Robust weak-measurement protocol for Bohmian velocities

We present a protocol for measuring Bohmian, or the mathematically equivalent hydrodynamic, velocities based on an ensemble of two position measurements, defined from a positive operator-valued measure, separated by a finite time interval. The protocol is very accurate and robust as long as the first measurement uncertainty divided by the finite time interval between measurements is much larger than the Bohmian velocity, and the system evolves under flat potential between measurements. The difference between the Bohmian velocity of the unperturbed state and the measured one is predicted to be much smaller than $1\%$ in a large range of parameters. Counterintuitively, the measured velocity is that at the final time and not a time-averaged value between measurements.

Figure 1

(a) Velocity distribution ($v$, black solid line; $v_e$, green dashed line) and (b) quantum current density ($J$, black solid line; $J_e$, green dashed line) for an electron in a double-slit experiment at $t_s=11$ ps, ${\sigma }_s=0.2$ nm, and ${\sigma }_w=150$ nm. Insets (a') and (b') are the total error ${\varepsilon }_s\left(x_s\right)+{\varepsilon }_w\left(x_s\right)$ in the highlighted position interval for the velocity and current, respectively. Inset (c) is ${\left|\Psi \right|}^2$ at $t_s=11$ ps.

Figure 2

Relative error ${\varepsilon }_w$ integrated over all positions $x_s$ as a function of ${\sigma }_w$ and $\tau$ for ${\sigma }_s=0.2$ nm for the numerical test represented in Fig. 1. The black line bounds the region for $\varepsilon \left({\sigma }_w\right)\le 1\%$, and the red (gray) line is the analytical error for the value $\tau =1$ ps.

Figure 3

(left) Region of relative error ${\varepsilon }_w<1\%$ and (right) region of relative error ${\varepsilon }_s<1\%$. Solid lines are the boundaries for the velocity, and dashed lines are the boundaries for the quantum current. The dotted line bounds the ${\sigma }_s^2/\tau$ region.