An experimental apparatus, suitable for simulating quantum tunneling, consists of a waveguide in which evanescent waves originate for frequencies below the cut-off one. In this situation, the undersized waveguide behaves like a quanto-mechanical potential barrier (unidimensional barrier for electromagnetic waves), and is thus suitable for simulating quanto-mechanical tunneling (Fig. 1). Moreover, the analogy goes beyond this qualitative description, since optical and quantum tunneling are described by closely-related wave equations. In fact, the Helmholtz equation for the propagation of a scalar field (electric or magnetic component of the wave) is formally identical to the time-independent Schrödinger equation for the motion of a particle in a classically forbidden region. The only difference lies in the dispersion relation, which reflects the different time dependence in the Schrödinger equation with respect to the d'Alembert equation. This difficulty can easily be surmounted once the dispersion relations are properly taken into account and suitable substitutions are considered. However, there is a basic difference between quantum and optical tunneling: since optical tunneling is not a single particle process, it can be probed in a non-invasive way, in contrast to tunneling particle for which any kind of measurements is invasive and therefore modifies the motion.

Early experiments date from 1989, and were performed, for microwave
frequencies (carrier frequency of about 10 GHz, modulation frequency of about
1000 Hz), by a direct measurement of
time delay in traveling the undersized waveguide (Fig. 2).
Measurements were performed for two different types of pulses: delay time for a
step-function signal, and delay time for a beat envelop of two signals at very
close frequencies. In both cases, the experimental results are best described
by the quanto-mechanical model known as “phase time”,
or Hartman, model.
This model, which analytically describes the group delay (τ_{ϕ}_{ }in
Fig. 3 ) , is characterized by the fact
that, in the tunneling region, the time
delay is extremely small and the group velocity is “superluminal”, that is,
greater than the speed of light *c*.
This kind of behavior, namely a group velocity that is greater than
light velocity, was predicted at the beginning of the last century: in the
presence of anomalous dispersion, group velocity
overcomes the speed of light. However, Sommerfeld showed that this
effect cannot be extended to the signal velocity which, in any case, should be limited to* c*.
It is not clear whether this superluminality
concerns only tunneling processes or, more generally, is present in each kind
of process in which evanescent waves are present.

**Tunneling and complex time.**

Delay time measurements performed on the beat envelope of two
slightly different frequencies can be usefully employed in order to obtain information on the real and imaginary parts of a complex
time, which, in accordance with a theoretical model, could be suitable for the
description of tunneling processes. The
quantity that is directly measured is the time τ_{ϕ}_{ }_{ }(Fig. 4). However,
because of the almost quadratic characteristic of the detectors, the
ratio of the
components of the amplitude of the beat (measured after and before the narrowing, respectively) is
proportional to the product of the square root of the transmission coefficient *T*_{1}^{1/2 }*T*_{2}^{1/2 } at the two frequencies υ₁ and υ₂ . Thus, for a given frequency υ₁, by varying υ₂ and by determining with a fitting procedure the slope of the curve *T*_{2}^{1/2
} ^{ }as a function of υ in the neighborhood of υ₁, we obtain the time τ_{z}_{ }= ∂ (ln *T*^{1/2})
/ ∂w. The two
times τ_{ϕ} and τ_{z} represent the real and imaginary parts of a complex time that describes this
kind of process.

Editing by D.
Mugnai (Department: Structures
of Matter and Spectroscopy)