Abstract

Advances of quantum control technology have led to nearly perfect single-qubit control of nuclear spins and atomic hyperfine ground states. In contrast, quantum control of strong optical transitions, even for free atoms, are far from being perfect. Developments of such quantum control appears to be limited by available laser technology for generating isolated, sub-nanosecond optical waveforms with 10's of GHz programming bandwidth. Here we propose a simple and robust method for the desired pulse shaping, based on precisely stacking multiple delayed picosecond pulses. Our proof-of-principal demonstration leads to arbitrarily shapeable optical waveforms with 30 GHz bandwidth and 100 ps duration. We confirm the stability of the waveforms by interfacing the pulses with laser-cooled atoms, resulting in “super-resolved” spectroscopic signals. This pulse shaping method may open exciting perspectives in quantum optics, and for fast laser cooling and atom interferometry with mode-locked lasers.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (3)

N. Picque and T. W. Hansch, “Frequency comb spectroscopy,” Nat. Photonics 13(3), 146–157 (2019).
[Crossref]

D. Heinrich, M. Guggemos, M. Guevara-Bertsch, M. I. Hussain, C. F. Roos, and R. Blatt, “Ultrafast coherent excitation of a $^{40}$Ca$^+$40 ion,” New J. Phys. 21, 073017 (2019).
[Crossref]

X. Long, S. S. Yu, A. M. Jayich, and W. C. Campbell, “Suppressed Spontaneous Emission for Coherent Momentum Transfer,” Phys. Rev. Lett. 123(3), 033603 (2019).
[Crossref]

2017 (1)

J. D. Wong-Campos, S. A. Moses, K. G. Johnson, and C. Monroe, “Demonstration of two-atom entanglement with ultrafast optical pulses,” Phys. Rev. Lett. 119(23), 230501 (2017).
[Crossref]

2016 (2)

G. H. Low, T. J. Yoder, and I. L. Chuang, “Methodology of Resonant Equiangular Composite Quantum Gates,” Phys. Rev. X 6, 041067 (2016).
[Crossref]

C. E. Rogers and P. L. Gould, “Nanosecond pulse shaping at 780 nm with fiber-based electro-optical modulators and a double-pass tapered amplifier,” Opt. Express 24(3), 2596–2606 (2016).
[Crossref]

2015 (4)

X. Rong, J. P. Geng, F. Z. Shi, Y. Liu, K. B. Xu, W. C. Ma, F. Kong, Z. Jiang, Y. Wu, and J. F. Du, “Experimental fault-tolerant universal quantum gates with solid-state spins under ambient conditions,” Nat. Commun. 6(1), 8748 (2015).
[Crossref]

M. O. Scully, “Single photon subradiance: Quantum control of spontaneous emission and ultrafast readout,” Phys. Rev. Lett. 115(24), 243602 (2015).
[Crossref]

J. F. Leger, C. Ventalon, B. Mathieu, S. Dieudonne, and L. Bourdieu, “Fast spatial beam shaping by acousto-optic diffraction for 3D non-linear microscopy,” Opt. Express 23, 28191 (2015).
[Crossref]

F. D. Fuller and J. P. Ogilvie, “Experimental Implementations of Two-Dimensional Fourier Transform Electronic Spectroscopy,” Annu. Rev. Phys. Chem. 66(1), 667–690 (2015).
[Crossref]

2014 (4)

G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, “Correction of arbitrary field errors in population inversion of quantum systems by universal composite pulses,” Phys. Rev. Lett. 113(4), 043001 (2014).
[Crossref]

A. Dunning, R. Gregory, J. Bateman, N. Cooper, M. Himsworth, J. A. Jones, and T. Freegarde, “Composite pulses for interferometry in a thermal cold atom cloud,” Phys. Rev. A 90(3), 033608 (2014).
[Crossref]

J. Mizrahi, B. Neyenhuis, K. G. Johnson, W. C. Campbell, C. Senko, D. Hayes, and C. Monroe, “Quantum control of qubits and atomic motion using ultrafast laser pulses,” Appl. Phys. B 114(1-2), 45–61 (2014).
[Crossref]

A. M. Jayich, A. C. Vutha, M. T. Hummon, J. V. Porto, and W. C. Campbell, “Continuous all-optical deceleration and single-photon cooling of molecular beams,” Phys. Rev. A 89(2), 023425 (2014).
[Crossref]

2012 (3)

J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Line-by-line pulse shaping with spectral resolution below 890 MHz,” Opt. Express 20(3), 3110–3117 (2012).
[Crossref]

