Abstract

Ultrafast (femtosecond) interferometric pump–probe techniques can be used to measure rates of population and quantum phase decay in complicated media such as liquids and solids. However, the levels probed in such systems are often inhomogeneously broadened or are part of a continuum of states. The use of broadband ultrafast lasers thus results in multiple levels being excited and detected. The inherent averaging that is due to this effect can alter the measured coherent response, thus affecting the information that can be retrieved on the phase decay. The importance of these effects is considered for the representative case of two-photon photoemission from metals. The effects of (i) continuum excitation; (ii) excitation from the Fermi level, i.e., a spectral step function; (iii) excitation from broadened levels with a finite width; and (iv) photoelectron energy analyzer resolution are determined.

© 2000 Optical Society of America

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References

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  1. H. Petek and S. Ogawa, “Femtosecond time-resolved two-photon photoemission studies of electron dynamics in metals,” Prog. Surf. Sci. 56, 239–310 (1997).
    [CrossRef]
  2. S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339–1342 (1997).
    [CrossRef]
  3. B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Ausenegg, “Particle plasmon decay time determination by measuring the optical near field’s autocorrelation: influence of inhomogeneous line broadening,” Appl. Phys. B 69, 223–227 (1999).
    [CrossRef]
  4. U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
    [CrossRef]
  5. V. Blanchet, C. Nicole, M. A. Bouchéne, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2716–2719 (1997).
    [CrossRef]
  6. I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
    [CrossRef]
  7. V. Blanchet, M. A. Bouchéne, and B. Girard, “Temporal coherent control in the photoionization of Cs2: theory and experiment,” J. Chem. Phys. 108, 4862–4876 (1998).
    [CrossRef]
  8. V. Blanchet, M. A. Bouchéne, O. Cabrol, and B. Girard, “One-color coherent control in Cs2. Observation of 2.7 fs beats in the ionization signal,” Chem. Phys. Lett. 233, 491–499 (1995).
    [CrossRef]
  9. R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, UK, 1983).
  10. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).
  11. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).
  12. F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
    [CrossRef]
  13. J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
    [CrossRef] [PubMed]
  14. S. Ogawa, H. Nagano, and H. Petek, “Optical decoherence and quantum beats in Cs/Cu(111),” Surf. Sci. 427–428, 34–38 (1999).
    [CrossRef]

1999

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Ausenegg, “Particle plasmon decay time determination by measuring the optical near field’s autocorrelation: influence of inhomogeneous line broadening,” Appl. Phys. B 69, 223–227 (1999).
[CrossRef]

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

S. Ogawa, H. Nagano, and H. Petek, “Optical decoherence and quantum beats in Cs/Cu(111),” Surf. Sci. 427–428, 34–38 (1999).
[CrossRef]

1998

I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
[CrossRef]

V. Blanchet, M. A. Bouchéne, and B. Girard, “Temporal coherent control in the photoionization of Cs2: theory and experiment,” J. Chem. Phys. 108, 4862–4876 (1998).
[CrossRef]

1997

U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
[CrossRef]

V. Blanchet, C. Nicole, M. A. Bouchéne, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2716–2719 (1997).
[CrossRef]

H. Petek and S. Ogawa, “Femtosecond time-resolved two-photon photoemission studies of electron dynamics in metals,” Prog. Surf. Sci. 56, 239–310 (1997).
[CrossRef]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339–1342 (1997).
[CrossRef]

1995

V. Blanchet, M. A. Bouchéne, O. Cabrol, and B. Girard, “One-color coherent control in Cs2. Observation of 2.7 fs beats in the ionization signal,” Chem. Phys. Lett. 233, 491–499 (1995).
[CrossRef]

1993

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Ausenegg, F. R.

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Ausenegg, “Particle plasmon decay time determination by measuring the optical near field’s autocorrelation: influence of inhomogeneous line broadening,” Appl. Phys. B 69, 223–227 (1999).
[CrossRef]

Bacher, G.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Blanchet, V.

V. Blanchet, M. A. Bouchéne, and B. Girard, “Temporal coherent control in the photoionization of Cs2: theory and experiment,” J. Chem. Phys. 108, 4862–4876 (1998).
[CrossRef]

V. Blanchet, C. Nicole, M. A. Bouchéne, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2716–2719 (1997).
[CrossRef]

V. Blanchet, M. A. Bouchéne, O. Cabrol, and B. Girard, “One-color coherent control in Cs2. Observation of 2.7 fs beats in the ionization signal,” Chem. Phys. Lett. 233, 491–499 (1995).
[CrossRef]

Bouchéne, M. A.

