## Strong-field quantum control of 2 + 1 photon absorption of atomic sodium |

Optics Express, Vol. 19, Issue 3, pp. 2266-2277 (2011)

http://dx.doi.org/10.1364/OE.19.002266

Acrobat PDF (926 KB)

### Abstract

We demonstrate ultrafast coherent control of multiphoton absorption in a dynamically shifted energy level structure. In a three-level system that models optical interactions with sodium atoms, we control the quantum interference of sequential 2 + 1 photons and direct three-photon transitions. Dynamic change in energy levels predicts an enormous enhancement of |7*p*〉-state excitation in the strong-field regime by a negatively chirped pulse. In addition, the |4*s*〉-state excitation is enhanced symmetrically by nonzero linear chirp rates given as a function of laser peak intensity and laser detuning. Experiments performed by ultrafast shaped-pulse excitation of ground-state atomic sodium verifies the various strong-field contributions to |3*s*〉-|7*p*〉 and |3*s*〉-|4*s*〉 transitions. The result suggests that for systems of molecular level understanding adiabatic control approach with analytically shaped pulses becomes a more direct control than feedback-loop black-box approaches.

© 2011 Optical Society of America

## 1. Introduction

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*p*〉-state excitation by a negatively chirped pulse in the strong-field regime. In addition, the |4

*s*〉-state excitation is enhanced symmetrically by nonzero linear chirp rates given as a function of laser peak intensity and laser detuning. We first obtained analytical formulas for the strong-field contributions to |3

*s*〉-|7

*p*〉 and |3

*s*〉-|4

*s*〉 transitions and then verified them by experiments performed with programmed laser pulses.

## 2. Theoretical considerations

*g*〉, |

*e*〉, |

*f*〉} as [31]

*S*(

_{g}*t*),

*S*(

_{e}*t*), and

*S*(

_{f}*t*) represent the dynamic Stark shifts, Ω(

*t*) and Ω

*(*

_{ef}*t*) are the two-photon and one-photon Rabi frequencies, respectively, Δ

_{1}is the two-photon detuning defined by Δ

_{1}= 2

*ν*−

*ω*+

_{e}*ω*, and Δ

_{g}_{2}is the one-photon detuning, Δ

_{2}=

*ν*−

*ω*+

_{f}*ω*, where

_{e}*ω*,

_{g}*ω*, and

_{e}*ω*are the respective energies of the |

_{f}*g*〉, |

*e*〉, and |

*f*〉 states.

*π*(

*t*) is the phase of the laser field relative to its central frequency,

*ν*. The Hamiltonian can be alternatively expressed as using the transformation

*Ĥ*

^{(T)}=

*T̂*

^{†}

*ĤT̂*−

*ih̄T̂*

^{†}

*dT̂/dt*, where

*T̂*=

*e*

^{i[Δ1t+2π(t)]}|g〉〈

*g| + |e*〉〈

*e|*+

*e*

^{−i[Δ2t+π(t)]}|

*f*〉〈

*f*|. Furthermore, the transformation matrix

*T̂′*=

*e*

^{−i[∫tSg(u)du+Δ1t+2π(t)]}|

*g*〉〈g| +

*e*

^{−i∫tSe(u)du}|

*e*〉〈

*e*| +

*e*

^{−i[∫tSf(u)du−Δ2t−π(t)]}|

*f*〉〈

*f*| transforms the diagonal terms in Eq. (2) into the off-diagonal phase terms, making the Hamiltonian where

*Q*

_{1}(

*t*) = − ∫

^{t}*S*(

_{eg}*u*)

*du*+ Δ

_{1}

*t*+ 2

*π*(

*t*) and

*Q*

_{2}(

*t*) = − ∫

^{t}*S*(

_{fe}*u*)

*du*+ Δ

_{2}

*t*+

*π*(

*t*).

*S*(

_{eg}*t*) and

*S*(

_{fe}*t*) are the level-shift parameters given as

*S*(

_{eg}*t*) =

*S*(

_{e}*t*) −

*S*(

_{g}*t*) and

*S*(

_{fe}*t*) =

*S*(

_{f}*t*) −

*S*(

_{e}*t*).