S. Odedra and S. Wimperis, “Improved background suppression in 1H MAS NMR using composite pulses,” J. Magn. Reson. 221, 41–50 (2012).
[Crossref]

T. Ichikawa, M. Bando, Y Kondo, and M. Nakahara, “Geometric aspects of composite pulses,” Philos. Trans. R. Soc., A 370(1976), 4671–4689 (2012).
[Crossref]

2011 (4)

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

T. Mansuryan, M. Kalashyan, J. Lhermite, E. Suran, V. Kermene, A. Barthelemy, and F. Louradour, “Compact direct space-to-time pulse shaping with a phase-only spatial light modulator,” Opt. Lett. 36(9), 1635–1637 (2011).
[Crossref]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5(12), 770–776 (2011).
[Crossref]

D. Jacob, E. Mimoun, L. De. Sarlo, M. Weitz, J. Dalibard, and F. Gerbier, “Production of sodium Bose Einstein condensates in an optical dimple trap,” New J. Phys. 13(6), 065022 (2011).
[Crossref]

2010 (2)

O. Mendoza-Yero, G. Munguez-Vega, J. Lancis, and V. Climent, “Diffractive pulse shaper for arbitrary waveform generation,” Opt. Lett. 35(4), 535–537 (2010).
[Crossref]

A. Monmayrant, S. Weber, and B. Chatel, “A newcomers guide to ultrashort pulse shaping and characterization,” J. Phys. B: At., Mol. Opt. Phys. 43(10), 103001 (2010).
[Crossref]

2009 (4)

A. D. Bristow, X. Dai, D. Karaiskaj, C. Carlsson, T. Zhang, K. R. Hagen, R. Jimenez, and S. T. Cundiffa, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref]

A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81(3), 1051–1129 (2009).
[Crossref]

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2d ft electronic spectroscopy of Biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[Crossref]

F. Frei, A. Galler, and T. Feurer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. 130(3), 034302 (2009).
[Crossref]

2008 (3)

B. J. Sussman, R. Lausten, and A. Stolow, “Focusing of light following a 4-f pulse shaper: Considerations for quantum control,” Phys. Rev. A 77(4), 043416 (2008).
[Crossref]

C. B. Huang, Z. Jiang, D. E.. Leaird, J. Caraquitena, and A. M. Weiner, “Spectral line-by-line shaping for optical and microwave arbitrary waveform generations,” Laser Photonics Rev. 2(4), 227–248 (2008).
[Crossref]

S. Zhdanovich, E. A. Shapiro, M. Shapiro, J. W. Hepburn, and V. Milner, “Population Transfer between Two Quantum States by Piecewise Chirping of Femtosecond Pulses: Theory and Experiment,” Phys. Rev. Lett. 100, 103004 (2008).
[Crossref]

2007 (3)

X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf, “Strong optical forces from adiabatic rapid passage,” Phys. Rev. A: At., Mol., Opt. Phys. 75(1), 011402 (2007).
[Crossref]

B. Dromey, M. Zepf, M. Landreman, K. O’Keeffe, T. Robinson, and S. M. Hooker, “Generation of a train of ultrashort pulses from a compact birefringent crystal array,” Appl. Opt. 46(22), 5142–5146 (2007).
[Crossref]

P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127(21), 214307 (2007).
[Crossref]

2005 (2)

N. Khanejaa, T. Reissb, C. Kehletb, T. Schulte-Herbruggen, and S. J. Glaser, “Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms,” J. Magn. Reson. 172(2), 296–305 (2005).
[Crossref]

Z. Jiang, D. E. M. Leaird, and A Weiner, “Line-by-line pulse shaping control for optical arbitrary waveform generation,” Opt. Express 13(25), 10431–10439 (2005).
[Crossref]

2003 (2)

P. Tian, D. Keusters, S. Yoshifumi, and W. S. Warren, “Femtosecond Phase-Coherent Two-Dimensional Spectroscopy,” Science 300(5625), 1553–1555 (2003).
[Crossref]

D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Rep. 374, 385–481 (2003).
[Crossref]

2002 (1)

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear raman spectroscopy and microscopy,” Nature 418(6897), 512–514 (2002).
[Crossref]

2001 (3)

G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72(5), 627–630 (2001).
[Crossref]

C. J. Hawthorn, K. P. Weber, and R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum. 72(12), 4477–4479 (2001).
[Crossref]