V. Blanchet, M. A. Bouchéne, and B. Girard, “Temporal coherent control in the photoionization of Cs2: theory and experiment,” J. Chem. Phys. 108, 4862–4876 (1998).
[CrossRef]

V. Blanchet, C. Nicole, M. A. Bouchéne, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2716–2719 (1997).
[CrossRef]

V. Blanchet, M. A. Bouchéne, O. Cabrol, and B. Girard, “One-color coherent control in Cs2. Observation of 2.7 fs beats in the ionization signal,” Chem. Phys. Lett. 233, 491–499 (1995).
[CrossRef]

Cabrol, O.

V. Blanchet, M. A. Bouchéne, O. Cabrol, and B. Girard, “One-color coherent control in Cs2. Observation of 2.7 fs beats in the ionization signal,” Chem. Phys. Lett. 233, 491–499 (1995).
[CrossRef]

Fauster, T.

I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
[CrossRef]

U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
[CrossRef]

Feldmann, J.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Girard, B.

V. Blanchet, M. A. Bouchéne, and B. Girard, “Temporal coherent control in the photoionization of Cs2: theory and experiment,” J. Chem. Phys. 108, 4862–4876 (1998).
[CrossRef]

V. Blanchet, C. Nicole, M. A. Bouchéne, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2716–2719 (1997).
[CrossRef]

V. Blanchet, M. A. Bouchéne, O. Cabrol, and B. Girard, “One-color coherent control in Cs2. Observation of 2.7 fs beats in the ionization signal,” Chem. Phys. Lett. 233, 491–499 (1995).
[CrossRef]

Göbel, E. O.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Hägele, A.

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

Hartmann, C.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Heberle, A. P.

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339–1342 (1997).
[CrossRef]

Hertel, I. V.

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

Höfer, U.

I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
[CrossRef]

U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
[CrossRef]

Koch, M.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Krenn, J. R.

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Ausenegg, “Particle plasmon decay time determination by measuring the optical near field’s autocorrelation: influence of inhomogeneous line broadening,” Appl. Phys. B 69, 223–227 (1999).
[CrossRef]

Lamprecht, B.

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Ausenegg, “Particle plasmon decay time determination by measuring the optical near field’s autocorrelation: influence of inhomogeneous line broadening,” Appl. Phys. B 69, 223–227 (1999).
[CrossRef]

Leitner, A.

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Ausenegg, “Particle plasmon decay time determination by measuring the optical near field’s autocorrelation: influence of inhomogeneous line broadening,” Appl. Phys. B 69, 223–227 (1999).
[CrossRef]

Lutz, H. O.

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

Meier, F.

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

Meier, T.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Nagano, H.

S. Ogawa, H. Nagano, and H. Petek, “Optical decoherence and quantum beats in Cs/Cu(111),” Surf. Sci. 427–428, 34–38 (1999).
[CrossRef]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339–1342 (1997).
[CrossRef]

Nickel, H.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Nicole, C.

V. Blanchet, C. Nicole, M. A. Bouchéne, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2716–2719 (1997).
[CrossRef]

Ogawa, S.

S. Ogawa, H. Nagano, and H. Petek, “Optical decoherence and quantum beats in Cs/Cu(111),” Surf. Sci. 427–428, 34–38 (1999).
[CrossRef]

H. Petek and S. Ogawa, “Femtosecond time-resolved two-photon photoemission studies of electron dynamics in metals,” Prog. Surf. Sci. 56, 239–310 (1997).
[CrossRef]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339–1342 (1997).
[CrossRef]

Petek, H.

S. Ogawa, H. Nagano, and H. Petek, “Optical decoherence and quantum beats in Cs/Cu(111),” Surf. Sci. 427–428, 34–38 (1999).
[CrossRef]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339–1342 (1997).
[CrossRef]

H. Petek and S. Ogawa, “Femtosecond time-resolved two-photon photoemission studies of electron dynamics in metals,” Prog. Surf. Sci. 56, 239–310 (1997).
[CrossRef]

Reuss, C.

I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
[CrossRef]

U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
[CrossRef]

Schäfer, W.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Schreiber, E.

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

Schulz, C. P.

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

Schweizer, H.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Shumay, I. L.

I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
[CrossRef]

U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
[CrossRef]

Stienkemeier, F.

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

Thomann, U.