*t*)

*τ ≪*1 and Ω

*(*

_{ef}*t*)

*τ ≪*1. For the |

*f*〉 state, the probability amplitude is given as the second-order Dyson series: where

*Q*

_{1}(

*t*) and

*Q*

_{2}(

*t*) are the atomic phases induced from laser detuning, level shift, and the temporal phase of laser pulses. The probability amplitude of the |

*e*〉 state is given by

*g*〉, |

*e*〉 bases.

*e*〉 and |

*f*〉 state is determined by the engineered quantum interference of the transition probabilities. In the following section, we discuss the derivation and testing of approximate behaviors of the excited energy levels of atomic sodium induced by shaped pulses, especially in the strong-field interaction regime where the structure of the energy levels is strongly altered during the pulse interaction.

## 3. Experimental

*s*〉 energy-state by non-resonant two-photon absorption of 777-nm light. The energy difference between the |4

*s*〉 and |7

*p*〉 states is 781 nm, and the transition between these states is single-photon resonant with the laser light. Because the atoms interacted with a sub-picosecond optical pulse of broad wavelengths, including both resonant wavelengths, 777 and 781 nm, the |4

*s*〉 and |7

*p*〉 states were simultaneously excited. The resulting wave function is a coherent superposition state of |3

*s*〉, |4

*s*〉, and |7

*p*〉. Sodium atoms were prepared in a heated optical cell. The vapor pressure of sodium in the solid phase is given by [32

32. D. A. Steck, Alkali D Line Data, http://steck.us/alkalidata/

_{10}

*P*= 2.881 + 5.298 − 5603/

_{v}*T*, where the temperature,

*T*, is in Kelvin, and the vapor pressure,

*P*, is in Torr; thus, the density of sodium atoms was 2.0 × 10

_{v}^{17}/

*m*

^{3}in the heated cell at 423 K. The lifetimes of the excited sodium atoms are on the order of a few tens of nanoseconds. The ionization probability at the explored laser intensity of 10

^{13}W/cm

^{2}was 1000 times smaller than the |4

*s*〉 − |7

*p*〉 transition. For the interaction of sodium atoms with short optical pulses of a few picoseconds, the decay processes and ionization processes could be ignored during the excitation; therefore, the |3

*s*〉, |4

*s*〉, and |7

*p*〉 energy states formed the three-state model system under consideration.

*μ*J of energy were shaped by an actively controlled acousto-optic programmable dispersive filter [33

33. F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. **25**, 575–577 (2000). [CrossRef]

*μm*, and the Rayleigh range was 4.0 mm. The focused spot in the vapor cell was imaged by a telescope, and the fluorescence was collected by a photomultiplier (PMT, Hamamatsu R1527P). The two-photon Rabi coupling was Ω(0)=−10.25 Trad/s at the laser peak, and the Rabi coupling between the |4

*s*〉 and |7

*p*〉 states was Ω

*(0) = 5.45 Trad/s. For the two-photon rotating wave approximation, we verified that |Δ|*

_{er}*≪ ω*−

_{jg}*ν ≃ ω*+

_{je}*ν*. At a 777-nm center wavelength, the two-photon detuning was nearly zero, and the given condition was satisfied. At 800 nm, |Δ|

*≈*140

*×*Trad/s and

*ω*−

_{jg}*ν ≫*700

*×*Trad/s, so the approximation condition was satisfied.

