Experimental Demonstration of Uncertainty Relations for the Triple Components of Angular Momentum

The uncertainty principle is considered to be one of the most striking features in quantum mechanics. In the textbook literature, uncertainty relations usually refer to the preparation uncertainty which imposes a limitation on the spread of measurement outcomes for a pair of noncommuting observables. In this work, we study the preparation uncertainty for the angular momentum, especially for spin-$1/2$. We derive uncertainty relations encompassing the triple components of angular momentum and show that, compared with the relations involving only two components, a triple constant $2/\sqrt{3}$ often arises. Intriguingly, this constant is the same for the position and momentum case. Experimental verification is carried out on a single spin in diamond, and the results confirm the triple constant in a wide range of experimental parameters.

Figure 1

Geometric analog of uncertainty relations (2)–(5). The equilateral triangle has vertices $A$, $B$, $C$ and a point $P$ inside. The lengths of the line segments $PA$, $PB$, and $PC$ are denoted by $|PA|$, $|PB|$, and $|PC|$, respectively. The areas of the triangles $PAB$, $PBC$, and $PCA$ are denoted by $|△PAB|$, $|△PBC|$, and $|△PCA|$, respectively. These geometric quantities have the same relations as the inequalities (2)–(5) under the correspondence $|PA|\leftrightarrow \mathrm{\Delta }{S}_x$, $|PB|\leftrightarrow \mathrm{\Delta }{S}_y$, $|PC|\leftrightarrow \mathrm{\Delta }{S}_z$ and $|△PAB|\leftrightarrow |⟨S_z⟩|/4$, $|△PBC|\leftrightarrow |⟨S_x⟩|/4$, $|△PCA|\leftrightarrow |⟨S_y⟩|/4$.

Figure 2

Experimental system and method. (a) Negatively charged NV center in diamond and the electronic energy level structure. The NV center consists of a substitutional nitrogen atom and a neighboring vacancy. The electronic ground state ${}^3{A}_2$ is a triplet state, where two levels are encoded as a qubit and exploited in the experiment. (b) Quantum circuit for qubit control and measurement. (c) Pulse sequence for implementing the quantum circuit. In the experiment, such a process is repeated four million times.

Figure 3

Experimental results. (a) Bloch sphere and the vector ${\mathbf{r}}_1$ with $\theta =\arctan \sqrt{2}$. (b) Expectations of $S_x$, $S_y$, and $S_z$. The red, olive, and blue curves, in turn, represent the theoretical values of $⟨S_x⟩$, $⟨S_y⟩$, and $⟨S_z⟩$. The corresponding scattered points represent the experimental values. (c) Experimental demonstration for the uncertainty relation of the multiplicative form. The solid orange, solid green, and dashed green curves, in turn, represent the theoretical values of Pro0, Pro1, and Pro2, where $\mathrm{Pro}0=\mathrm{\Delta }{S}_x\mathrm{\Delta }{S}_y\mathrm{\Delta }{S}_z$, $\mathrm{Pro}1={|{\tau }^3⟨S_x⟩⟨S_y⟩⟨S_z⟩/8|}^{1/2}$, and $\mathrm{Pro}2={|⟨S_x⟩⟨S_y⟩⟨S_z⟩/8|}^{1/2}$. The orange and green scattered points represent the experimental value of Pro0 and Pro1, respectively. (d) Experimental demonstration for the uncertainty relations of the additive form. The solid magenta, solid violet, and dashed violet curves, in turn, represent the theoretical values of Sum0, Sum1, and Sum2, where $\mathrm{Sum}0={(\mathrm{\Delta }{S}_x)}^2+{(\mathrm{\Delta }{S}_y)}^2+{(\mathrm{\Delta }{S}_z)}^2$, $\mathrm{Sum}1=\tau (|⟨S_x⟩|+|⟨S_y⟩|+|⟨S_z⟩|)/2$, and $\mathrm{Sum}2=(|⟨S_x⟩|+|⟨S_y⟩|+|⟨S_z⟩|)/2$. The magenta and violet scattered points represent the experimental value of Sum0 and Sum1, respectively. Error bars represent $\pm 1$ s.d.

Figure 4

Experimental results. (a) Bloch sphere and the vector ${\mathbf{r}}_2$ with $\phi =\pi /4$. (b) Expectations of $S_x$, $S_y$, and $S_z$. (c) Experimental demonstration for the uncertainty relation of the multiplicative form. (d) Experimental demonstration for the uncertainty relations of the additive form. All the symbols have the same meaning as those in Fig. 3.