I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
[CrossRef]

U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
[CrossRef]

Thomas, P.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

von Plessen, G.

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

Wallauer, W.

I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
[CrossRef]

U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
[CrossRef]

Appl. Phys. B

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Ausenegg, “Particle plasmon decay time determination by measuring the optical near field’s autocorrelation: influence of inhomogeneous line broadening,” Appl. Phys. B 69, 223–227 (1999).
[CrossRef]

Chem. Phys. Lett.

V. Blanchet, M. A. Bouchéne, O. Cabrol, and B. Girard, “One-color coherent control in Cs2. Observation of 2.7 fs beats in the ionization signal,” Chem. Phys. Lett. 233, 491–499 (1995).
[CrossRef]

J. Chem. Phys.

V. Blanchet, M. A. Bouchéne, and B. Girard, “Temporal coherent control in the photoionization of Cs2: theory and experiment,” J. Chem. Phys. 108, 4862–4876 (1998).
[CrossRef]

Phys. Rev. B

I. L. Shumay, U. Höfer, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Lifetimes of image-potential states on Cu(100) and Ag(100) measured by femtosecond time-resolved two-photon photoemission,” Phys. Rev. B 58, 13974–13981 (1998).
[CrossRef]

Phys. Rev. Lett.

V. Blanchet, C. Nicole, M. A. Bouchéne, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2716–2719 (1997).
[CrossRef]

F. Stienkemeier, F. Meier, A. Hägele, H. O. Lutz, E. Schreiber, C. P. Schulz, and I. V. Hertel, “Coherence and relaxation in potassium-doped helium droplets studied by femtosecond pump–probe spectroscopy,” Phys. Rev. Lett. 83, 2320–2323 (1999).
[CrossRef]

J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave packets,” Phys. Rev. Lett. 70, 3027–3030 (1993).
[CrossRef] [PubMed]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339–1342 (1997).
[CrossRef]

Prog. Surf. Sci.

H. Petek and S. Ogawa, “Femtosecond time-resolved two-photon photoemission studies of electron dynamics in metals,” Prog. Surf. Sci. 56, 239–310 (1997).
[CrossRef]

Science

U. Höfer, I. L. Shumay, C. Reuss, U. Thomann, W. Wallauer, and T. Fauster, “Time-resolved coherent photoelectron spectroscopy of quantized electronic states on metal surfaces,” Science 277, 1480–1482 (1997).
[CrossRef]

Surf. Sci.

S. Ogawa, H. Nagano, and H. Petek, “Optical decoherence and quantum beats in Cs/Cu(111),” Surf. Sci. 427–428, 34–38 (1999).
[CrossRef]

Other

R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, UK, 1983).

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).

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Figures (7)

Fig. 1
Fig. 1

a) Schematic of two-photon photoemission. On left side, sequential two-photon excitation first lifts an electron to the intermediate level, leaving behind an excited hole. The second photon lifts the excited electron above the vacuum level at which it can be detected. It is also possible to have a direct two-photon excitation from the ground state to the final state. b) Schematic of three-level system and the parameters representing the time scales for decay of coherences between levels, i.e., the T2 terms, and population decay, i.e., T11.

Fig. 2
Fig. 2

Simulated I2PC (solid curve) to show effects of different optical Bloch equation parameters. Net displacement in the wings from the laser second-order autocorrelation (19-fs electric field FWHM hyperbolic-secant-squared pulse, ωl=1.55 eV: heavy dashed curve) is due to the incoherent lifetime of the intermediate state, T11=30 fs. The oscillatory envelope outside the laser autocorrelation is due to the finite decay time for coherences between the levels, in this case T201=T212=5 fs and T202=20 fs. The longer T202 lifetime results in oscillations in the wings at twice the frequency, i.e., 2ω, of the laser fundamental. The inset shows the I2PC magnified by 2 compared with an I2PC with the same parameters but with a shift in the final-state energy of Δ2=40 meV (offset vertically to highlight differences). The slightly different driving frequency results in a phase shift of the oscillations. The relative phase shift between the Δ2=40-meV and 0-meV traces for the 2ω oscillations (thin dashed curve) demonstrates that they are completely out of phase for a delay of 60 fs.