*p*〉 excited population, we used the |7

*s*〉-|3

*p*〉 transition. (Fig. 1) The 475-nm fluorescence signal was measured as a function of the control parameters of the shape programming. In addition, the 3

*p*〉 − |3

*s*〉 transition signal was used for the combined sum of |7

*p*〉 and |4

*s*〉 populations, both of which were excited by shaped laser pulses. We note that the Einstein coefficients were

*A*

_{7}

_{p}_{−3}

*= 7.96*

_{s}*×*10

^{4}/

*s*and

*A*

_{7}

_{p}_{−7}

*= 1.70*

_{s}*×*10

^{5}/

*s*, and 57% of the |7

*p*〉 population reached the |3

*p*〉 state [34

34. D. E. Keller, A. E. Kramida, J. R. Fuhr, L. Podobedova, and W. L. WIese, in NIST atomic Spectra database, NIST Standard Reference Database Version 4, http://physics.nist.gov/PhysRefData/ASD/

*s*〉-|3

*p*〉 and |3

*p*〉-|3

*s*〉 transitions, respectively, and were individually collected via spectral bandpass filters placed in front of the PMT. The signal intensity of the PMT was kept in the well-calibrated linear detection range, and the |7

*p*〉 excited state, maintained below the saturation limit, was linearly mapped with the fluorescence signal.

## 4. Results and discussion

*T*

_{0}of a transform-limited pulse is about 37 fs. For a linearly chirped pulse which interacts with Sodium atoms, actual Gaussian pulse width is given by

*a*

_{2}is chirp rate in frequency-domain. In time-domain, the frequency sweeping of a linearly chirped pulse is given as a function of time by

*π*(

*t*) = 2

*βt*, where

*τ*(

*a*

_{2})/

*T*

_{0}times relative to that of a transform-limited pulse. Thus, the dynamic Stark shifts induced by linearly chirped pulses become smaller than those by a transform limited pulse.

### 4.1. Control of sodium |7p〉-state excitation

*s*〉 to |

*f*〉 = |7

*p*〉 states. Three possible paths are indicated in Fig. 2(a). For a positively chirped pulse, the photon frequency increased as a function of time; therefore, the |3

*s*〉 − |7

*p*〉 transition was possible along path (III), which is a direct transition path from |3

*s*〉 to |7

*p*〉 that does not pass by the |4

*s*〉 state. On the other hand, for a negatively chirped pulse, the photon frequency decreased, and two paths are possible: a sequential path (I) and a direct path (II). Considering the fact that path (III) is an inefficient excitation, it is thought that the |7

*p*〉 atoms were generated more effectively by a negatively chirped pulse than by a positively chirped pulse.

*p*〉 state, obtained as a function of the chirp parameter, can be understood in a time-frequency schematic. Figures 2(b) and (c) show one-photon (red) and two-photon (blue) spectrograms of chirped laser pulses that are plotted in two-dimensional time space and frequency. They are overlaid with the resonant frequency shifts of |4

*s*〉-|7

*p*〉 and |3

*s*〉-|4

*s*〉 transitions (dashed lines). Figure 2(b) shows that the |3

*s*〉-|4

*s*〉 transition occurred first and the |4

*s*〉-|7

*p*〉 occurred later; thus, a sequential excitation along the |3

*s*〉 → |4

*s*〉 → |7

*p*〉 path was satisfied. On the other hand, for a positively chirped pulse shown in Fig. 2(c), because the |3

*s*〉-|4

*s*〉 resonance frequency was up-shifted during the optical interaction, the two-photon spectrum overlaps with the resonance line after the temporal center of the pulse. As a result, the |3

*s*〉-|4

*s*〉 and |4

*s*〉-|7

*p*〉 transitions occurred in a time-reversal sequence, indicating that the sequential excitation was not possible. Thus, sequential excitation is not possible for a positively chirped pulse, and only non-sequential excitations, e.g., along path (III) in Fig. 2(a), are possible.

*p*〉 state and verify the results with corresponding experiments.

### 4.2. Calculation of the sequential and direct |7p〉 excitations

*p*〉 state is obtained from Eq. (4). To calculate the transition probability amplitudes for the sequential and direct excitation paths, we separated Eq. (4) into two parts: resonant and non-resonant. In the perturbative interaction regime, the probability amplitudes corresponding to these two parts can be written, respectively, as [25

25. C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. **96**, 063603 (2006). [CrossRef] [PubMed]

*℘*is the Cauchy principal value. The subscripts “res” and “nonres” denote the resonant and non-resonant excitations, and

*δ*and (

_{eg}*δ*are the maximum amplitudes of the total level shifts of the |3

_{fe}*s*〉 − |4

*s*〉 and |7

*p*〉 − |4

*s*〉 transitions, respectively, which are calculated at the temporal peak of a transform-limited pulse.

