Observation of Geometric Phases for Mixed States using NMR Interferometry

Examples of geometric phases abound in many areas of physics. They offer both fundamental insights into many physical phenomena and lead to interesting practical implementations. One of them, as indicated recently, might be an inherently fault-tolerant quantum computation. This, however, requires one to deal with geometric phases in the presence of noise and interactions between different physical subsystems. Despite the wealth of literature on the subject of geometric phases very little is known about this very important case. Here we report the first experimental study of geometric phases for mixed quantum states. We show how different they are from the well-understood, noiseless, pure-state case.

Figure 1

A quantum network describing the experiment. The top horizontal line represents an auxiliary spin half particle, or an auxiliary qubit, labeled as qubit $a$. The bottom line represents a qubit labeled as $b$, in state ${\rho }_b$ which undergoes a cyclic evolution induced by a unitary operation $U$. We choose our reference basis, for qubits $a$ and $b$, to be states $|\uparrow ⟩$ and $|\downarrow ⟩$. They describe the spin state aligned with or against a static magnetic field $B_0$ applied in the $z$ direction. In this basis $|\pm ⟩=(1/\sqrt{2})(|\uparrow ⟩\pm |\downarrow ⟩)$ thus the initial state of the auxiliary qubit is $|+{⟩}_a$. Projectors $|\pm ⟩⟨\pm |$ can also be written as $\frac{\mathrm{1}}{\mathrm{2}}(1\pm {\sigma }_x)$.

Figure 2

The cyclic path $ABCDA$ subtends the solid angle $\Omega$ on the Bloch sphere. The solid angle can be changed by varying $\theta$—the angle of inclination between the $x,y$ plane and the $ABC$ plane (or the $ADC$ plane). In the experiment the Bloch vectors of different lengths $r$ follow the path identical to the $ABCDA$ but right below it, at the distance $r$ from the center of the sphere. The two Bloch vectors corresponding to the two eigenvectors of the density operator are the unit vectors $\pm \overrightarrow{r}/r$. In the cyclic evolution one of them follows the path $ABCDA$ and subtends the solid angle $\Omega$ and the other follows the path symmetrically on the opposite side of the sphere and subtends the solid angle $-\Omega$.

Figure 3

Summary of experimentally determined geometric phase $\gamma$ as a function of purity of the mixed state for three different solid angles $\Omega$. The solid lines correspond to the theoretical result: $\gamma =-\arctan (r\tan \frac{\Omega }{2})$.