adiabatic transfer processes TECHNISCHE UNIVERSITÄT KAISERSLAUTERN Lecture 9 Coherent light-matter interaction: Optically driven adiabatic transfer processes K. Bergmann Lecture course - Riga, fall 2013

Informal voluntary test (2) for “Optically driven adiabatic transfer processes” Nov. 22, 2013 Identification (Name or any code – for anonymity): ……………………………………… (1) Name key conditions (properties of the radiation field, such as intensity, bandwidth, detuning, pulse shape) required for observing the phenomenon called “coherent population return (CPR)” ? coherence, detuning larger than natural linewidth, smooth pulse shape and sufficiently high intensity to guarantee adiabatic following (2) What physical mechanism causes the phenomenon of electro-magnetically induced transparency (EIT) ? Interference of amplitudes of transition dipole moment to the two adiabatic states related to the coupling of two bare states by the (strong) radiation field Write down the RWA-Hamiltonian for a three level system interacting with two lasers field (P- and S-laser) for the case of two-photon resonance. 1  Calculate the eigenvalues of the 3-level RWA Hamiltonian, see (4), for the case of one-photon resonance, i.e. P = S = 0. ½

down-chirp, best with intensity peak (5) Consider the multi-level system shown below. The ground state level can be coupled by a frequency-variable laser to three excited state. The transform limited bandwidth of the laser is large enough (i.e. the pulse length is short enough) to be able to excite all three level at once. What needs to be done when only the upper one of the three levels shall be populated ? Explain the process is the frame of adiabatic energy, with needed frequency, pulse timing and pulse duration shown. down-chirp, best with intensity peak at the time when the crossing with the upper state occurs  E

signing up for the exam today and on Friday comment regarding the exam 90 minutes about 8 – 10 problems passing grade guaranteed if 1/3 of maximum points reached suggestions for the preparation: check the questions for each lecture get acquainted with the determination of eigenvalues and eigenfunctions work in small teams and learn with and from each other signing up for the exam today and on Friday

summary of 8th lecture – the five phases of STIRAP The five phases of the transfer process Rabi frequencies 1 3 2 WP WS 4 eigenvalues mixing angle populations AT by S P = 0 EIT for P by S transfer AP EIT for S by P AT by P S = 0

summary of 8th lecture – the five phases of STIRAP The five phases of the transfer process Rabi frequencies 1 3 2 WP WS 4 eigenvalues <a±|ao>   maximum here ∙ mixing angle populations largest rate of change of mixing angle = highest risk of non-adiabatic losses relevant for understanding optimal delay of pulses for maximum transfer largest AT splitting = best protection against non-adiabatic losses

summary of 8th lecture - adiabaticity criterion ∙ „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 d |a+> |ao> d|ao> Comment: d|ao> is a vector of arbitrarily short length. therefore both |ao> and ao> + d|ao> are of length „unity“ , as required. alternatively, one may consider the angle between ao> and d|ao> to be 90o - , with  being an arbitrarily small angle

adiabaticity criterion for transfer within an open 3-level system the goals for this lecture adiabaticity criterion for transfer within an open 3-level system criterion for pulsed lasers (in terms of energy) modification for non-transforme limited radiation dependence on geometry examples what determines the optimum pulse delay ? further experimental results Lecture 10: extension to multilevel systems (e.g. matter wave mirrors) plus: the new concept of SCRAP (Stark chirped rapid adiabatic passage) including the concept of “adiabatic elimination” ( 3-LS  2-LS)

2eff T >>>(1/T) 4.2 Conditions for adiabatic evolution – rewritten in terms of pulse energy 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 ! (see next lecture: adiabatic elimination) energy required for „adiabatic evolution“ increases as 1/T if e.g. E = 1 J is o.k. for a 10 ns pulse, E = 1 J needed for 10 fs pulse for a shorter pulse, stronger coupling is required to guarantee adiabatic following (AF) higher pulse energy is needed higher pulse energy may lead to undesirable multi-photon processes AF condition is difficult (if not impossible) to meet for e.g. fs-pulses

