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Europhysics Letters (EPL)

Instanton Bounces as Real Physical Processes

Daniela Mugnai, Anedio Ranfagni

Published May 1, 1988• 20 citations
Publisher

Abstract

According to the instanton method, the decay from a metastable state is described by a sequence of bounces, each of them represents an attempt to traverse the energy barrier. Analogously to Büttiker-Landauer tunnelling time interpretation, it is suggested that a real time is spent by the system also during the attempts of tunnelling (instanton bounces).

DOI
10.1209/0295-5075/6/1/001

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According to the instanton method, the decay from a metastable state is described by a sequence of bounces, each of them represents an attempt to traverse the energy barrier. Analogously to Büttiker-Landauer tunnelling time interpretation, it is suggested that a real time is spent by the system also during the attempts of tunnelling (instanton bounces).
In the case of decay rate problem, the potential barrier separates the potentialminimum from the continuum where the system is energetically allowed to go to infinity [see Figure 1.1(b)]. The general formulation developed in Chapter 6 for tunneling splitting can be applied to the present problem with some modifications [44]. The instanton trajectoryq0(τ ) starts in the infinitely remote past at τ = −∞ frompotentialminimum and returns back to the minimum in the infinitely remote future at τ = ∞. This periodic orbit always goes through a turning point on the potential contour V (q) = 0, which can be assumed to occur at τ = 0.
The authors analyse tunnelling through time-modulated barriers. The numerical results for the transmitted sideband intensities, obtained with wave-packet simulations, are in quantitative agreement with the analytical results of Buttiker and Landauer (1982, 1985). For opaque barriers the asymmetry of the sideband intensities can be characterized with a single frequency whose inverse Buttiker and Landauer interpret as the traversal time. This simple behaviour does not hold in general, and the authors discuss some of the difficulties one meets in an attempt to generalise the one-parameter description to arbitrary barriers. Furthermore, they examine analytically the transmitted wavefunction in the adiabatic limit, and show that the related density and current involve both the dwell time and the Buttiker time, defined in the context of a Larmor clock. This suggests the possibility that experiments probing different aspects of the transmitted wavefunction may yield different, but complementary, information about the tunnelling process.
The old question of "How long does it take to tunnel through a barrier?" has acquired new urgency with the advent of techniques for the fabrication of semiconductor structures in the nanometer range. For the restricted problem of tunneling in a scattering configuration, a coherent picture is now emerging. The dwell time ${\ensuremath{\tau}}_{D}$ has the status of an exact statement of the time spent in a region of space, averaged over all incoming particles. The phase times ${\ensuremath{\tau}}_{T}^{\ensuremath{\phi}}$ and ${\ensuremath{\tau}}_{R}^{\ensuremath{\phi}}$ are defined separately for transmitted and reflected particles. They are asymptotic statements on completed scattering events and include self-interference delays as well as the time spent in the barrier. Consequently, neither the dwell time nor the phase times can answer the question of how much time a transmitted (alternatively, reflected) particle spent in the barrier region. Our discussion of this question relies on a few simple criteria: (1) The average duration of a physical process must be real. (2) Since transmission and reflection are mutually exclusive events, the times ${\ensuremath{\tau}}_{T}$ and ${\ensuremath{\tau}}_{R}$ spent in the barrier region are, if they exist, conditional averages. Consequently, they must obey the identity ${\ensuremath{\tau}}_{D}=T{\ensuremath{\tau}}_{T}+R{\ensuremath{\tau}}_{R}$, where TTT and RRR are the transmission and reflection probabilities, respectively. The existence of this identity distinguishes tunneling in a scattering configuration from tunneling out of a metastable state. (3) Any proposed ${\ensuremath{\tau}}_{T}$ and ${\ensuremath{\tau}}_{R}$ must meet every requirement that can be constructed from ${\ensuremath{\tau}}_{D}$. On the basis of (2), the naively extrapolated phase times, as well as the B\"uttiker-Landauer time, must be rejected. The local Larmor times, as introduced by Baz', satisfy (2), but not every criterion of type (3). The local Larmor clock is therefore unreliable. The asymptotic Larmor clock shows the phase times, as it should. Finally, the inverse characteristic frequency of an oscillating barrier cannot always be defined. It is shown not to represent the duration of the tunneling process. This leaves the dwell time and the phase times as the only well-established times in this context. It also leaves open the question of the length of time a transmitted particle spends in the barrier region. It is not clear that a generally valid answer to this question exists.
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 τBLτ_{BL}τBL​ in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein τBLτ_{BL}τBL​ 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.