Exotic Bohmian arrival times of spin-1/2 particles: An analytical treatment

It is well known that orthodox quantum mechanics does not make unambiguous predictions for the statistics in arrival time (or time-of-flight) experiments. Bohmian mechanics (or de Broglie–Bohm theory) offers a distinct conceptual advantage in this regard, owing to the well-defined concepts of point particles and trajectories embedded in this theory. We revisit a recently proposed experiment [S. Das and D. Dürr, Sci. Rep. 9, 2242 (2019)], the numerical analysis of which revealed a striking spin dependence in the (Bohmian) time-of-arrival distributions of a spin-1/2 particle. We present here a mathematically tractable variant of the same experiment, where the predicted effects can be established rigorously. We also obtain some results that can be compared with experiment.

Figure 1

Collection of Bohmian trajectories of a spin-0 particle passing through a double-slit interferometer, with initial positions sampled randomly from the initial ${\left|\psi \right|}^2$ distribution (dots). Most trajectories are reflected back (not shown). Inset: magnified view of the near field region. Figure courtesy of Kellers [19].

Figure 2

Schematic drawing of the experimental setup. The barrier at $d$ is switched off at $t=0$ and arrival times are monitored at $z=L$.

Figure 3

Arrival time histogram ${\mathrm{\Pi }}_{\mathtt{Bohm}}^{{\mathrm{\Psi }}_0}\left(\tau \right)$ versus (dimensionless) arrival time $\frac{\hslash \tau }{m{d}^2}$ for the spin-up-down wave function: the detector is placed at $L=100d$, and we have set $\beta =0$ with $\omega ={10}^3\frac{\hslash }{m{d}^2}$. The histogram was generated with $8\times {10}^5$ Bohmian trajectories whose initial points were randomly drawn from the Born $|{\mathrm{\Psi }}_0{|}^2$ distribution. All Bohmian trajectories in this case strike the detector before $t={\tau }_{\mathtt{max}}\approx 42.9\frac{m{d}^2}{\hslash }$. An infinite train of self-similar smaller lobes, separated by distinct “no-arrival windows,” is seen below $\tau =15.9\frac{m{d}^2}{\hslash }$ (dashed line).

Figure 4

Two real branches of $W\left(x\right)$: $W_{-1}\left(x\right)$ (dashed); $W_0\left(x\right)$ (solid). The thick red line emphasizes the interval $[-1/e,0)$, the range of the function $g(Y_0,Z_0)$.

Figure 5

Schematic plot of $\xi \left(t\right)$ against $t$, showing the extreme values ${\xi }_s$ and ${\xi }_b$. Note that $\xi \left(0\right)=Z_0$ and $\dot{\xi }\left(t\right)=Y\left(t\right)=0$ at the instants $t_1,t_2,...,t_5$. The solid (dashed) parts of the curve correspond to $Y\left(t\right)>0$ ($<0$).

Figure 6

Typical Bohmian trajectory of a spin-1/2 particle with wave function ${\mathrm{\Psi }}_{\stackrel{\updownarrow }{}}$ and initial position ${\mathit{R}}_0=0.3\widehat{\mathit{x}}+0.1\widehat{\mathit{y}}+0.5\widehat{\mathit{z}}$, the $x$ coordinate of which is a constant of motion. The trajectory is plotted for the time interval [0,20] with $\omega =50$.

Figure 7

Graphs of ${\mathrm{\Pi }}_{\stackrel{\uparrow }{}}$ vs $\tau$ for select values of $L$. The distribution stretches over larger arrival times with increasing $L$.

Figure 8

Schematic plot of $Z\left(t\right)$ vs $t$ for a spin-up-down Bohmian trajectory, enveloped between the dashed curves ${\xi }_s\sqrt{1+t^2}$ and ${\xi }_b\sqrt{1+t^2}$. The trajectory intersects $z=L$ at the instants $t_1$ ($=\tau$), $t_2$, and $t_3$, which lie in the interval $[t_s,t_b]$, in accordance with (55). $t^{\prime }$ denotes the first instant after $t_s$ at which the trajectory touches the upper envelope.

Figure 9

Up-down arrival time histograms for select values of $L$ and $\omega =500$. Each histogram is constructed from $\approx {10}^5$ Bohmian trajectories, whose initial conditions were sampled randomly from the initial ${\left|\mathrm{\Psi }\right|}^2$ distribution (43). It should be noted that for every $L$ there exists a maximal arrival time ${\tau }_{\mathtt{max}}$.

Figure 10

Graphs of mean first arrival time $\langle \tau \rangle$ and standard deviation $\mathrm{\Delta }$ vs $L$ for a fixed $\omega =500$. The spin-up statistics, unlike the up-down ones, are independent of $\omega$. The maximum arrival time ${\tau }_{\mathtt{max}}$ of the up-down distribution is also depicted here, which lies in the shaded region permitted by inequality (59).

Figure 11

Up-down arrival time histograms for select values of $\omega$ and $L=50$. The histograms approach ${\mathrm{\Pi }}_s\left(\tau \right)$, the distribution of $t_s$ (thick gray curve), as $\omega \to \infty$, while ${\tau }_{\mathtt{max}}\to \sqrt{L^2-1}\approx 50$. The angle subtended at the foot of the distribution, the podal angle, $\gamma \approx {tan}^{-1}(4.16/L^2)$.

Figure 12

Comparison of the theoretically calculated podal angle of the limiting distribution ($\omega \to \infty$) with numerical estimates for two (large) values of $\omega$.