Time operators in stroboscopic wave-packet basis and the time scales in tunneling

We demonstrate that the time operator that measures the time of arrival of a quantum particle into a chosen state can be defined as a self-adjoint quantum-mechanical operator using periodic boundary conditions and applied to wave functions in energy representation. The time becomes quantized into discrete eigenvalues; and the eigenstates of the time operator, i.e., the stroboscopic wave packets introduced recently [Phys. Rev. Lett. 101, 046402 (2008)], form an orthogonal system of states. The formalism provides simple physical interpretation of the time-measurement process and direct construction of normalized, positive definite probability distribution for the quantized values of the arrival time. The average value of the time is equal to the phase time but in general depends on the choice of zero time eigenstate, whereas the uncertainty of the average is related to the traversal time and is independent of this choice. The general formalism is applied to a particle tunneling through a resonant tunneling barrier in one dimension.

Figure 1

Form of the potential energy used for demonstration of the tunneling time scales in 1D. The potential has two delta-shaped barriers of strength $\lambda$ at both ends and a constant value $u$ in between. Varying the latter gives access to various transport regimes, from resonant tunneling to opaque tunneling. The energy band of the stroboscopic wave-packet representation is indicated by arrows.

Figure 2

Amplitude (solid red line) and the phase (dashed green line) of the transmission amplitude $t(\varepsilon )$ for a scattering state in the potential given in Fig. 1 for $u=0.3$ (upper panel) and $u=0.65$ (lower panel). Since the energy band consists of energies $\varepsilon \in (0.2,0.6)$, the smaller $u$ corresponds to resonant transport, whereas the larger $u$ gives predominantly an opaque tunneling regime.

Figure 3

Probability densities of projected states $\mathrm{\varphi ̃}(x)$ for $x_R=100$ into which the particle arrives, for two different potential barriers $u$. In contrast to the unprojected state $\varphi (x)$, the arrival states are distorted and contain significant weight on both sides off the potential barrier (the amplitude for the state for $u=0.65$ on the right is ${10}^5$ times magnified so that it is visible) localized within $x\in (0,10)$.

Figure 4

Probabilities of arrival into state $\mathrm{\varphi ̃}(x)$ at time ${\tau }_m$ for different transport regimes. In resonant transport ($u=0.3$), due to good localization of the state, we see several maxima giving multiple bounces between the $\delta$-function barriers; increasing $u$ into the tunneling regime, we first see very broad distribution ($u=0.53$) with its width characterized by the Büttiker-Landauer time, evolving into a relatively narrow saturated distribution arriving earlier than the free particle (the Hartman effect).