*E*(

*ω*) is the Fourier transform of the electric field profile, a spectrally chirped Gaussian pulse, given by where

*T*is the Gaussian width of a transform-limited pulse, and

_{o}*a*

_{2}is the linear chirp parameter in the spectral domain.

*s*〉 → |7

*p*〉 excitation was obtained by substituting Eq. (8) into Eq. (6) as where

*δ*

_{2ph}(

*ν*) =

*ω*+

_{eg}*δ*− 2

_{eg}*ν*, and the one-photon detuning is defined as

*δ*

_{1ph}(

*ν*) =

*ω*+

_{fe}*δ*−

_{fe}*ν*. Alternatively, Eq. (9) can be written as where three-photon detuning is defined as

*δ*

_{3ph}(

*ν*) =

*ω*+

_{fg}*δ*− 3

_{fg}*ν*, and

*δ*is the structure factor,

_{s}*δ*=

_{s}*ω*+

_{fe}*δ*− (

_{fe}*ω*+

_{fg}*δ*)/3 that is independent of

_{fg}*ν*.

*p*〉 is the absolute square of Eq. (10). The result is a symmetric function of chirp

*a*

_{2}, which contradicts the prediction in Sec. 4.1 that the sequential excitation should be an asymmetric function of the chirp. However, there is indeed another sequential path in the non-resonant excitation path in Eq. (7). Although the resonant excitation (

*a*

_{f,res}) contributed only to the sequential transition path (|3

*s*〉

*→ |*4

*s*〉

*→ |*7

*p*〉), the non-resonant part (

*a*

_{f,nonres}) contributed to both the sequential and direct paths, i. e., where

*ω ≈*(

*ω*+

_{fg}*δ*)

_{fg}*/*3. Then, the denominator of the integrand in Eq. (7) can be treated as a constant, and it is simple to show that

*ω ≈ ω*+

_{fe}*δ*, in Eq. (7). Equation (7) is written as

_{fe}*O*(

*ω*−

*ω*+

_{fe}*δ*)

_{fe}^{2}, the non-resonant sequential excitation is given by

*x*)=+1 (−1) for

*x*> 0 (

*x*< 0), is due to the contour integral given as a function of the sign of

*δ*

_{s}a_{2}. The result is valid in the chirp range of

*T*is 37 fs, and the approximation

_{o}*a*

_{2}| > 1369 fs

^{2}. In sodium,

*δ*= −8.3 Trad/s, and as a result,

_{s}*a*

_{2}> 0. Therefore, the net sequential excitation path to |7

*p*〉 in Eq. (11) vanishes for a positively chirped pulse because the resonant and non-resonant contributions cancel each other.

### 4.3. Verification of shaped-pulse |7p〉-state excitation

*p*〉 excitation was experimentally tested as a function of the chirp rate

*a*

_{2}. Figure 3 shows the calculation of the net transition probability (solid line) of |3

*s*〉 − |7

*p*〉 given in Eq. (11), the components of which were obtained from Eq. (10), Eq. (12), and Eq. (14) as a function of the linear chirp rate. The laser (transform-limit) peak intensity was kept at

*I*= 3.0

*×*10

^{1}1 W/cm

^{2}, and the chirp rate

*a*

_{2}was varied in the range [−1.0, 1.0]

*×*10

^{4}fs

^{2}. As predicted in the schematic picture in Sec. 4.1, the excitation was significantly enhanced by negatively chirped pulses because the sequential excitation path along |3

*s*〉 − |4

*s*〉 − |7

*p*〉 was zero for positively chirped pulses. The direct transition from |3

*s*〉 − |7

*p*〉, which is a symmetric function in Eq. (12), was 10 times smaller than the sequential transition for the tested laser peak intensity. For comparison, the sequential and direct excitation probabilities are plotted using dashed and dot-dash lines, respectively. For the numerical calculation, the dynamic Stark shift of the |3