M. Garwood and L. DelaBarre, “ADVANCES IN MAGNETIC RESONANCE, The Return of the Frequency Sweep: Designing Adiabatic Pulses for Contemporary NMR,” J. Magn. Reson. 153(2), 155–177 (2001).
[Crossref]

2000 (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[Crossref]

1999 (1)

1998 (3)

1997 (2)

P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. 140(4-6), 245–249 (1997).
[Crossref]

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schetze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56(5), R3354–R3357 (1997).
[Crossref]

1995 (1)

T. G. M Freegarde, J. Walz, and T. W. Hansch, “Confinement and manipulation of atoms using short laser pulses,” Opt. Commun. 117(3-4), 262–267 (1995).
[Crossref]

1993 (1)

1992 (1)

1983 (1)

C. Froehly, B. Colombeau, and M. Vampouille, “Shaping and analysis of picosecond light pulses,” Prog. Opt. 20, 63–153 (1983).
[Crossref]

Bando, M.

T. Ichikawa, M. Bando, Y Kondo, and M. Nakahara, “Geometric aspects of composite pulses,” Philos. Trans. R. Soc., A 370(1976), 4671–4689 (2012).
[Crossref]

Barthelemy, A.

Bateman, J.

A. Dunning, R. Gregory, J. Bateman, N. Cooper, M. Himsworth, J. A. Jones, and T. Freegarde, “Composite pulses for interferometry in a thermal cold atom cloud,” Phys. Rev. A 90(3), 033608 (2014).
[Crossref]

Blatt, R.

D. Heinrich, M. Guggemos, M. Guevara-Bertsch, M. I. Hussain, C. F. Roos, and R. Blatt, “Ultrafast coherent excitation of a $^{40}$Ca$^+$40 ion,” New J. Phys. 21, 073017 (2019).
[Crossref]

Bloch, I.

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schetze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56(5), R3354–R3357 (1997).
[Crossref]

Bourdieu, L.

Bristow, A. D.

A. D. Bristow, X. Dai, D. Karaiskaj, C. Carlsson, T. Zhang, K. R. Hagen, R. Jimenez, and S. T. Cundiffa, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref]

Campbell, W. C.

X. Long, S. S. Yu, A. M. Jayich, and W. C. Campbell, “Suppressed Spontaneous Emission for Coherent Momentum Transfer,” Phys. Rev. Lett. 123(3), 033603 (2019).
[Crossref]

A. M. Jayich, A. C. Vutha, M. T. Hummon, J. V. Porto, and W. C. Campbell, “Continuous all-optical deceleration and single-photon cooling of molecular beams,” Phys. Rev. A 89(2), 023425 (2014).
[Crossref]

J. Mizrahi, B. Neyenhuis, K. G. Johnson, W. C. Campbell, C. Senko, D. Hayes, and C. Monroe, “Quantum control of qubits and atomic motion using ultrafast laser pulses,” Appl. Phys. B 114(1-2), 45–61 (2014).
[Crossref]

Caraquitena, J.

C. B. Huang, Z. Jiang, D. E.. Leaird, J. Caraquitena, and A. M. Weiner, “Spectral line-by-line shaping for optical and microwave arbitrary waveform generations,” Laser Photonics Rev. 2(4), 227–248 (2008).
[Crossref]

Carlsson, C.

A. D. Bristow, X. Dai, D. Karaiskaj, C. Carlsson, T. Zhang, K. R. Hagen, R. Jimenez, and S. T. Cundiffa, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref]

Chang, C.-C.

Chang, D. E.

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Chatel, B.

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S. Zhdanovich, E. A. Shapiro, M. Shapiro, J. W. Hepburn, and V. Milner, “Population Transfer between Two Quantum States by Piecewise Chirping of Femtosecond Pulses: Theory and Experiment,” Phys. Rev. Lett. 100, 103004 (2008).
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Appl. Phys. B (2)

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J. Chem. Phys. (2)

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J. Phys. B: At., Mol. Opt. Phys. (1)

A. Monmayrant, S. Weber, and B. Chatel, “A newcomers guide to ultrashort pulse shaping and characterization,” J. Phys. B: At., Mol. Opt. Phys. 43(10), 103001 (2010).
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C. B. Huang, Z. Jiang, D. E.. Leaird, J. Caraquitena, and A. M. Weiner, “Spectral line-by-line shaping for optical and microwave arbitrary waveform generations,” Laser Photonics Rev. 2(4), 227–248 (2008).
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Nat. Commun. (1)

