A clear consensus on how long it takes a particle to tunnel through a potential barrier has never been so urgently required, since the electron dynamics in strong-field ionization can be resolved on attosecond time-scale in experiment and the exact nature of the tunneling process is the key to trigger subsequent attosecond techniques. Here a general picture of tunneling time is suggested by introducing a quantum travel time, which is defined as the ratio of the travel distance to the expected value of the velocity operator under the barrier. Specially, if applied to rectangular barrier tunneling, it can retrieve the Büttiker-Landauer time $τ_{BL}$ in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein $τ_{BL}$ can not be well defined. In the case of strong-field tunneling process, with the help of the newly defined time, the tunneling delay time measured by attoclock experiment can be interpreted as a travel time spent by the electron to tunnel from a point under the barrier to the tunnel exit. In addition, a peculiar oscillation structure in the wavelength dependence of tunneling delay time in deep tunneling regime is observed, which is beyond the scope of adiabatic tunneling picture. This oscillation structure can be attributed to the interference between the ground state tunneling channel and the excited states tunneling channels.A clear consensus on how long it takes a particle to tunnel through a potential barrier has never been so urgently required, since the electron dynamics in strong-field ionization can be resolved on attosecond time-scale in experiment and the exact nature of the tunneling process is the key to trigger subsequent attosecond techniques. Here a general picture of tunneling time is suggested by introducing a quantum travel time, which is defined as the ratio of the travel distance to the expected value of the velocity operator under the barrier. Specially, if applied to rectangular barrier tunneling, it can retrieve the Büttiker-Landauer time $τ_{BL}$ in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein $τ_{BL}$ can not be well defined. In the case of strong-field tunneling process, with the help of the newly defined time, the tunneling delay time measured by attoclock experiment can be interpreted as a travel time spent by the electron to tunnel from a point under the barrier to the tunnel exit. In addition, a peculiar oscillation structure in the wavelength dependence of tunneling delay time in deep tunneling regime is observed, which is beyond the scope of adiabatic tunneling picture. This oscillation structure can be attributed to the interference between the ground state tunneling channel and the excited states tunneling channels.

arXiv:1903.06897Mar 16, 20190 citations

The Quantum Physics of Time Travel

David Deutsch, Michael Lockwood

Scientific AmericanMar 1, 199493 citations

Quantum Theory Looks at Time Travel

Daniel M. Greenberger, Karl Svozil

arXiv:quant-ph/0506027Springer eBooksDec 3, 20058 citations

A one-dimensional barrier and time-dependent tunnelling

K W H Stevens

There has recently been a growth in interest in electron tunnelling through barriers, and in particular in the question of whether it is possible to talk of a tunnelling time and, if so, what is it? A brief account of a study of the signal velocity of wavepackets travelling into a rectangular barrier was given recently, and the present paper is an extended version of this theory. In particular it is shown that the signal velocity of a free electron is the same as its group velocity, which in turn is the same as that which a classical particle particle of the same energy would have. It is also shown that when the wavepacket enters a classically forbidden region the signal velocity continues to have a meaning, and indeed a pulse-like wavefunction travels very much like a light pulse in a dispersive and attenuating medium. The expression for the signal velocity is symmetrical with respect to the barrier height, so that pulses well below the barrier travel faster, though with greater attenuation, than pulses near the top of the barrier. This result agrees with conclusions reached by some other workers. It is also shown that an electron responds to an accelerating field in the barrier as would be expected for a classical particle. It is argued that the not uncommon statement that the non nor eigenstates of a free particle describe a beam of particles is incorrect for electrons, for such a solution can at most describe two electrons. The theory indicates that by using orthogonal wavepackets it is possible to construct many-electron wavefunctions which do describe beams of electrons, and then free electrons travel with the group (and signal) velocity.There has recently been a growth in interest in electron tunnelling through barriers, and in particular in the question of whether it is possible to talk of a tunnelling time and, if so, what is it? A brief account of a study of the signal velocity of wavepackets travelling into a rectangular barrier was given recently, and the present paper is an extended version of this theory. In particular it is shown that the signal velocity of a free electron is the same as its group velocity, which in turn is the same as that which a classical particle particle of the same energy would have. It is also shown that when the wavepacket enters a classically forbidden region the signal velocity continues to have a meaning, and indeed a pulse-like wavefunction travels very much like a light pulse in a dispersive and attenuating medium. The expression for the signal velocity is symmetrical with respect to the barrier height, so that pulses well below the barrier travel faster, though with greater attenuation, than pulses near the top of the barrier. This result agrees with conclusions reached by some other workers. It is also shown that an electron responds to an accelerating field in the barrier as would be expected for a classical particle. It is argued that the not uncommon statement that the non nor eigenstates of a free particle describe a beam of particles is incorrect for electrons, for such a solution can at most describe two electrons. The theory indicates that by using orthogonal wavepackets it is possible to construct many-electron wavefunctions which do describe beams of electrons, and then free electrons travel with the group (and signal) velocity.

Journal of Physics C Solid State PhysicsJul 10, 198383 citations

The quantum time machine

Justin Mullins

The New ScientistNov 1, 20100 citations

Quantum-mechanical calculation of travel time across a potential barrier and the problem of classical limit

D. Sen, A. N. Basu, S. Sengupta

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BFeb 1, 19950 citations

τ_{BL}

in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein

τ_{BL}

can not be well defined. In the case of strong-field tunneling process, with the help of the newly defined time, the tunneling delay time measured by attoclock experiment can be interpreted as a travel time spent by the electron to tunnel from a point under the barrier to the tunnel exit. In addition, a peculiar oscillation structure in the wavelength dependence of tunneling delay time in deep tunneling regime is observed, which is beyond the scope of adiabatic tunneling picture. This oscillation structure can be attributed to the interference between the ground state tunneling channel and the excited states tunneling channels.