Geometric manipulation of the quantum states of two-level atoms

Manipulation of the quantum states of two-level atoms has been investigated using laser-controlled geometric phase change, which has the potential to build robust quantum logic gates for quantum computing. For a qubit based on two electronic transition levels of an atom, two basic quantum operations that can make any universal single qubit gate have been designed employing resonant laser pulses. An operation equivalent to a phase gate has been demonstrated using ${\mathrm{Tm}}^{3+}$ doped in a yttrium aluminum garnet crystal.

Figure 1

(a) Schematic of an electronic transition of a two-level atom with the wave function $\mid \psi ⟩$ driven by a laser field ${\Omega }_0\cos ({\omega }_{L}t+\theta )$ (b) Bloch vector $\vec{r}$ representing the wave function $\mid \psi ⟩$ and its precession about the Rabi vector $\vec{\Omega }$ on the Bloch sphere.

Figure 2

Bloch vector rotation about axis $\widehat{3}$ driven by two resonant $\pi$ pulses, the corresponding Rabi vectors with relative phase $\delta ∕2$, are labeled as ${\vec{\Omega }}_1$ and ${\vec{\Omega }}_2$, respectively. (a) The path of Bloch vector ${\vec{r}}_{-3}$ representing wave function $\mid 0⟩$ enclosing a solid angle, $2(\pi -\delta ∕2)$. (b) The path of the Bloch vector ${\vec{r}}_3$ representing $\mid 1⟩$ enclosing solid angle, $-2(\pi -\delta ∕2).$

Figure 3

Bloch vector rotation about axis $\widehat{2}$ driven by three resonant pulses, the corresponding Rabi vectors are labeled as ${\vec{\Omega }}_1,$ ${\vec{\Omega }}_2,$ and ${\vec{\Omega }}_3,$ respectively. ${\vec{\Omega }}_1$ and ${\vec{\Omega }}_3$ represent two $\pi ∕2$ pulses with the same phase while ${\vec{\Omega }}_2$ is a $\pi$ pulse with a relative phase of $\pi +\delta ∕2$. The path of Bloch vector ${\vec{r}}_2$ representing wave function, $\mid i⟩=(1∕\sqrt{2})(\mid 0⟩+\mid i⟩)$ encloses solid angle, $\delta ∕2.$

Figure 4

Regular pulse sequence of a photon echo process. (b) Pulse sequence with two control pulses to perform rotation, $U_3\left(\delta \right)$, where the rotation angle $\delta$ is controlled by the relative phase $\delta ∕2$ between the two control pulses. The rotation results in the echo field phase change by $\delta$. (c) Schematic of the experimental setup.

Figure 5

Phase change of the photon echoes vs the control phase. Experimental results (dots) were calculated from 409.6-ns segments of echo signals (see the three examples shown in the insets) using FFT. The theoretical values, $\delta$, (solid line) are twice of the control phase $\delta ∕2$.