X. Rong, J. P. Geng, F. Z. Shi, Y. Liu, K. B. Xu, W. C. Ma, F. Kong, Z. Jiang, Y. Wu, and J. F. Du, “Experimental fault-tolerant universal quantum gates with solid-state spins under ambient conditions,” Nat. Commun. 6(1), 8748 (2015).
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Nat. Photonics (2)

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5(12), 770–776 (2011).
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N. Picque and T. W. Hansch, “Frequency comb spectroscopy,” Nat. Photonics 13(3), 146–157 (2019).
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Nature (2)

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N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear raman spectroscopy and microscopy,” Nature 418(6897), 512–514 (2002).
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New J. Phys. (2)

D. Heinrich, M. Guggemos, M. Guevara-Bertsch, M. I. Hussain, C. F. Roos, and R. Blatt, “Ultrafast coherent excitation of a $^{40}$Ca$^+$40 ion,” New J. Phys. 21, 073017 (2019).
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Opt. Commun. (3)

T. G. M Freegarde, J. Walz, and T. W. Hansch, “Confinement and manipulation of atoms using short laser pulses,” Opt. Commun. 117(3-4), 262–267 (1995).
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Opt. Lett. (5)

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Phys. Rev. A: At., Mol., Opt. Phys. (1)

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Phys. Rev. Lett. (5)

X. Long, S. S. Yu, A. M. Jayich, and W. C. Campbell, “Suppressed Spontaneous Emission for Coherent Momentum Transfer,” Phys. Rev. Lett. 123(3), 033603 (2019).
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S. Zhdanovich, E. A. Shapiro, M. Shapiro, J. W. Hepburn, and V. Milner, “Population Transfer between Two Quantum States by Piecewise Chirping of Femtosecond Pulses: Theory and Experiment,” Phys. Rev. Lett. 100, 103004 (2008).
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M. O. Scully, “Single photon subradiance: Quantum control of spontaneous emission and ultrafast readout,” Phys. Rev. Lett. 115(24), 243602 (2015).
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G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, “Correction of arbitrary field errors in population inversion of quantum systems by universal composite pulses,” Phys. Rev. Lett. 113(4), 043001 (2014).
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J. D. Wong-Campos, S. A. Moses, K. G. Johnson, and C. Monroe, “Demonstration of two-atom entanglement with ultrafast optical pulses,” Phys. Rev. Lett. 119(23), 230501 (2017).
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Phys. Rev. X (1)

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Prog. Opt. (1)

C. Froehly, B. Colombeau, and M. Vampouille, “Shaping and analysis of picosecond light pulses,” Prog. Opt. 20, 63–153 (1983).
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Rev. Mod. Phys. (1)

