Abstract

The question of how long a tunneling particle spends inside the barrier region has remained unresolved since the early days of quantum mechanics. The main theoretical contenders, such as the Buttiker–Landauer, Eisenbud–Wigner, and Larmor time, give contradictory answers. On the other hand, recent attempts at reconstructing valence electron dynamics in atoms and molecules have entered a regime where the tunneling time genuinely matters. Here, we compare the main competing theories of tunneling time against experimental measurements using the attoclock in strong laser field ionization of helium atoms. The attoclock uses a close to circularly polarized femtosecond laser pulse, mapping the angle of rotation of the laser field vector to time similar to the hand of a watch. Refined attoclock measurements reveal a real (not instantaneous) tunneling delay time over a large intensity regime, using two independent experimental apparatus. Only two theoretical predictions are compatible within our experimental error: the Larmor time and the probability distribution of tunneling times constructed using a Feynman Path Integral formulation. The latter better matches the observed qualitative change in tunneling time over a wide intensity range, and predicts a broad tunneling time distribution with a long tail. The implication of such a probability distribution of tunneling times, as opposed to a distinct tunneling time, would imply that one must account for a significant, though bounded and measurable, uncertainty as to when the hole dynamics begin to evolve. We therefore expect our results to impact the reconstruction of attosecond electron dynamics following tunnel ionization.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2014 (1)

A. S. Landsman, U. Keller, “Tunneling time in strong field ionization,” J. Phys. B 47, 204024 (2014).

2013 (1)

A. S. Landsman, C. Hofmann, A. Pfeiffer, C. Cirelli, U. Keller, “A unified approach to probing Coulomb effects in tunnel ionization for any ellipticity of laser light,” Phys. Rev. Lett. 111, 263001 (2013).

2012 (5)

C. H. R. Ooi, W. Ho, A. D. Bandrauk, “Photoionization spectra by intense linear, circular, and elliptic polarized lasers,” Phys. Rev. A 86, 023410 (2012).
[Crossref]

J. M. Dahlström, A. L’Huillier, A. Maquet, “Introduction to attosecond delays in photoionization,” J. Phys. B 45, 183001 (2012).
[Crossref]

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref]

M. Lein, “Atomic physics: Electrons get real,” Nature 485, 313–314 (2012).
[Crossref]

A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abusamha, L. B. Madsen, U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
[Crossref]

2011 (3)

A. Wirth, M. T. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, E. Goulielmakis, “Synthesized light transients,” Science 334, 195–200 (2011).
[Crossref]

K. Klünder, J. M. Dahlström, M. Gisselbrecht, T. Fordell, M. Swoboda, D. Guenot, P. Johnsson, J. Caillat, J. Mauritsson, A. Maquet, R. Taïeb, A. L’Huillier, “Probing single-photon ionization on the attosecond time scale,” Phys. Rev. Lett. 106, 143002 (2011).
[Crossref]

S. Nagele, R. Pazourek, J. Feist, K. Doblhoff-Dier, C. Lemell, K. Tökési, J. Burgdörfer, “Time-resolved photoemission by attosecond streaking: extraction of time information,” J. Phys. B 44, 081001 (2011).
[Crossref]

2010 (4)

O. Smirnova, M. Ivanov, “Ultrafast science towards a one-femtosecond film,” Nat. Phys. 6, 159–160 (2010).
[Crossref]

M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdorfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, V. S. Yakovlev, “Delay in photoemission,” Science 328, 1658–1662 (2010).
[Crossref]

R. Murray, W. K. Liu, M. Y. Ivanov, “Partial Fourier-transform approach to tunnel ionization: atomic systems,” Phys. Rev. A 81, 023413 (2010).
[Crossref]

E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, F. Krausz, “Real-time observation of valence electron motion,” Nature 466, 739–743 (2010).
[Crossref]

2009 (3)

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref]

M. Wollenhaupt, M. Krug, J. Kohler, T. Bayer, C. Sarpe-Tudoran, T. Baumert, “Three-dimensional tomographic reconstruction of ultrashort free electron wave packets,” Appl. Phys. B 95, 647–651 (2009).
[Crossref]

H. Akagi, T. Otobe, A. Staudte, A. Shiner, F. Turner, R. Dörner, D. M. Villeneuve, P. B. Corkum, “Laser tunnel ionization from multiple orbitals in HCl,” Science 325, 1364–1367 (2009).
[Crossref]

2008 (3)

M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavicic, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320, 1478–1482 (2008).
[Crossref]

P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
[Crossref]

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Büttiker, U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
[Crossref]

2007 (1)

M. Uiberacker, T. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature 446, 627–632 (2007).
[Crossref]

2005 (1)

