Abstract

We apply the method of quantum state tomography for the reconstruction of classical laser pulses. The scheme is based on linear inversion, has no need for iterative inversion algorithm or deconvolution, and accounts for partial coherence. The reconstruction protocol is successfully tested on amplitude and phase shaped femtosecond pulses.

© 2016 Optical Society of America

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References

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  1. D. J. Bradley, “Generation and measurement of picosecond pulses,” J. Phys. Chem. 82, 2259–2268 (1978).
    [Crossref]
  2. D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
    [Crossref] [PubMed]
  3. K. W. DeLong, R. Trebino, and D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B 11, 1595 (1994).
    [Crossref]
  4. C. Iaconis and I. a. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
    [Crossref]
  5. C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashortoptical pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
    [Crossref]
  6. D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 1–4 (2015).
    [Crossref]
  7. D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
    [Crossref] [PubMed]
  8. V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775 (2004).
    [Crossref] [PubMed]
  9. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
    [Crossref] [PubMed]
  10. K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
    [Crossref]
  11. L. Cohen, “Time-Frequency Distributions - A Review,” Proc. IEEE 77, 941–981 (1989).
    [Crossref]
  12. J. Paye, “The Chronocyclic Representation of Ultrashort Light Pulses,” IEEE J. Quantum Electron. 28, 2262–2273 (1992).
    [Crossref]
  13. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308 (2009).
    [Crossref]
  14. C. Bourassin-Bouchet and M. E. Couprie, “Partially coherent ultrafast spectrography,” Nat Commun. 6, 6465 (2015).
    [Crossref] [PubMed]
  15. W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36 (1991).
    [Crossref]
  16. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
    [Crossref]
  17. J. Schlienz and G. Mahler, “Description of entanglement,” Phys. Rev. A 52, 4396–4404 (1995).
    [Crossref] [PubMed]
  18. C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
    [Crossref]
  19. P. Faist and R. Renner, “Practical and Reliable Error Bars in Quantum Tomography,” Phys. Rev. Lett. 117, 1–6 (2016).
    [Crossref]
  20. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [Crossref]
  21. T. Feurer, “Laser Beam Propagation Generation and Propagation of Customized Light,” in “Laser Beam Propagation Generation and Propagation of Customized Light,” A. Forbes, ed. (CRC Press, 2014), pp. 77–112.
    [Crossref]
  22. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934 (2001).
    [Crossref]
  23. S. Linden, H. Giessen, and J. Kuhl, “XFROG - A New Method for Amplitude and Phase Characterization of Weak Ultrashort Pulses,” Phys. Status Solidi B 206, 119–124 (1998).
    [Crossref]

2016 (1)

P. Faist and R. Renner, “Practical and Reliable Error Bars in Quantum Tomography,” Phys. Rev. Lett. 117, 1–6 (2016).
[Crossref]

2015 (4)

D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 1–4 (2015).
[Crossref]

C. Bourassin-Bouchet and M. E. Couprie, “Partially coherent ultrafast spectrography,” Nat Commun. 6, 6465 (2015).
[Crossref] [PubMed]

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
[Crossref] [PubMed]

2009 (1)

2004 (1)

2001 (2)

P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934 (2001).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

1999 (1)

C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashortoptical pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[Crossref]

1998 (2)

S. Linden, H. Giessen, and J. Kuhl, “XFROG - A New Method for Amplitude and Phase Characterization of Weak Ultrashort Pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[Crossref]

C. Iaconis and I. a. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[Crossref]

1995 (1)

J. Schlienz and G. Mahler, “Description of entanglement,” Phys. Rev. A 52, 4396–4404 (1995).
[Crossref] [PubMed]

1994 (1)

1993 (2)

1992 (1)

J. Paye, “The Chronocyclic Representation of Ultrashort Light Pulses,” IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[Crossref]

1991 (1)

W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36 (1991).
[Crossref]

1989 (1)

L. Cohen, “Time-Frequency Distributions - A Review,” Proc. IEEE 77, 941–981 (1989).
[Crossref]

1982 (1)

K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[Crossref]

1978 (1)

D. J. Bradley, “Generation and measurement of picosecond pulses,” J. Phys. Chem. 82, 2259–2268 (1978).
[Crossref]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Beck, M.