Fig. 3
Fig. 3

Effects of analyzer resolution on I2PC oscillatory envelope. Inset shows level scheme for discrete initial and intermediate states and a continuum of photoelectron states. The energy analyzer samples a finite range, ΔE, of the continuum. The calculated I2PC envelopes are for (A) a second-harmonic autocorrelation of the 19-fs electric field FWHM hyperbolic-secant-squared pulse, (B) I2PC averaged over 20-meV analyzer resolution for T11=T201=5 fs and T202=T212=100 fs, and (C) I2PC for infinitely sharp analyzer resolution and same parameters as at (B). Note that analyzer resolution can completely wash out coherent oscillatory structure for time scalesh/ΔE.

Fig. 4
Fig. 4

Examples of one- and two-photon photoexcitation from a continuum. Here the solid curves represent the resonant transition, while the dashed and dotted–dashed curves represent levels and distributions that are sampled to the red (Δ1R) and blue (Δ1B) of the resonant frequency, respectively. a) In the discrete one-photon case, Δ1 represents the position of the final level with respect to the laser frequency envelope. b) For the continuum one-photon case, Δ1 represents the shift of Eobs with respect to the peak of the final photoelectron distribution from a specific ground-state energy. Averaging over Δ1 is the same as averaging over all ground state energies. c) The two-photon continuum case is similar, except that averaging over the ground-state energies is the same as averaging over Δ2. For each Δ2, it is also necessary to average over all intermediate state energies, i.e., Δ1.

Fig. 5
Fig. 5

Simulated interferograms for discrete and continuum two-photon excitation with a constant density of states. Both interferograms are for a system with the same decay times (T11=25 fs, T201=T212=20 fs, T202=50 fs), but one is averaged over a continuum of possible initial and intermediate states with a discrete final state, while the other is for three discrete levels. The averaging that is due to the continuum washes out the interference fringes outside the laser autocorrelation trace. However, the incoherent population decay time results in a net offset in the wings, compared with the laser autocorrelation. Simulations are for a pulse with ωl=1.55 eV and a 19-fs FWHM electric field Gaussian envelope.

Fig. 6
Fig. 6

a) Observed and b) simulated I2PC from the Fermi edge of 50 K cesiated Cu(100) with 1.55-eV excitation. The observed trace is much broader than the laser second-harmonic autocorrelation. In addition, the phase shift between the laser autocorrelation and the I2PC demonstrates that the interferogram is due predominately to electrons driven by the higher-frequency components of the laser pulse. The observed I2PC can be completely reproduced by the continuum model. In a least-squares fit of the data, this model reveals extremely long phase-coherence times of >100 fs between the ground and final state.

Fig. 7
Fig. 7

Continuum excitation from the Fermi level. The population of the initial states changes rapidly with energy at the Fermi edge, skewing the final distribution of energies to electrons that have been predominately excited by higher-frequency components of the laser pulse. Averaging results in an incomplete cancellation of the oscillatory envelope from coherent terms, and the skewed distribution results in an apparent shift in the oscillation frequency from ωl.

Equations (15)

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ckt=-iΔkck+Ck+1,kck+1+Dk-1,kck-1-ckTspont.
ρ11t=()-2Tspont+1T11ρ11.
(c2c1*)t=()-i(Δ2-Δ1)+12T12+12T11+1T212c2c1*.
Itot(τ)=f(Δ2)g(Δ1=0, Δ2)I(τ, Δ1=0, Δ2)dΔ2,
Itot(τ)=g(Δ1=Δ1+Δ2, Δ2)×I(τ, Δ1=Δ1+Δ2, Δ2)dΔ1dΔ2,
Itot(τ)=f(Eobs-2ωl-Δ2)g(Δ1=Δ1+Δ2, Δ2)×I(τ, Δ1=Δ1+Δ2, Δ2)dΔ1dΔ2,
E(t)=12[˜ exp(iωlt)+˜* exp(-iωlt)].
˜(t)=(t)exp[iϕ(t)],
dij=|d|ϕi|rˆ|ϕj,
ρ11t=i2d01(˜*c0c1*-˜c1c0*)+i2d12(˜c2c1*-˜*c1c2*)-1T11ρ11,
ρ22t=i2d12(˜*c1c2*-˜c2c1*)-1T12ρ22,
ρ00t=-ρ11t-ρ22t,
(c1c0*)t=i2d01˜*(ρ00-ρ11)+i2d12˜c2c0*-iΔ1+12T11+1T201c1c0*,
(c2c1*)t=i2d12˜*(ρ11-ρ22)-i2d01˜c2c0*-i(Δ2-Δ1)+12T12+12T11+1T212c2c1*,
(c2c0*)t=i2˜*(d12c1c0*-d01c2c1*)-iΔ2+12T12+1T202c2c0*.

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