to |<1|a>| and to coefficient of |2>, i.e. proportional sin  cos  → for small  (large 1) → leak ~ will be added to the lecture material for full analytic solutions with both 1 ≠ 0 and 2 ≠ 0 allowed, see Fewell, Shore, Bergmann: Australian J. Phys. 50, 281 – 308 (1997)

Example: three-level system 3.3.3 relevance of the zero-energy eigenvalue 2 1 3  c  a Example: three-level system relevance of the zero-energy eigenvalue |ao> = co1|1> + co2|2> + co3 |3> underlines the relevance of the two-photon resonance  = 0 c2  0

S P 3.3.4 Variation of <1| ao>|2 with pulse overlap 2 The movie shows the variation of the adiabatic state coordinateswith respect to the bare state coordinates as a function of the overlap between S and P laser When we have initially ΩS≠ 0 and ΩP = 0 then  = 0, therefore |ao>, |1> and |> are all parallel: only the adiabatic state |ao> is populated File: STATEVECTORMOVIE100808

3.3.5 STIRAP features - examples Ne* 3P1 3P0 3,1P1 |1ñ |3ñ |2ñ S P final level population radiation from |2> transfer metastable levels radiation from |2> interaction time >> rad

3.3.5 preview of STIRAP features – examples, the Ne* level scheme the carrier of a suitable level system: Ne* intermediate  transfer thermal population depleted by optical pumping probing ↑ ↑ final, J = 2 initial, J = 0 ← detection

WS WP 3.3.5 preview of STIRAP features - examples Ne*, Na2, SO2, NO, … particle beam 1 3 2 WP WS 4 the „signature“ of the STIRAP process mixing angle changes smoothly from 0o to 90o mixing angle = 45o

4. Stimulated Raman Adiabatic Passage (STIRAP) 4.1 Details of the transfer process 4.2 Conditions for adiabatic evolution 4.3 The transfer process in the state vector picture 4.4 Experimental verification

WS WP WS WP 4.1 Details of the STIRAP-transfer process The building blocks for the STIRAP process are now on hand: process to be completed in a time short compared to lifetime of |3> (e.g. Rydberg states) 3-level-system eigenvalues and eigenfunctions and: EIT – AT – AP electro-magn. ind. transp. – Autler Townes – adiabatic passage 1 3 2 WP WS 4 …..putting them together for: 1 3 2 WP WS 4 a lambda level-system a ladder level-system

WP WS 4.1 Details of the STIRAP-transfer process Rabi frequencies 3 2 WP WS 4 Rabi frequencies tan  = P / S adiabatic energies mixing angle - populations

4.1 Details of the STIRAP-transfer process 2 adiabatic states degeneracy bare states 3 tan  = P / S 4 1 -

WS WP Wp = 0 4.1 Details of the STIRAP-transfer process Autler-Townes 2 Autler-Townes WS WP Wp = 0 3 tan  = P / S 4 1 |1> -

WS WP Wp  0 4.1 Details of the STIRAP-transfer process EIT at work 2 EIT at work WS WP Wp  0 3 tan  = P / S P radiation NOT absorbed 4 1 |1> -

WS WP 4.1 Details of the STIRAP-transfer process transfer - 2 3 tan  = P / S 4 1 |1> -

WS WP WS  0 4.1 Details of the STIRAP-transfer process EIT at work 2 EIT at work WS WP WS  0 3 tan  = P / S S radiation NOT absorbed 4 1 |1> -

WS WP WS = 0 4.1 Details of the STIRAP-transfer process Autler-Townes 2 Autler-Townes WS WP WS = 0 3 tan  = P / S 4 1 |1> |3> -

| <a+ | ao > | << | o - + | 4.2 Conditions for adiabatic evolution The reasoning follows closely what was presented for the 2-level systems eigen-states of Hamiltonian are: |ao>, |a+>, |a-> eigen-energies are: o + - population initially in |ao> and it should stay there Hamiltonian is time-dependent, non-adiabatic coupling unavoidable : |ao> |a+> and |a-> quantum mechanics: non-adiabatic (or „diabatic“) coupling is small, if | <a+ | ao > | << | o - + | ∙

| <a+ | ao > | << | + - o | 4.2 Conditions for adiabatic evolution | <a+ | ao > | << | + - o | ∙ ∙ ∙ - ∙ when is valid ∙ then also is valid

| <ao | a+ > | << | o - + | 4.2 Conditions for adiabatic evolution | <ao | a+ > | << | o - + | ∙ ∙ see ref. 2 d |a+> d |ao> ∙ |ao> ∙ „local“ averaged over pulse duration  T: < |  | > = ½  / T ∙ << eff eff T >> 1 „global“: Ref. 2

adiabatic evolution if: eff  >> 1 4.2 Conditions for adiabatic evolution eff =  P2 + S2 2 WS WP 3 tan  = P / S 4 1 |1> |3>  adiabatic evolution if: eff  >> 1

4.2 a mechanical analog of STIRAP quantum states |1> |2> |3> quantum states |1> |2> |3> File: Pendel mit Kopplung File: Pendel ohne Kopplung amplitude of vibration corresponds to level population

2eff T >>>(1/T) 4.2 Conditions for adiabatic evolution ∙ „local“ eff T >> 1 „global“: applicable when (phase) fluctuations during the interaction period are small The global condition suffices for nearly transforme limited pulses (constant phase and smooth envelope) The local condition must be evaluated for pulses with fluctuating phase (and/or deviation from smooth evolution of the envelope) model needed Global adiabaticity criterion in terms of pulse energy pulse energy eff T >> 1 2eff T2 >>> 1 2eff T >>>(1/T) problems with large Stark shift ! energy required for „adiabatic evolution“ increases as 1/T if e.g. E = 1 J is o.k. for a 10 ns pulse, E = J needed for 10 fs pulse

4.2 Transfer process in the state vector picture - adiabatic evolution evolution is adiabatic : File: ADIA observe the blue arrow

4.2 Transfer process in the state vector picture – nonadiabatic evolution evolution is NOT adiabatic : File: NADIA observe the blue arrow

Questions related to the topics discussed in lecture 7 (7.1) Write down the 3-level RWA Hamiltonian for the case that the frequencies of the two fields satisfy the conditions for two-photon resonance, i.e. the energy difference between the P-field and S-field photons equal the energy difference of the levels 1 and 3. (7.2) Verify the validity of the eigenfunctions |a+> and |a-> for the three- level system (two-photon detuning 2 = 0). (7.3) Calculate and plot the variation of the three-level splitting for a Gaussian pulse and one-photon detuning 1 smaller, equal, larger and much larger than the chosen Rabi frequency. (7.4) How does the contribution of (the initially populated) bare state |1> to the adiabatic state |ao> vary with the overlap of S and P pulse ? (7.5) Discuss the „five phases“ of the STIRAP process, each showing prodeminantly the importance of Auter-Townes splitting, EIT, adiabatic passage (7.6) Show that we have a zero-energy eigenvalue, whenever two-photon resonance is valid, independent of the one-photon detuning

end of 7th lecture Coherent light-matter interaction: Optically driven adiabatic transfer processes end of 7th lecture