Bourassin-Bouchet, C.

C. Bourassin-Bouchet and M. E. Couprie, “Partially coherent ultrafast spectrography,” Nat Commun. 6, 6465 (2015).
[Crossref] [PubMed]

Bradley, D. J.

D. J. Bradley, “Generation and measurement of picosecond pulses,” J. Phys. Chem. 82, 2259–2268 (1978).
[Crossref]

Brenner, K. H.

K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[Crossref]

Brügmann, M. H.

D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 1–4 (2015).
[Crossref]

D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
[Crossref] [PubMed]

Cohen, L.

L. Cohen, “Time-Frequency Distributions - A Review,” Proc. IEEE 77, 941–981 (1989).
[Crossref]

Couprie, M. E.

C. Bourassin-Bouchet and M. E. Couprie, “Partially coherent ultrafast spectrography,” Nat Commun. 6, 6465 (2015).
[Crossref] [PubMed]

Dantus, M.

DeLong, K. W.

Dorrer, C.

Faist, P.

P. Faist and R. Renner, “Practical and Reliable Error Bars in Quantum Tomography,” Phys. Rev. Lett. 117, 1–6 (2016).
[Crossref]

Feurer, T.

D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 1–4 (2015).
[Crossref]

D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
[Crossref] [PubMed]

T. Feurer, “Laser Beam Propagation Generation and Propagation of Customized Light,” in “Laser Beam Propagation Generation and Propagation of Customized Light,” A. Forbes, ed. (CRC Press, 2014), pp. 77–112.
[Crossref]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Giessen, H.

S. Linden, H. Giessen, and J. Kuhl, “XFROG - A New Method for Amplitude and Phase Characterization of Weak Ultrashort Pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[Crossref]

Gu, X.

Gühne, O.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Iaconis, C.

C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashortoptical pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[Crossref]

C. Iaconis and I. a. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[Crossref]

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Kane, D. J.

Kimmel, M.

Kleinmann, M.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

Knips, L.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

Kuhl, J.

S. Linden, H. Giessen, and J. Kuhl, “XFROG - A New Method for Amplitude and Phase Characterization of Weak Ultrashort Pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[Crossref]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Linden, S.

S. Linden, H. Giessen, and J. Kuhl, “XFROG - A New Method for Amplitude and Phase Characterization of Weak Ultrashort Pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[Crossref]

Lozovoy, V. V.

Mahler, G.

J. Schlienz and G. Mahler, “Description of entanglement,” Phys. Rev. A 52, 4396–4404 (1995).
[Crossref] [PubMed]

Moroder, T.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Neethling, P.

D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 1–4 (2015).
[Crossref]

O’Shea, P.

Pastirk, I.

Paye, J.

J. Paye, “The Chronocyclic Representation of Ultrashort Light Pulses,” IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[Crossref]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Raymer, M. G.

Renner, R.

P. Faist and R. Renner, “Practical and Reliable Error Bars in Quantum Tomography,” Phys. Rev. Lett. 117, 1–6 (2016).
[Crossref]

Richart, D.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

Rohwer, E.

D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
[Crossref] [PubMed]

D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 1–4 (2015).
[Crossref]

Schlienz, J.

J. Schlienz and G. Mahler, “Description of entanglement,” Phys. Rev. A 52, 4396–4404 (1995).
[Crossref] [PubMed]

Schwemmer, C.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

Spangenberg, D.

D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
[Crossref] [PubMed]

D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 1–4 (2015).
[Crossref]

Trebino, R.

Walmsley, I.

C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashortoptical pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[Crossref]

Walmsley, I. A.

Weinfurter, H.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Wódkiewicz, K.

K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[Crossref]

Wong, V.

Zurek, W. H.

