Single-particle steering and nonlocality: The consecutive Stern-Gerlach experiments

Quantum nonlocality and quantum steering are fundamental correlations of quantum systems which cannot be created using classical resources only. Nonlocality describes the ability to influence the possible results of measurements carried out in distant systems, in quantum steering where Alice remotely steers Bob's state. Research in nonlocality and steering is of fundamental interest for the development of quantum information and in many applications requiring nonlocal resources like quantum key distribution. On the other hand, the Stern-Gerlach experiment holds an important place in the history, development, and teaching of quantum mechanics and quantum information. In particular, the thought experiment of consecutive Stern-Gerlach experiments is commonly used to exemplify the concept of noncommutativity between quantum operators. However, to the best of our knowledge, the consecutive Stern-Gerlach experiments have not been treated in a fully fledged quantum manner yet, and it is a widely accepted idea that atoms crossing consecutive Stern-Gerlach experiments follow classical paths. Here we demonstrate that two consecutive Stern-Gerlach experiments generate nonlocality and steering, and these nonlocal effects strongly modify our usual understanding of this experiment. Also, we discuss the implications of this result and its relation with entanglement. This suggests the use of quantum correlations, of particles possessing mass, to generate nonlocal tasks using this experiment.

Figure 1

Scheme of the consecutive Stern-Gerlach experiments. The inhomogeneity of the magnetic field in the first experiment is in direction $x$, which is associated with a Hamiltonian, $\widehat{H_2}$, and an evolution operator, $\widehat{U_2}$; the inhomogeneity of the magnetic field of the second is in direction $z$, with its respective Hamiltonian, $\widehat{H_3}$, and evolution operator, $\widehat{U_3}$.

Figure 2

CSGE featuring the Einstein's boxes; see Ref. [21]. The red box is the oven, in violet a SGE in the $x$ direction is depicted, in blue the SGE in $z$ direction is shown, and the red and green dots represent the fact that there are not classical trajectories; see Refs. [30, 31]. Alice could communicate with Bob by using the classical channel in magenta. Moreover, Alice is in full control of the SGE by using the classical channel in yellow, and she possess the ability to turn it on and to choose between a single or $N$ atoms. The red dots end in the $z$ axis in Tokyo with Alice and in Paris with Bob. The green dots in the $x$ axis arrive at suitable places where there are two friends of Alice.

Figure 3

Correlation function for the $Z-\theta$ pair taking the fixed dimensionless quantities as $X=0.1$, $P_x=0.129$, $P_z=0.049$, ${\tau }_2=6.8$, ${\tau }_3=2.6$, $k_2=0.3$, $k_3=0.3$, $x_0=4$, and $z_0=4$.

Figure 4

Plot of $B_{\text{CHSH}}$ for the primed quantities $Z^{\prime }=2.4$ and ${\theta }^{\prime }=\frac{\pi }{5}$. We found a minimum of $-2.62405$ for this function; in this way, there exists a violation for the inequiality (12) by an amount of 0.62405, i.e., around $75\%$ of the maximal amount of violation, $\approx 0.8284$, given by the Cirel'son's bound [57, 64].

Figure 5

Entanglement entropy ($E$) of the SGE for various values of $k$ trough the time $\tau$. In the CSGE, we have $k_{2,3}$ and ${\tau }_{2,3}$, which are equivalent to these nonlabeled constants. In this way, we can directly compare the temporal behavior of the quantum correlations and entanglement in the CSGE, relying on the parameter $k$. Plot is similar to the one given in Ref. [32].