J. Breidbach, L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Phys. Rev. Lett. 94, 033901 (2005).
[Crossref]

2004 (1)

N. Yamada, “Unified derivation of tunneling times from decoherence functionals,” Phys. Rev. Lett. 93, 170401 (2004).
[Crossref]

2001 (1)

G. L. Yudin, M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A 64, 013409 (2001).
[Crossref]

2000 (1)

R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, H. Schmidt-Bocking, “Cold target recoil ion momentum spectroscopy: a ‘momentum microscope’ to view atomic collision dynamics,” Phys. Rep. 330, 95–192 (2000).
[Crossref]

1997 (1)

D. H. Parker, A. T. J. B. Eppink, “Photoelectron and photofragment velocity map imaging of state-selected molecular oxygen dissociation/ionization dynamics,” J. Chem. Phys. 107, 2357 (1997).
[Crossref]

1994 (2)

D. Sokolovski, S. Brouard, J. N. L. Connor, “Traversal-time wave-function analysis of resonance and nonresonance tunneling,” Phys. Rev. A 50, 1240–1256 (1994).
[Crossref]

R. Landauer, T. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. 66, 217 (1994).
[Crossref]

1993 (1)

A. M. Steinberg, P. G. Kwiat, R. Y. Chia, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[Crossref]

1989 (1)

E. H. Hauge, J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, 917–936 (1989).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

1987 (1)

D. Sokolovski, L. M. Baskin, “Traversal time in quantum scattering,” Phys. Rev. A 36, 4604–4611 (1987).
[Crossref]

1986 (1)

M. V. Ammosov, N. B. Delone, V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

1983 (1)

M. Büttiker, “Larmor precession and the traversal time for tunneling,” Phys. Rev. B 27, 6178–6188 (1983).
[Crossref]

1982 (1)

M. Buttiker, R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[Crossref]

1966 (1)

A. M. Perelomov, V. S. Popov, M. V. Terentev, “Ionization of atoms in an alternating electric field,” Zh. Eksp. Teor. Fiz. 50, 1393 (1966).

1965 (1)

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

1955 (1)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[Crossref]

Abusamha, M.

A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abusamha, L. B. Madsen, U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8, 76–80 (2012).
[Crossref]

Aharonov, Y.

Y. Aharonov, D. Z. Albert, L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

Akagi, H.

H. Akagi, T. Otobe, A. Staudte, A. Shiner, F. Turner, R. Dörner, D. M. Villeneuve, P. B. Corkum, “Laser tunnel ionization from multiple orbitals in HCl,” Science 325, 1364–1367 (2009).
[Crossref]

Alahmed, Z. A.

A. Wirth, M. T. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, E. Goulielmakis, “Synthesized light transients,” Science 334, 195–200 (2011).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

Azzeer, A. M.

A. Wirth, M. T. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, E. Goulielmakis, “Synthesized light transients,” Science 334, 195–200 (2011).
[Crossref]

E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, F. Krausz, “Real-time observation of valence electron motion,” Nature 466, 739–743 (2010).
[Crossref]

M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdorfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, V. S. Yakovlev, “Delay in photoemission,” Science 328, 1658–1662 (2010).
[Crossref]

Bandrauk, A. D.

C. H. R. Ooi, W. Ho, A. D. Bandrauk, “Photoionization spectra by intense linear, circular, and elliptic polarized lasers,” Phys. Rev. A 86, 023410 (2012).
[Crossref]

Bandulet, H. C.

M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavicic, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320, 1478–1482 (2008).
[Crossref]

Baskin, L. M.

D. Sokolovski, L. M. Baskin, “Traversal time in quantum scattering,” Phys. Rev. A 36, 4604–4611 (1987).
[Crossref]

Baumert, T.

M. Wollenhaupt, M. Krug, J. Kohler, T. Bayer, C. Sarpe-Tudoran, T. Baumert, “Three-dimensional tomographic reconstruction of ultrashort free electron wave packets,” Appl. Phys. B 95, 647–651 (2009).
[Crossref]

Bayer, T.

M. Wollenhaupt, M. Krug, J. Kohler, T. Bayer, C. Sarpe-Tudoran, T. Baumert, “Three-dimensional tomographic reconstruction of ultrashort free electron wave packets,” Appl. Phys. B 95, 647–651 (2009).
[Crossref]

Biegert, J.

P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, U. Keller, “Attosecond angular streaking,” Nat. Phys. 4, 565–570 (2008).
[Crossref]

Breidbach, J.

J. Breidbach, L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Phys. Rev. Lett. 94, 033901 (2005).
[Crossref]

Brouard, S.