W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36 (1991).
[Crossref]

Adv. Opt. Photon. (1)

IEEE J. Quantum Electron. (2)

C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashortoptical pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[Crossref]

J. Paye, “The Chronocyclic Representation of Ultrashort Light Pulses,” IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. (1)

D. J. Bradley, “Generation and measurement of picosecond pulses,” J. Phys. Chem. 82, 2259–2268 (1978).
[Crossref]

Nat Commun. (1)

C. Bourassin-Bouchet and M. E. Couprie, “Partially coherent ultrafast spectrography,” Nat Commun. 6, 6465 (2015).
[Crossref] [PubMed]

Opt. Commun. (1)

K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[Crossref]

Opt. Lett. (6)

Phys. Rev. A (3)

D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 1–4 (2015).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

J. Schlienz and G. Mahler, “Description of entanglement,” Phys. Rev. A 52, 4396–4404 (1995).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 1–6 (2015).
[Crossref]

P. Faist and R. Renner, “Practical and Reliable Error Bars in Quantum Tomography,” Phys. Rev. Lett. 117, 1–6 (2016).
[Crossref]

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Phys. Status Solidi B (1)

S. Linden, H. Giessen, and J. Kuhl, “XFROG - A New Method for Amplitude and Phase Characterization of Weak Ultrashort Pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[Crossref]

Phys. Today (1)

W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36 (1991).
[Crossref]

Proc. IEEE (1)

L. Cohen, “Time-Frequency Distributions - A Review,” Proc. IEEE 77, 941–981 (1989).
[Crossref]

Other (1)

T. Feurer, “Laser Beam Propagation Generation and Propagation of Customized Light,” in “Laser Beam Propagation Generation and Propagation of Customized Light,” A. Forbes, ed. (CRC Press, 2014), pp. 77–112.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Schematic of the experimental setup. A Ti:Sa oscillator is pumped by a Verdi laser. A mirror (M0) sends the beam through a prism compressor (N-SF11 equilateral dispersive prism P1 and P2, mirror M1). A periscope (P) manipulates the polarization and guarantees the phase-matching condition in the SHG crystal. The pulses are sent through a grating compressor consisting of two transmission grating (G1 and G2, from LightSmyth 1503.76 grooves/mm), two plane mirrors (M2 and M5), and two cylindrical mirrors (M3 and M4, 600 mm radius). At the symmetry plane an SLM allows to shape the spectrum. The pulses are focused by the lens L1 in a nonlinear crystal (PPKTP), and the generated SHG light is collimated (lens L2) and detected by a photodiode (PD). The remaining laser light is filtered by a shortpass and a bandpass filters (F).

Fig. 2
Fig. 2

(a) Spectrum at the output of the Ti:Sa oscillator, fiber coupled (fiber core 200 µm) and measured with the OSA (Yokogawa AQ6370). (b) Autocorrelation function ACF(τ): measurement (blue) with a Gaussian fit (red) at the tip-tip distance d = 209.6 mm. (c) Wisth of ACF(τ) as a function of the tip-tip distance of the prism compressor. The blue points indicate the measured widths, and the red curve is a fit of a quadratic function. (d) The folded geometry of the prism compressor leads to a spatially dispersed spectrum. Here, the dispersion in the vertical direction y is shown. The red line indicates the change of center wavelength.

Fig. 3
Fig. 3

(a) Measured intensities of the upconversion signal for the 496 measurements needed for the pulse reconstruction of Fig. 4. (b) Example (for measurement number 50) of the amplitude (black continuous curve) and phase (red dashed curve) of the applied complex transfer function on the 640 pixels of the SLM. The asymmetry is due to the non-linear pixel to wavelength calibration.

Fig. 4
Fig. 4

Reconstruction results for different chirped pulses. From top to bottom: positive linear chirp (β = 500 rad/fs), negative linear chirp (β = −500 rad/fs), and an incoherent mixture of both. (a) Modulus and phase of each entry ρ ^ i j of the density matrix reconstructed using Eq. (8). (b) Plot of the eigenvalues λn of ρ ^ and reconstructed spectra (normalized to one), using only the eigenvector corresponding to the largest eigenvalue λd−1. (c) Normalized Wigner function according to Eq. (23). The dotted line indicate the expected chirp of β = ±500 fs2; Marginal (blue) and expected (red) spectra from the Wigner function.

