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 (τϕ 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 τϕ (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 T11/2 T21/2 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 T21/2 as a function of υ in the neighborhood of υ₁, we obtain the time τz = ∂ (ln T1/2) / ∂w. The two times τϕ and τz 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)