*s*〉 state was

*S*= −32.8(

_{g}*I/I*) Trad/s, determined by couplings with |

_{o}*p*〉 states, where

*I*is the reference laser intensity,

_{o}*I*= 1.0

_{o}*×*10

^{11}W/cm

^{2}. The shift of the |4

*s*〉 state was

*S*= 28.9(

_{e}*I/I*) Trad/s at the same intensity. Thus, the net frequency shift of the two-photon transition was positive. The |7

_{o}*p*〉 state was shifted by couplings with the |

*s*〉 and |

*d*〉 states as well as with continuum states. It is known that the presence of a continuum increases the energy of the excited state by the ponderomotive energy given by

*S*(

_{r}*t*) =

*e*

^{2}

*E*

^{2}(

*t*)

*/*4

*mν*

^{2}, where

*ν*is the laser frequency, and

*m*is the mass of an electron. As a result, the |7

*p*〉 state was up-shifted

*S*= 8.7(

_{r}*I/I*) Trad/s, and the net frequency shift of the|4

_{o}*s*〉-|7

*p*〉 transition was negative

*S*= −20.2(

_{re}*I/I*) Trad/s. Also, because the laser beam had a Gaussian spatial intensity distribution, the total excitation probability was calculated as the sum of local excitations, i.e.,

_{o}*a*

_{2}and laser peak intensity

*I*is compared with the experimentally tested results in Fig. 4. The |7

*p*〉 excited atoms were measured by monitoring the |7

*s*〉 − |3

*p*〉 fluorescence as a function of laser (transform-limited) peak intensity and the chirp parameter (

*a*

_{2}). The laser peak intensity,

*I*, was varied from zero to

*I*= 6.0

*×*10

^{11}W/cm

^{2}, and

*a*

_{2}was varied in the range of [−1.0, 1.0]

*×*10

^{4}fs

^{2}. The net transition probability, which is the sum of the calculated sequential and direct transitions shown in Fig. 4(a), showed excellent agreement with the experimental results in Fig. 4(b).

### 4.4. Chirped-pulse excitation of |4*s*〉-state atoms

*s*〉 atoms. Figure 5 shows the calculation and experimental results of the excitation probability of |4

*s*〉-state atoms. It is evident from Eq. (5) that the dominant excitation to the |4

*s*〉 state was the direct two-photon absorption from the |3

*s*〉 state. The presence of the |7

*p*〉 state affected this excitation in terms of the third-order Dyson series, and the new excitation path was a four-photon sequential excitation along |3

*s*〉

*→ |*4

*s*〉

*→ |*7

*p*〉

*→ |*4

*s*〉. Therefore, the excitation probability amplitude of |4

*s*〉 atoms is given by where the direct two-photon transition

*p*〉 state is negligible. Therefore, the excitation probability is given by where

*A*=

*δ*

_{2ph}(

*ν*)

*τ*, and

*s*〉-|4

*s*〉 transition caused off-resonance to the two-photon excitation. As the dynamic Stark shift, which is stored in the parameter

*A*, increased, the term exp (−

*A*

^{2}/2(1 +

*B*

^{2}))) became important, and the net probability had a local minimum at a chirp rate of zero.

*s*〉 state was determined by the sum of the contributions of the sequential and direct paths, but the sequential transition, dotted lines in Fig. 5(a), only contributed at negative chirp rates and was 1000 times smaller than the direct transition. The calculated excitation probability, given as a function of the linear chirp rate of the shaped pulses at various peak intensities in Fig. 5(b), showed good agreement with the experimental data in Fig. 5(c). A dip was observed at the zero chirp rate. In the experiment, because the pulse energy was fixed, the pulse duration (pulse peak intensity) was shortest (maximum) at the zero chirp rate. The strong peak intensity at the zero chirp rate induced strong off-resonance, reducing the absorption, as expected from Eq. (19). The net excitation probability showed a nearly symmetric function of chirp rate and nearly vanished at zero chirp.