Fig. 5
Fig. 5

Reconstruction results with projections for amplitude modulated pulses. Two frequency parts are isolated by the transfer function as in Eq. (31). (a) Modulus and phase of each entry ρ ^ i j of the density matrix reconstructed using Eq. (8). (b) Plot of the eigenvalues λn of ρ ^ and reconstructed spectra (normalized to one), using only the eigenvector corresponding to the largest eigenvalue λd−1. (c) Normalized Wigner function according to Eq. (23). The dotted line indicate the expected center frequencies at Ω = ±0.04 rad/fs.; Marginal (blue) and expected (red) spectra from the Wigner function.

Equations (35)

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ρ ^ = n p n | Ψ n Ψ n |
ρ ^ = j λ j | Φ j Φ j | .
ρ ^ = 1 d d + 1 2 j = 1 d 2 1 r j σ ^ j ,
r j = Tr { σ ^ j ρ ^ } .
σ ^ j = k = 1 2 d 2 d b j , k m ^ k ,
| m k = j = 0 d 1 u k , j | u j .
s k Tr { m ^ k ρ ^ } .
ρ ^ = 1 d d + 1 2 N j = 1 d 2 1 σ ^ j k = 1 2 d 2 d b j , k s k .
c j = P j | Ψ = a b d Ω Ψ ( Ω ) P j * ( Ω ) ,
C ( Ω a , Ω b ) = 0 ( Ω a ) 0 * ( Ω b ) .
C ( Ω a , Ω b ) = n p n Ψ n * ( Ω b ) Ψ n ( Ω a ) .
ρ ^ p u l s e ( Ω a , Ω b ) = Ω a | ρ ^ p u l s e | Ω b
= n p n Ω a | Ψ n Ψ n | Ω b
= C ( Ω a , Ω b ) .
ρ ^ p u l s e ( Ω a , Ω b ) = k , l ρ k , l P k * ( Ω a ) P l ( Ω b ) ,
ρ k , l = j λ j ν j ( k ) * ν j ( l ) .
W ( t , Ω ) = d Ω 2 π C ( Ω + Ω 2 , Ω Ω 2 ) e i Ω t .
μ = d Ω d Ω | C ( Ω + Ω 2 , Ω Ω 2 ) | 2 [ d Ω C ( Ω , Ω ) ] 2 , 0 μ 1 .
μ = d Ω d Ω | ρ ^ p u l s e ( Ω + Ω 2 , Ω Ω 2 ) | 2
= d Ω a d Ω b | ρ ^ p u l s e ( Ω a , Ω b ) | 2
= Tr { ρ ^ 2 } .
W ( t , Ω ) = d Ω 2 π ρ ^ p u l s e ( Ω + Ω 2 , Ω Ω 2 ) e i Ω t ,
W ( t , Ω ) = d Ω 2 π j = 0 d 1 λ j [ k = 0 d 1 ν j ( k ) P k ( Ω + Ω 2 ) ] [ l = 0 d 1 ν j ( l ) P l ( Ω Ω 2 ) ] e i Ω t .
M k ( Ω ) = j u k , j P j ( Ω ) ,
S k = | a b d Ω Ψ ( Ω ) M k * ( Ω ) | 2 .
S S H G | d Ω ( Ω ) ( Ω ) | 2 ,
( Ω ) = 0 ( Ω ) M ( Ω ) .
P n ( x ) = 2 n k = 0 n ( 1 ) k x k ( n k ) ( n + k 1 2 n )
e ( Ω ) = 1 2 [ S ( Ω ) + S ( Ω ) ] , o ( Ω ) = 1 2 [ S ( Ω ) S ( Ω ) ] ,
| f ( Ω ) | 2 = | e ( Ω ) 2 o ( Ω ) 2 | 2 = S ( Ω ) S ( Ω ) .
S ( Ω ) = 1 2 ( e i ( Ω Ω 0 ) 2 σ 2 + e i ( Ω + Ω 0 ) 2 σ 2 ) .
| m j j ± 1 2 ( | u j ± | u j ) , | m j j ± i 1 2 ( | u j ± i | u j ) , | m j j | u j .
σ ^ j 2 1 = 2 j ( j 1 ) ( k = 1 j 1 m ^ k k ( j 1 ) m ^ j j ) ,
σ ^ ( j 1 ) 2 + 2 ( j 1 ) = m ^ j j + m ^ j j ,
σ ^ ( j 1 ) 2 + 2 j 1 = m ^ j j + i m ^ j j i ,

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