Transient effects on electron spin observation

In an earlier publication we addressed the problem of splitting an electron beam in the Stern-Gerlach experiment. In contrast to arguments put forward in the early days of quantum theory, we concluded that there are no issues of principle preventing the observation of electron spin during free flight. In that paper, however, we considered only a sudden switch off of the separating magnetic field. In this work we consider the possible effects of finite switching times at the beginning and the end of the interaction period. We consider a model where the coupling between the electron and the field is time dependent. As a result of the time dependence, the field also acquires an electric component, but this seems to cause no significant change of our conclusions. On the other hand, the smooth change of the interaction enforces the same longitudinal velocity on the electron both at the beginning and end of the interaction period because of conservation laws; this effect was missing in our earlier calculations. As the electrons are supposed to travel as a beam, this feature helps by restoring the beam quality after the interaction.

Figure 1

(a) Cross section through an idealized set-up for a Stern-Gerlach measurement on an electron beam using a pulsed magnetic field. The shaped electron beam (shown as an ellipse with the arrow) passes between four wires which make a transient magnetic quadrupole field when a pulse of current flows (with directions indicated by $•$ and $\times$). (b) The same section showing the detail of the field pattern around the wave packet. In both (a) and (b) the width of the wave packet is shown exaggerated.

Figure 2

The time-dependent magnetic-field gradient $a\left(t\right)$ as given by Eqs. (7, 4) with $a_0=1$ and the pulse center at $t_c=0$. The rise times are $\tau =\mathrm{0.002\hspace{0.17em}52},\mathrm{0.006\hspace{0.17em}30},$ and 0.0126 with pulse width $t_p=0.0427$. The value of $t_p$ is chosen to match the interaction time $t_0$ in Ref. 4

Figure 3

The resolution $s^2$, Eq. (26), as a function of $t_c$. The solid line shows the numerical calculation of Eq. (27) for the pulse τ=0.006 30 in Fig. 2. The dashed line is the result (30) for the square pulse.

Figure 4

Wave packets for the electron, in momentum space, shown after the field pulse. Parameters for the wave packet are (see Ref. 4) ${\Omega }^2=9.48$, $\Delta y\sim 0.103$, and $\Delta z\sim 20.7$. The parameters for the pulse are $t_p=0.0427$, and (a) $\tau =\mathrm{0.002\hspace{0.17em}52}$, $t_c=0.0302$; (b) $\tau =\mathrm{0.006\hspace{0.17em}30}$, $t_c=0.0365$; and (c) $\tau =0.0126$, $t_c=0.0554$.

Figure 5

Cross section at $p_y=0$ of the different momentum distributions for the same parameters as in Fig. 4.

Figure 6

Real space (rather than momentum space) picture of the wave packet shown in Fig. 4a. The time, $t=0.061$, is after the field pulse, as is seen in Fig. 2 with $t_c=0.0302$ rather than $t_c=0$. At this time no spatial splitting of the spin components is yet seen.

Figure 7

Different forward velocities for the wave packet. Curves (a), (b), and (c) show the quantum-mechanical result with $\tau =\mathrm{0.002\hspace{0.17em}52}$, $\tau =\mathrm{0.006\hspace{0.17em}30},$ and $\tau =0.0126$, respectively. These and other parameters are as in Fig. 4. The dashed lines show the velocity from the corresponding classical simulations with $20\times {10}^6$ ensemble members.

Figure 8

Particle momentum in $x$ and $z$ directions for both spin components with time parametrized. Both components start at $⟨p_z⟩=0$, but then move in opposite $⟨p_z⟩$ directions as $v_x$ falls and then rises. The final values indicate the diverging angle of the emerging electron beam spin components. The parameters for (a)–(c) are as in Fig. 4. The absolute values of the splitting in the three cases are (a) $⟨p_z⟩=0.043$, (b) $⟨p_z⟩=0.043$, and (c) $⟨p_z⟩=0.045$. The dashed line is the result from a classical simulation with parameters as in curve (b) $\left(\tau =\mathrm{0.006\hspace{0.17em}30}\right)$ and $20\times {10}^6$ ensemble members.