Instantaneous velocity during quantum tunnelling

TL;DR

This work provides a time-dependent picture of tunnelling by computing the instantaneous velocity inside a barrier using Bohmian mechanics. It shows that the velocity decays from large initial values to near zero in evanescent states while the barrier density rises to a stationary profile, and derives explicit barrier-width–dependent velocity formulas that vanish as the barrier grows. The results resolve apparent contradictions between nonzero entry motion and zero steady-state current, highlight the difference between current- and density-based velocity measures, and lay out experimental paths for time-resolved tunnelling observables. The combination of 1D TDSE simulations and a coupled-waveguide model offers a concrete dynamical framework for interpreting and testing time-resolved tunnelling phenomena.

Abstract

Quantum tunnelling, a hallmark phenomenon of quantum mechanics, allows particles to pass through the classically forbidden region. It underpins fundamental processes ranging from nuclear fusion and photosynthesis to the operation of superconducting qubits. Yet the underlying dynamics of particle motion during tunnelling remain subtle and are still the subject of active debate. Here, by analyzing the temporal evolution of the tunnelling process, we show that the particle velocity inside the barrier continuously relaxes from a large initial value toward a smaller one, and may even approach zero in the evanescent regime. Meanwhile, the probability density within the barrier gradually builds up before reaching its stationary profile, in contrast to existing inherently. In addition, starting from the steady-state equations, we derive an explicit relation between the particle velocity and the barrier width, and show that the velocity in evanescent states approaches zero when the barrier is sufficiently wide. These findings resolve the apparent paradox of a vanishing steady-state velocity coexisting with a finite particle density. We point out that defining an effective speed from the probability density, rather than from the probability current, can lead to spuriously nonzero "stationary speed," as appears to be the case in Ref. [Nature 643, 67 (2025)]. Our work establishes a clear dynamical picture for the formation of tunnelling flow and provides a theoretical foundation for testing time-resolved tunnelling phenomena.

Figures (5)