D. Sokolovski, S. Brouard, J. N. L. Connor, “Traversal-time wave-function analysis of resonance and nonresonance tunneling,” Phys. Rev. A 50, 1240–1256 (1994).
[Crossref]

Bruner, B. D.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref]

Burgdorfer, J.

M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdorfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, V. S. Yakovlev, “Delay in photoemission,” Science 328, 1658–1662 (2010).
[Crossref]

Burgdörfer, J.

S. Nagele, R. Pazourek, J. Feist, K. Doblhoff-Dier, C. Lemell, K. Tökési, J. Burgdörfer, “Time-resolved photoemission by attosecond streaking: extraction of time information,” J. Phys. B 44, 081001 (2011).
[Crossref]

Buttiker, M.

M. Buttiker, R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[Crossref]

Büttiker, M.

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Büttiker, U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
[Crossref]

M. Büttiker, “Larmor precession and the traversal time for tunneling,” Phys. Rev. B 27, 6178–6188 (1983).
[Crossref]

Caillat, J.

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E. Goulielmakis, Z. H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, F. Krausz, “Real-time observation of valence electron motion,” Nature 466, 739–743 (2010).
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Phys. Rep. (1)

R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, H. Schmidt-Bocking, “Cold target recoil ion momentum spectroscopy: a ‘momentum microscope’ to view atomic collision dynamics,” Phys. Rep. 330, 95–192 (2000).
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A. M. Steinberg, P. G. Kwiat, R. Y. Chia, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
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Science (5)

A. Wirth, M. T. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, E. Goulielmakis, “Synthesized light transients,” Science 334, 195–200 (2011).
[Crossref]

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, M. Büttiker, U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Science 322, 1525–1529 (2008).
[Crossref]

M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavicic, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320, 1478–1482 (2008).
[Crossref]

H. Akagi, T. Otobe, A. Staudte, A. Shiner, F. Turner, R. Dörner, D. M. Villeneuve, P. B. Corkum, “Laser tunnel ionization from multiple orbitals in HCl,” Science 325, 1364–1367 (2009).
[Crossref]

M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdorfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, V. S. Yakovlev, “Delay in photoemission,” Science 328, 1658–1662 (2010).
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Sov. Phys. JETP (2)

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Zh. Eksp. Teor. Fiz. (1)

A. M. Perelomov, V. S. Popov, M. V. Terentev, “Ionization of atoms in an alternating electric field,” Zh. Eksp. Teor. Fiz. 50, 1393 (1966).

Other (1)

A. S. Landsman, U. Keller, “Attosecond science and the tunneling time problem,” Phys. Rep., doi: 10.1016/j.physrep.2014.09.002 (in press).
[Crossref]

Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Measuring electron momenta distributions using VMIS. (a) Fitting of peaks in electron momenta distribution with a Gaussian at lower intensity and clockwise polarization. (b) Higher-intensity and counterclockwise polarization. (c) Experimental setup. (d) Reconstruction of full 3D electron momenta distribution using projections in the momentum space. 
Fig. 2.
Fig. 2. Measured offset angle θmπ/2, from which tunneling time is extracted. Black line and dots correspond to the Coulomb correction obtained using the TIPIS model [14] with single trajectory and classical trajectory Monte Carlo (CTMC) simulations, respectively (more in Section 2 of Supplement 1). The error bars are the result of a geometrical sum of the errors from the fit of the angular distribution of the electrons and the error of the polarization characterization. For the measurements performed with the COLTRIMS apparatus, an additional source of uncertainty comes from the thermal spread of the gas jet (more in Section 1 of Supplement 1).
Fig. 3.
Fig. 3. Comparison of theory to experiment. (a) Five theoretical predictions of tunneling time compared to experiment. (b) Same as (a), but zoomed into the lower time delay. (c) Potential resulting from the combined Coulomb–laser field. (d) Tunneling time as a function of barrier width. The measured tunneling time is significantly smaller than the free propagation time for electrons with kinetic energies characteristic of electron motion in Helium: for example, around 40 as and 230 as for tunneling time and free propagation time, respectively, for barrier width of 13 au.
Fig. 4.
Fig. 4. Normalized distribution where each probability value corresponds to an integer number of attoseconds of tunneling time. Top, 3D probability distribution of tunneling times for intensities ranging from 1 to 7.5×1014W/cm2, corresponding to field strengths of 0.04–0.11 au. The highly oscillatory structure is due to interference and shows tunneling probability peaking sharply at discrete values as tunneling time increases. (a) Probability distribution at intensity=1.625×1015W/cm2, FWHM80 as, and skewness=0.9. (b) Probability distribution at intensity=6.5×1014W/cm2, FWHM50 as, and skewness=1.09.

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