## 5. conclusion

*s*〉 state to the |7

*p*〉 state. The dressed-state picture for the three-photon interaction with the three-level model system predicted that the resonant and non-resonant contributions in the sequential excitation interfered destructively and canceled each other for positively chirped pulses. Both analytic formulas and experimental results showed that a negatively chirped pulse enhanced the |7

*p*〉 population because the sequential path was opened by a negatively chirped pulse. In addition, the |4

*s*〉-state excitation was enhanced symmetrically by nonzero linear chirp rates given as a function of laser peak intensity and laser detuning. Experiments verified the various strong-field contributions to |3

*s*〉-|7

*p*〉 and |3

*s*〉-|4

*s*〉 transitions. The transition amplitude formulas analytically obtained by considering the atomic Hamiltonian showed good agreement with the experiment. The result suggests that adiabatic control approach with analytically shaped pulses provides a more direct control of systems of molecular level understandings than feedback-loop black-box approaches do.

## Acknowledgments

## References and links

1. | R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. |

2. | K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. |

3. | M. Shapiro and P. Brumer, |

4. | D. J. Tanner and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. |

5. | A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. |

6. | D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Rep. |

7. | H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science |

8. | A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science |

9. | D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature |

10. | T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B |

11. | Z. Zheng and A. M. Weiner, “Coherent control of second harmonic generation using spectrally phase coded femtosecond waveforms,” Chem. Phys. |

12. | D. Meshulach and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulse,” Phys. Rev. A. |

13. | Z. Amitay, A. Gandman, L. Chuntonov, and L. Rybak, “Multichannel selective femtosecond coherent control based on symmetry properties,” Phys. Rev. Lett. |

14. | N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. |

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16. | J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express |

17. | T. C. Weinacht, J. Ahn, and P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature |

18. | M. C. Stowe, A. Pe’er, and J. Ye, “Control of four-level quantum coherence via discrete spectral shaping of an optical frequency comb,” Phys. Rev. Lett. |

19. | V. Blanchet, C. Nicole, M.-A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. |

20. | M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D |

21. | B. Chatel, J. Degert, S. Stock, and B. Girard, “Competition between sequential and direct paths in a two-photon transition,” Phys. Rev. A |

22. | S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic cesium: an analytical control approach,” Opt. Express |

23. | S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guerin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A |

24. | N. Dudovich, T. Polack, A. Pe’er, and Y. Silberberg, “Simple route to strong-field coherent control,” Phys. Rev. Lett. |

25. | C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. |

26. | H. Suchowski, A. Natan, B. D. Bruner, and Y. Silberberg, “Spatio-temporal coherent control of atomic systems: weak to strong field transition and breaking of symmetry in 2D maps,” J. Phys. B:At. Mol. Opt. Phys. |

27. | E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. |

28. | M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, “Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses,” Appl. Phys. B |

29. | S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a three-level system using shaped ultrafast laser pulses,” Phys. Rev. Lett. |

30. | T. Bayer, M. Wollenhaupt, C. Sarpe-Tudoran, and T. Baumert, “Robust photon locking,” Phys. Rev. Lett. |

31. | P. Meystre and M. Sargent III, |

32. | D. A. Steck, Alkali D Line Data, http://steck.us/alkalidata/ |

33. | F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. |

34. | D. E. Keller, A. E. Kramida, J. R. Fuhr, L. Podobedova, and W. L. WIese, in NIST atomic Spectra database, NIST Standard Reference Database Version 4, http://physics.nist.gov/PhysRefData/ASD/ |

**OCIS Codes**

(190.4180) Nonlinear optics : Multiphoton processes

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 8, 2010

Revised Manuscript: January 18, 2011

Manuscript Accepted: January 18, 2011

Published: January 24, 2011

**Citation**

Sangkyung Lee, Jongseok Lim, Chang Yong Park, and Jaewook Ahn, "Strong-field quantum control of 2 + 1 photon absorption of atomic sodium," Opt. Express **19**, 2266-2277 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2266

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