• Figure 1: Schematic illustration of the particle dynamics upon encountering a potential barrier. (a) When a particle's energy exceeds the barrier height, the particle easily traverses the barrier, analogous to a boat moving over a weir when the water level is higher than the barrier. (b) When a particle’s energy lies below the barrier height, quantum tunnelling permits it to enter the classically forbidden region, where its velocity may slow to nearly zero. This behavior is analogous to a boat whose water level lies below a weir: although classical intuition would forbid passage, the boat can nonetheless cross with a finite probability and momentarily come to rest within the barrier. (c) Once the wave is reflected, the nearly stationary "boat" inside the barrier is driven backward, propagating out of the barrier and regaining its pre-barrier speed. (d) The incident wave packet is modeled as a pulse with Gaussian rising and falling edges, featuring a full width at half maximum of $t_{\mathrm{FWHM}} = 1.14\,\mathrm{ns}$ and a flat-top duration of $t_{\mathrm{flat}} = 0.86\,\mathrm{ns}$. The barrier width is denoted by $a$. In the limit $a \to \infty$, the barrier becomes semi-infinite. (e) A schematic of the probability density distribution $\rho$ of a particle encountering the barrier. This work presents the time evolution of the particle velocity $v$ inside the barrier and elucidates its dependence on the barrier width $a$.
• Figure 2: Instantaneous Bohmian velocity of a one-dimensional wave packet encountering a potential barrier. The wave packet propagates along the $x$-axis, and the pseudocolor plot shows the spatial distribution of the probability density at $t = 1.09\,\mathrm{ns}$. Each plot is normalised to its maximum value. The left edge of each pseudocolor panel corresponds to the entrance of the barrier, while the right edge corresponds to $x = L_x$. The barrier height is fixed at $V_0 = 0.538\,\mathrm{meV}$. Panels (a), (b), and (c) correspond to energy detunings $\Delta = E - V_0 = 0.1\,\mathrm{meV}$, $0\,\mathrm{meV}$, and $-0.1\,\mathrm{meV}$, respectively. In each panel, three markers--red circles, green triangles, and blue squares--denote three spatial locations inside the barrier, and the time evolution of the Bohmian velocity at these positions is plotted using curves with the corresponding markers. In (a), the three positions have depths of $0.425\,\mu\mathrm{m}$, $128.79\,\mu\mathrm{m}$, and $256.30\,\mu\mathrm{m}$. In (b), the depths are $0.39\,\mu\mathrm{m}$, $39.42\,\mu\mathrm{m}$, and $78.45\,\mu\mathrm{m}$. In (c) and (d), the depths are $0.35\,\mu\mathrm{m}$, $2.82\,\mu\mathrm{m}$, and $8.45\,\mu\mathrm{m}$. The gray shaded region corresponds to the time period during which the pulse flat-top interacts with the barrier, from $0.86\,\mathrm{ns}$ to $1.71\,\mathrm{ns}$. Panel (d) uses the same detuning as (c), $\Delta = -0.1\,\mathrm{meV}$, but incorporates a finite particle lifetime of $\Gamma^{-1} = 270\,\mathrm{ps}$, corresponding to a dissipation rate of $\Gamma \approx 3.7\,\mathrm{GHz}$.
• Figure 3: Evolution of a one-dimensional wave packet in two coupled waveguides after encountering a potential barrier. (a) Potential profiles of waveguide 1 (WG1, solid blue) and waveguide 2 (WG2, dashed blue), together with the inter-waveguide coupling strength $C$ (orange diamonds). WG1 has $V_1 = 0$ for $x < 0$ and $V_1 = V_0 = 0.538\,\mathrm{meV}$ for $x \ge 0$. WG2 has $V_2 = 2V_0$ for $x < 0$ and $V_2 = V_0$ for $x \ge 0$. The coupling is switched on at $x = 1.25\,\mu\mathrm{m}$, $1.14\,\mu\mathrm{m}$ and $1.02\,\mu\mathrm{m}$ for $\Delta = 0.1\,\mathrm{meV}$, $0\,\mathrm{meV}$ and $-0.1\,\mathrm{meV}$ respectively, with a strength of $26.22\,\mu\mathrm{eV}$, as illustrated on the right. (b-e) The pseudocolor plot shows the spatial distribution of the probability density in the two waveguides at $t = 1.09\,\mathrm{ns}$. Each plot is normalised to its maximum value. The energy detunings are $\Delta = 0.1\,\mathrm{meV}$ for (b), $\Delta = 0$ for (c), and $\Delta = -0.1\,\mathrm{meV}$ for (d) and (e). The gray shaded region corresponds to the time period during which the pulse flat-top interacts with the barrier, from $0.86\,\mathrm{ns}$ to $1.71\,\mathrm{ns}$. For each panel, the time-dependent Bohmian velocities are plotted for two or three selected positions on each waveguide; red circles and blue squares mark the spatial locations, while filled and open versions of each marker denote the sampling points on waveguide 1 and waveguide 2, respectively. In (b), the positions are $x = 40.37\,\mu\mathrm{m}$ and $123.62\,\mu\mathrm{m}$. In (c), the positions are $x = 28.55\,\mu\mathrm{m}$ and $149.99\,\mu\mathrm{m}$. In (d) and (e), the positions are $x = 5.46\,\mu\mathrm{m}$ and $13.32\,\mu\mathrm{m}$. For WG2, an additional velocity trace is shown for the position $x = 2.05\,\mu\mathrm{m}$. A finite particle lifetime of $\Gamma^{-1} = 270\,\mathrm{ps}$ is applied in (d).
• Figure 4: Evolution of probability density distributions in the main and auxiliary waveguides for $\mathbf{ \Delta = -0.1\,{meV} }$. (a) Time evolution of the probability density $\rho_1$ in the main waveguide (WG1). (b) Time evolution of the probability density $\rho_2$ in the auxiliary waveguide (WG2). The red plane represents the cross-section at $x = 1.37\,\mu\mathrm{m}$, and the red curve shows the variation of the probability density at this position over time. Between $t = 0.86\,\mathrm{ns}$ and $t = 1.71\,\mathrm{ns}$, the flat-top portion of the incident pulse interacts with the barrier, during which the red curve becomes parallel to the time axis, indicating no change in probability density. (c) At $t = 1.09\,\mathrm{ns}$, the proportion of probability density in the auxiliary waveguide, defined as $\tilde{\rho}_2 = \rho_2/\left(\rho_1 + \rho_2\right)$, is plotted as a function of spatial distance. The data is fitted using $\tilde{\rho}_2 = {\left( {C_0\left(x - x_0\right)}/{v} \right)}^2$, yielding a "steady-state velocity" of particles in the auxiliary waveguide $v = 2292\,\mathrm{km/s}$ (with fitting parameter $x_0=-1.91\,\mu\mathrm{m}$ and the goodness of fit $R^2 = 0.9997$). This result contrasts with the zero velocity observed in the flat-top region of Fig. \ref{['fig:fig3']}(d).
• Figure 5: Time evolution of the effective potential. (a) For $\Delta = 0.1\,\mathrm{meV}$, the curves with green triangles and blue squares correspond to the time evolution of the effective potential $\tilde{V} = Q + V_0$ at positions $x = 106.71\,\mu\mathrm{m}$ and $x = 212.37\,\mu\mathrm{m}$, respectively. (b) For $\Delta = -0.1\,\mathrm{meV}$, the curves with green triangles and blue squares correspond to the time evolution of the effective potential $\tilde{V}$ at positions $x = 2.82\,\mu\mathrm{m}$ and $x = 8.45\,\mu\mathrm{m}$, respectively. In both panels, the energy $E$ and the barrier height $V_0$ are indicated by arrows. The blue shaded regions, with the gradient direction reversed, directly reflects the relative height of $E$ with respect to $V_0$. (c) Dependence of the steady-state velocity on the barrier width for ${\Delta = -0.05\,\mathrm{meV}}$. The five curves represent the function $v(x) = {v_{\mathrm{0}}}/{\left[1+\left({V_0}/{\left|\Delta\right|}\right)\sinh^2{\left(\kappa\left(x-a\right)\right)}\right]}$ evaluated at different spatial positions $x$. Colors indicate distinct locations along the $x$-axis, as shown by the color bar embedded within a potential barrier of width $a$. Numerical results are denoted by filled circles; see Methods for derivation details.