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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic for the diffractive multi-delay generation based picosecond pulse shaping setup in this work. (b) Schematic for a representative Fourier transform pulse shaping setup. (c) Schematic for a representative direct space-to-time pulse shaping setup. PBS: polarization beam-splitter. AOD: Acoustic Optical Deflector. AOM: Acoustic Optical Modulator. O.I.: Optical Isolator. HWP: half-wave plate. SLM: Spatial light modulator.
Fig. 2.
Fig. 2. Auto-correlation measurements of the shaped pulses with $N=2$ sub-pulses associated with the two AOM frequencies $\omega _{1,2}=2\pi \times 76, 92$ MHz; and $N=3$ sub-pulses, $\omega _{1,2,3}=2\pi \times 76, 84, 92$ MHz frequencies. Figures (a)(b) are representative auto-correlation signals. Figures (c)(d) give corresponding $I(t)$ . In Fig. (e), $\Delta \tau _{1,2}$ vs $\Delta \omega _{1,2}$ extracted from ten measurements similar to those in Figs. (a)(b) are plotted with red dots. The error bars represent the fit uncertainties. By fitting the data, the green line gives $\Delta \tau _{1,2}/\textrm {ps}\approx 3.0 \Delta \omega _{1,2}/(2\pi \times \textrm {MHz})$ , in excellent agreement with Eq. (1).
Fig. 3.
Fig. 3. Setup for “super-resolved” trap loss spectroscopy with shaped picosecond pulses. (a) Schematic diagram for shaped pulse excitation of $^{87}$ Rb D1 line and the absorption imaging setup. Here $\delta f_{\textrm {HFS},g}=6.8 $ GHz and $\delta f_{\textrm {HFS},e}=0.8 $ GHz are hyperfine splittings. We take the shaped waveforms with $N=2$ sub-pulses and $\Delta \tau _{1,2}=96$ ps as an example. The spectrum given by $I(\omega )=I_0(\omega )S(\omega )$ displays $S(\omega )=\sin ^2((\omega \Delta \tau _{1,2}+\Delta \varphi _{1,2})/2)$ interference. For the wide-band $I_0(\omega )$ , the sub-pulses sequence is with a transform-limited frequency resolution $\delta \omega \approx \pi /\Delta \tau _{1,2}$ . For weak and repeated excitations, “super-resolution” features appear when $S(\omega )$ at $\omega =\omega _{eg}\pm \pi \delta f_{\textrm {HFS},g}$ vanishes. Representative absorption images for the trap loss spectroscopy during a $\Delta \varphi _{1,2}=\varphi _2-\varphi _1$ scan over $4\pi$ (with $\Delta \varphi _{1,2}^\textrm {c}$ scanning over 2 $\pi$ ) are given in Fig. (b). The data is from a denser set in Fig. 4(b).
Fig. 4.
Fig. 4. Phase-scanning trap loss spectroscopy for shaped waveforms with $N=2$ (a,b), $N=6$ (c), and $N=3$ (d) sub-pulses. The inter sub-pulse delays are $\Delta \tau _{1,2}=24$ ps, 96 ps, 12 ps and 24 ps from (a) to (d). See main text for detailed descriptions.
Fig. 5.
Fig. 5. Reconstruction of shaped waveform intensity $I_\textrm {out}(t)$ and $\textrm {Re}[ E_\textrm {out}(t)]$ quadrature (in the $\omega _{eg}$ rotating frame) according to Eq. (2), for $N-$ sub-pulse array with uniform $\{A_i\}$ and specific phase combination $\{\varphi _i\}$ marked in Figs. 4.
Fig. 6.
Fig. 6. Typical $(\tau _\textrm {d})_\textrm {max}$ (associated with $z_\textrm {R,G}=\lambda F^2/\pi w^2$ ) and $\delta f_\textrm {G}$ (associated with $w_\textrm {G}= \lambda F/\pi w$ ) in the shaper scheme vs the Gaussian waist $w$ of the input laser near the AOD, for pulse shaper with various lens focal length $F$ ( $F=200$ mm, solid line; $F=750$ mm, dash line). We consider $\lambda =0.8$ $\mu$ m and a 2400 line/mm grating with grating constant $d=0.42$ $\mu$ m. The colored circles mark the limits in this work. With $\delta f_\textrm {G}$ we can estimate the delay resolution $(\tau _\textrm {d})_\textrm {min}=\pi /(2\delta f_\textrm {G})$ . For operation of the shaper in the “broadband regime” of grating diffraction, $\delta f_\textrm {G}$ sets the upper bound for the input pulse bandwidth $\delta f_{\textrm{in}}^\textrm {L}$ which is also equal to the modulation bandwidth $\delta f_{\textrm{M}}$ of the output waveforms with duration $\tau _{\textrm{c}}\leq (\tau _\textrm {d})_\textrm {max}+\tau$ .

Equations (9)

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τ i = ω i λ v s F π c 4 d 2 / λ 2 1 ,
E out ( t ) = κ i N E i , o u t ( t ) , = κ i N A i 2 e i φ i E in ( t + τ i ) .
s ( ω ) = κ i N A i 2 e i φ i e i ω τ i .
H eff = π δ f HFS , g ( | b b | | a a | ) + ( Δ e i Γ / 2 ) | e e | + Ω a ( t ) 2 | e a | + Ω b ( t ) 2 | e b | + h . c .
ρ ˙ = 1 i ( H eff ρ ρ H eff ) + C a ρ C a + C b ρ C b .
p e ( j ) = | Θ 0 | 2 4 ( ρ a a ( j ) S ( ω e g + π δ f HFS , g ) + ρ b b ( j ) S ( ω e g π δ f HFS , g ) ) .
Δ ρ a a ( j + 1 ) = | Θ 0 | 2 4 ( S ( ω e g + π δ f HFS , g ) ρ a a ( j ) Γ b Γ a + Γ b + S ( ω e g π δ f HFS , g ) ρ b b ( j ) Γ a Γ a + Γ b ) .
p e ( s s ) = | Θ 0 | 2 4 S ~ ( ω e g , π δ f HFS , g ) ,
S ~ ( ω , δ f ) = 2 S ( ω π δ f ) S ( ω + π δ f ) S ( ω π δ f ) + S ( ω + π δ f ) .

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