Further comments on Stimulated Raman Adiabatic Passage (STIRAP) further examples decay rate („leak“) through |a+> and |a-> states adiabatic conditions involving laser pulses the role of geometry/shape of the laser beam profiles

WS WP 3.3.5 STIRAP features - examples reminder: the „signature“ of the STIRAP process for on resonance tuning and short-lived state 2 Ne* 1 3 2 WP WS 4 lifetime of population in level 2 << interaction time

STIRAP-population transfer between Niveau- und Kopplungsschema STIRAP-population transfer between Praseodym-states in a Pr:YSO crystal 4.4.3 Experimental verification – in the solid state (Praseodym) 4,6 MHz 4,8 MHz preparation opt. pumping pump Stokes probe electronic lifetime >> T l = 606 nm 17,3 MHz 10,2 MHz T  4 K PP,S  80 mW,   300 m, P,S  2  0.7 MHz, duration of laser pulse T  20 s Ref. 77 eff T  90

4.4.3 Experimental verification – in the solid state (Praseodym) frequency generator AOM travelling acoustic wave acts as a phase grating AOM the frequency of light diffracted into the first diffraction order is shifted by AOM. diffraction efficiency is high (50 – 90%) under Bragg condition: Ref. 77 Bragg: diffraction angle = „reflection angle“

4.4.3 Experimental verification – in the solid state (Praseodym) lifetime of population in level 2 > interaction time transfer via the dark state transfer via the bright states |ao(t)> = cos (t) |1> – sin (t) |3> |a±(t)> = sin (t) |1> ± |2> + cos (t) |3> S before P  = 0o initially S after P  = 90o initially Ref. 77 eff T  90 PRL 99, 113003 (2007)

when lifetime of population in level 2 << interaction time: 4.4.3 Experimental verification - transfer via „bright“ state M = 0 J = 1 J = 2 M =0 J = 0 probe laser Na2 400 MHz 400 MHz off resonance S prior P transfer via dark state S after P transfer via bright state the maximum D dissappears for on-resonance tuning when lifetime of population in level 2 << interaction time: reducing the „leak“ for transfer via the „bright“ states by detuning from resonance Ref. 3

WP WS 3.3.2 3-level system: adiabatic states off one-photon resonance 1 3 2 WP WS 4 1  = 90o transfer via bright states |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 ~ leak rate increases with eff and decreases with |1|

4.2 Conditions for adiabatic evolution: variation with geometry Three different geometries eff  >> 1 elliptical (vertical) circular elliptical (horizontal) particles cross the laser beam near the axis x z power of the lasers in all three cases the same Questions: (a) Is the transfer efficiency (quality of adiabaticity criterion) in all cases the same (identical laser power !) … …. or is one geometry to be preferred ? If so, which one ? (b) Two circular laser beams: how would the transfer efficiency change, if the diameter of the laser beams is doubled (power unchanged): improve – unaffected – become worse ?

4.2 Conditions for adiabatic evolution: variation with delay and width |<1|ao>| < 1 S not >> P good coupling |<1|ao>| = 1, S >> P |<1|ao>| = 1 good coupling and thus good adiabatic following weaker coupling |<1|ao>| < 1 good coupling |<1|ao>| = 1 S not >> P weaker coupling

2eff T >>> 2eff T > ½ (1 + N2) 4.2 Conditions for adiabatic evolution: consequence of fluctuations pulses in the „real world“ of ns-lasers: the phase is not constant but characterized by the size and frequency of phase jumps: see autocorrelation function the field amplitude doesn´t vary smoothly; characterized by the power spectrum (ns-pulses) transforme limited pulse real or simulated pulse real or simulated pulse transforme limited pulse reported only 2eff T >>>  T > 10 2eff T > ½ (1 + N2) = measure for (phase) fluctuations Ref. 8

