Abstract The measurement of tunneling times in strong-field ionization has been the topic of much controversy in recent years, with the attoclock and Larmor clock being two of the main contenders for correctly reproducing these times. While the non-zero Larmor tunneling time has been unambiguously detected, the tunneling time measured by the attoclock appears to vanish in tunnel ionization of a hydrogen atom. By expressing the attoclock as the weak value of temporal delay, we extend its meaning beyond the traditional setup and connect it to the weak-value interpretation of the Larmor clock. Our approach allows us to derive the position-resolved attoclock tunneling time. We show how this time, while non-zero at the tunnel exit, vanishes at the detector, far away from the atom. Formally, this means that the attoclock does not measure the “local” Larmor time, but instead a “non-local” time closely related to the phase time.Abstract The measurement of tunneling times in strong-field ionization has been the topic of much controversy in recent years, with the attoclock and Larmor clock being two of the main contenders for correctly reproducing these times. While the non-zero Larmor tunneling time has been unambiguously detected, the tunneling time measured by the attoclock appears to vanish in tunnel ionization of a hydrogen atom. By expressing the attoclock as the weak value of temporal delay, we extend its meaning beyond the traditional setup and connect it to the weak-value interpretation of the Larmor clock. Our approach allows us to derive the position-resolved attoclock tunneling time. We show how this time, while non-zero at the tunnel exit, vanishes at the detector, far away from the atom. Formally, this means that the attoclock does not measure the “local” Larmor time, but instead a “non-local” time closely related to the phase time.

Communications PhysicsApr 17, 20261 citation

Larmor time and proper time

Shoju Kudaka, Shuichi Matsumoto

Physics Letters AOct 1, 20122 citations

Larmor-clock transmission times for resonant double barriers

C. Richard Leavens, G. C. Aers

Recent generalizations of B\"uttiker's analysis of the Larmor clock to any region ${z}_{1}$ \ensuremath{\le}z\ensuremath{\le} ${z}_{2}$ within an arbitrary one-dimensional barrier lead to two local transmission ``times'' for an incident electron of energy E=${\ensuremath{\Elzxh}}^{2}$ ${k}^{2}$ /2m. These are conveniently regarded as the real and (minus) the imaginary parts of a complex time ${\ensuremath{\tau}}_{T}^{V}$(k; ${z}_{1}$ , ${z}_{2}$ ). In this paper the properties of ${\ensuremath{\tau}}_{T}^{V}$(k; ${z}_{1}$ , ${z}_{2}$ ) are investigated for the double-rectangular-barrier potential V(z)= ${V}_{1}$ \ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z) + ${V}_{2}$ \ensuremath{\Theta}(z-b)\ensuremath{\Theta}(d-z). Results are presented for the dependence of the real and imaginary parts of the Larmor-clock transmission time on incident energy and barrier and well widths. The local transmission time ${\ensuremath{\tau}}_{T}^{V}$(k; ${z}_{1}$ , ${z}_{2}$ ) is real for symmetric double barriers when the energy of the incident electron is exactly on resonance. Remarkably, the behavior of the real quantity ${\ensuremath{\tau}}_{T}^{V}$(k;0, z\ensuremath{\ll}a) for this special case provides further evidence for the importance of the imaginary part of the local Larmor-clock transmission time for the corresponding (isolated) single barrier V(z)= ${V}_{1}$ \ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z). At resonance, for symmetric double barriers, (real) local transmission speeds ${v}_{T}$ (z) can be very much in excess of the speed of light in the well region for z very close to a quasinode of the Schr\"odinger stationary-state wave function. It is proven that ${v}_{T}$ (z)\ensuremath{\le}c for a symmetric double barrier at resonance when the Dirac equation is used in place of the Schr\"odinger equation. On the other hand, it is shown that application of the Dirac equation to a single opaque rectangular barrier does not alter the well-known difficulty that a/Re[${\ensuremath{\tau}}_{T}^{V}$(k;0,a)] exceeds c for sufficiently large barrier width a.Recent generalizations of B\"uttiker's analysis of the Larmor clock to any region ${z}_{1}$\ensuremath{\le}z\ensuremath{\le}${z}_{2}$ within an arbitrary one-dimensional barrier lead to two local transmission ``times'' for an incident electron of energy E=${\ensuremath{\Elzxh}}^{2}$${k}^{2}$/2m. These are conveniently regarded as the real and (minus) the imaginary parts of a complex time ${\ensuremath{\tau}}_{T}^{V}$(k;${z}_{1}$,${z}_{2}$). In this paper the properties of ${\ensuremath{\tau}}_{T}^{V}$(k;${z}_{1}$,${z}_{2}$) are investigated for the double-rectangular-barrier potential V(z)=${V}_{1}$\ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z) +${V}_{2}$\ensuremath{\Theta}(z-b)\ensuremath{\Theta}(d-z). Results are presented for the dependence of the real and imaginary parts of the Larmor-clock transmission time on incident energy and barrier and well widths. The local transmission time ${\ensuremath{\tau}}_{T}^{V}$(k;${z}_{1}$,${z}_{2}$) is real for symmetric double barriers when the energy of the incident electron is exactly on resonance. Remarkably, the behavior of the real quantity ${\ensuremath{\tau}}_{T}^{V}$(k;0, z\ensuremath{\ll}a) for this special case provides further evidence for the importance of the imaginary part of the local Larmor-clock transmission time for the corresponding (isolated) single barrier V(z)=${V}_{1}$\ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z). At resonance, for symmetric double barriers, (real) local transmission speeds ${v}_{T}$(z) can be very much in excess of the speed of light in the well region for z very close to a quasinode of the Schr\"odinger stationary-state wave function. It is proven that ${v}_{T}$(z)\ensuremath{\le}c for a symmetric double barrier at resonance when the Dirac equation is used in place of the Schr\"odinger equation. On the other hand, it is shown that application of the Dirac equation to a single opaque rectangular barrier does not alter the well-known difficulty that a/Re[${\ensuremath{\tau}}_{T}^{V}$(k;0,a)] exceeds c for sufficiently large barrier width a.

