Penetration of arbitrary double potential barriers with probability unity: Implications for testing the existence of a minimum length

Quantum tunneling across double potential barriers is studied. With the assumption that the real space is a continuum, it is rigorously proved that large barriers of arbitrary shapes can be penetrated by low-energy particles with a probability of unity, i.e., realization of resonant tunneling (RT), by simply tuning the interbarrier spacing. The results are demonstrated by tunneling of electrons and protons, in which resonant and sequential tunneling are distinguished. The critical dependence of tunneling probabilities on the barrier positions not only demonstrates the crucial role of phase factors but also points to the possibility of ultrahigh accuracy measurements near resonance. By contrast, the existence of a nonzero minimum length puts upper bounds on the barrier size and particle mass, beyond which effective RT ceases. A scheme is suggested for dealing with the practical difficulties arising from the delocalization of particle position due to the uncertainty principle. This work opens a possible avenue for experimental tests of the existence of a minimum length based on atomic systems.

Figure 1

Schematics of resonant tunneling (RT) across double barriers. (a) Quantum interference of the incident and reflected matter waves. (b) Modulation of the energy levels of the quasibound states between the two barriers, by varying the interbarrier spacing $w$. (c) RT spectrum as a function of $w$ with a period of Δ ($=\frac{\pi }{k}$).

Figure 2

Schematics of (a) rectangular and (b) parabolic double barriers (DBs). The interbarrier spacing of resonant tunneling (RT), $w_n$, as a function of resonance number ($n_{\mathrm{RT}}$), for (c) electron and (d) proton across the DBs, at incident energy $E$ = 0.5 eV.

Figure 3

Tunneling spectrum of electrons across the rectangular double barrier (DB) shown in Fig. 2. Resonant tunneling (RT) at resonance level (a) $E_{\mathrm{RT}}=0.03\mathrm{eV}$ and (b) $E_{\mathrm{RT}}=0.5\mathrm{eV}$, as a function of interbarrier spacing $w$. Energy dependence of tunneling probability at fixed $w$, around (c) $E_{\mathrm{RT}}=0.03\mathrm{eV}$, $w=100008.49\AA$ and (d) $E_{\mathrm{RT}}=0.5\mathrm{eV}$, $w=100991.45\AA$. The data lines labeled by $P_1$, $P_{\mathrm{ST}}$, and $P_{\mathrm{RT}}$ correspond to tunneling through single barrier, sequential, and RT through the DB, respectively.

Figure 4

Tunneling spectrum of protons across the rectangular double barrier shown in Fig. 2. Resonance at (a) $E_{\mathrm{RT}}=0.5\mathrm{eV}$. Variations of tunneling probability with respect to small deviations from the resonant tunneling (RT) parameters: (b) Interbarrier spacing $w=(n_w-n_p)\times \mathrm{\Delta }l+w_n$, $n_p=596$, and $\mathrm{\Delta }l={10}^{-15}\AA$; (c) Incident energy at the vicinity of $E_{\mathrm{RT}}$, for $w=20.137016632763302\AA$, and the energy deviation $\varepsilon ={10}^{-10}$ eV. All digits of $w$ are meaningful.

Figure 5

Schematics of quantum tunneling of protons across single barrier (upper panel) and double barriers (lower panel) at the presence of resonant tunneling (RT). The probability of reflection is denoted by $P_{Ri}$ and tunneling by $P_{Ti}$, $i=1$, 2.

Figure 6

Tunneling properties of (a)–(c) electrons and (d) protons across homostructured (symmetric) and heterostructured (asymmetric) rectangular double barriers (DBs). (a) Variation of transmission coefficient $T$($E$) with the barrier width ratio $b$/$a$, at barrier height $V_1=V_2=1\mathrm{eV}$, and the interbarrier spacing $w=109.995\AA$. (b) Variation of $T$($E$) with the barrier height ratio $V_2/V_1$, at barrier width $a=b=1\AA$, and the interbarrier spacing $w=109.995\AA$. (c) Tunneling spectrum of electron across symmetric and asymmetric DBs with given interbarrier spacing $w=111.500\AA$, where the parameters $a=b=2\AA$, $V_1=V_2=1\mathrm{eV}$ for the symmetric DB, and the changes of δa = 0.2 Å, δb = 0.1 Å, $\delta V_1=0.1\mathrm{eV}$, and $\delta V_2=0.2$ eV for the asymmetric one. (d) Like (c) but for proton tunneling, where the interbarrier spacing $w=20.1596\AA$, $a=b=1\AA$, and $V_1=V_2=0.1\mathrm{eV}$ for the symmetric one, and the changes δa = 0.05 Å and $\delta V_1=0.005$ eV for the asymmetric DB. For (a)–(c), $T$(0.1 eV) = 1, and for (d), $T$(0.05 eV) = 1.

Figure 7

Calculated upper bounds ($V_{\max }$) of the barrier height of rectangular double barriers set by the Planck length for (a) electrons and (b) protons, as a function of barrier width. The values of $V_{\max }$ at barrier width of 6–10 $\AA$ are highlighted in the insets.

Figure 8

Schematic diagram for the kinetic energy distribution $\left[f\right(E\left)\right]$ of identical particles in microcanonical ensembles: Particle groups of high monochromaticity $[f\left(E\right)=g_1\left(E\right)]$, low monochromaticity $[f\left(E\right)=g_2\left(E\right)]$, and their superposition $[f\left(E\right)=g\left(E\right)=g_1\left(E\right)+g_2\left(E\right)]$.