4.3 Conditions for adiabatic evolution: consequence of fluctuations 100 % population evolution for a pulse which is: < 100 % smooth fluctuating statistics of transfer efficiency variation of transfer with Rabi frequency: fluctuations can be overcome by stronger coupling Ref. 8

4.3 Conditions for adiabatic evolution 2 1 3 P S If one of the frequencies fluctuates (or both): deviation from two-photon resonance no  = 0 eigenvalue no dark state loss by spontaneous emission However: if the fluctuations are (perfectly) correlated – and identically – in time: deviation from 1-photon resonance will still allow for 2  0 (2-photon resonance) fluctuation of 1 will lead to some non-adiabatic coupling, which decreases if the Rabi frequency increases 2 1 3 P S

4.3 Conditions for adiabatic evolution: consequence of fluctuations It is difficult (if not impossible) to guarantee perfect phase- correlation for radiation from two different laser sources 2 1 3 P S 2 1 3 P S It is possible to achieve perfect phase-correlation for radiation derived from a single laser source two-photon resonance maintained despite fluctuations, but …. M = -1 M = 0 M = +1 „white light STIRAP“ + - time dependence of 1 leads to some loss by non-adiabatic coupling Ref. 82 M. Auzinsh et.al. Phys.Rev. A78, 053415 (2008)

WP WS D2 = 0 P = S  = 45o tan = P / S = 1 4.3 The transfer process in the state-vector picture 1 3 2 WP WS 4 equal Rabi frequencies (or ratio ≠ 0 and ≠ ) D2 = 0 composition of |ao> P = S  = 45o tan = P / S = 1 no component |2> 50% of the population in dark (or trapping) state |ao> component |2> 25% of the population in each one of the bright states |a±> loss of population through optical pumping (decay to other levels)

4.3 The transfer process in the state-vector picture adiabatic state wave functions shown as state vectors referenced to the bare states P = 0  = 0 P = S  = 45o S = 0  = 90o

the solution of an optical puzzle: dark lines reminder Electromagnetically induced transparency – trapped states the solution of an optical puzzle: dark lines B x „dark resonance“ or „population trapping“ inhomogeneous B-field across cell vapor cell, sodium multimode laser spectral components of multimode laser couple m-states and form dark resonances observation in Pisa 1976 at specific location(s): B-field splitting = mode spacing gradient of B-field E Ref. 83

solution of the three-level puzzle Electromagnetically induced transparency – trapped states reminder M = 0 solution of the three-level puzzle induced by linearly polarized coherent radiation decay  depletion by optical pumping 2/3 of population remains in initial level z z J = 1  J = 0 transition as an example change of quantization axis E y y decomposition: linear polarization  circular-right + circular-left x M = ± 1 „dark states“ solve the puzzle: 50% of the population in M = -1 and M = + 1 remain in a dark state: ½ (the „missing“) 1/3 still: decay  depletion by optical pumping but now: 1/3 of population remains in initial level ??

Questions related to the topics discussed in lecture 9 (9.1) What are the conditions for efficient population transfer via a bright adiabatic state ? (9.2) Consider two circular laser beams (P and S): how would the transfer efficiency change, if the diameter of the laser beams is doubled (power unchanged), improve – unaffected – become worse ? (9.3) Related to (7.2): Consider elliptical shape of the laser (same laser power) and analyze the quality of global adiabaticity criterion. The long axis of the ellipse may be perpendicular or parallel to the direction of travel of the atoms (axis of the atomic beam). Compared to circular beams: is an elliptical beam in one or both of the orientations to be preferred to optimize the transfer efficiency ? (9.4) Influence of ratio of maximum Rabi frequency o on transfer: Given is a fixed width (in space or time) of the laser diameter and a suitable displace- ment or delay of the interation. You know that we have good transfer for Pmax  Smax. Would the transfer efficiency change (if yes: why and how) for Pmax >> Smax.

end of 9th lecture Coherent light-matter interaction: Optically driven adiabatic transfer processes end of 9th lecture