Physical review. B, Condensed matterSep 15, 198975 citations

Recent generalizations of B\"uttiker's analysis of the Larmor clock to any region

{z}_{1}

\ensuremath{\le}z\ensuremath{\le}

{z}_{2}

within an arbitrary one-dimensional barrier lead to two local transmission ``times'' for an incident electron of energy E=${\ensuremath{\Elzxh}}^{2}$

{k}^{2}

/2m. These are conveniently regarded as the real and (minus) the imaginary parts of a complex time ${\ensuremath{\tau}}_{T}^{V}$(k;

{z}_{1}

{z}_{2}

). In this paper the properties of ${\ensuremath{\tau}}_{T}^{V}$(k;

{z}_{1}

{z}_{2}

) are investigated for the double-rectangular-barrier potential V(z)=

{V}_{1}

\ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z) +

{V}_{2}

\ensuremath{\Theta}(z-b)\ensuremath{\Theta}(d-z). Results are presented for the dependence of the real and imaginary parts of the Larmor-clock transmission time on incident energy and barrier and well widths. The local transmission time ${\ensuremath{\tau}}_{T}^{V}$(k;

{z}_{1}

{z}_{2}

) is real for symmetric double barriers when the energy of the incident electron is exactly on resonance. Remarkably, the behavior of the real quantity ${\ensuremath{\tau}}_{T}^{V}$(k;0, z\ensuremath{\ll}a) for this special case provides further evidence for the importance of the imaginary part of the local Larmor-clock transmission time for the corresponding (isolated) single barrier V(z)=

{V}_{1}

\ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z). At resonance, for symmetric double barriers, (real) local transmission speeds

{v}_{T}

(z) can be very much in excess of the speed of light in the well region for z very close to a quasinode of the Schr\"odinger stationary-state wave function. It is proven that

{v}_{T}

(z)\ensuremath{\le}c for a symmetric double barrier at resonance when the Dirac equation is used in place of the Schr\"odinger equation. On the other hand, it is shown that application of the Dirac equation to a single opaque rectangular barrier does not alter the well-known difficulty that a/Re[${\ensuremath{\tau}}_{T}^{V}$(k;0,a)] exceeds c for sufficiently large barrier width a.