Single-laser-pulse implementation of arbitrary $ZYZ$ rotations of an atomic qubit

Arbitrary rotation of a qubit can be performed with a three-pulse sequence, for example, $ZYZ$ rotations. However, this requires precise control of the relative phase and timing between the pulses, making it technically challenging in optical implementation in a short time scale. Here we show any $ZYZ$ rotations can be implemented with a single laser pulse, that is, a chirped pulse with a temporal hole. The hole of this shaped pulse induces a nonadiabatic interaction in the middle of the adiabatic evolution of the chirped pulse, converting the central part of an otherwise simple $Z$ rotation to a $Y$ rotation, constructing $ZYZ$ rotations. The result of our experiment performed with shaped femtosecond laser pulses and cold rubidium atoms shows strong agreement with the theory.

Figure 1

Time dependence of Rabi frequency $\mathrm{\Omega }$ and mixing angle $\vartheta$ plotted for (a) a chirped pulse, (b) a chirped pulse with a temporal hole of width ${\tau }_h=0.1\tau$ and depth $k=0.65$, and (c) a chirped pulse with a temporal hole of width ${\tau }_h=0.1\tau$ and depth $k=9$. (d)–(f) Bloch vector evolution of the adiabatic states in the “detuning” interaction picture, corresponding to (a), (b), and (c), where the $x$ axis is the azimuthal angle of the Bloch sphere and the $y$ axis is the polar angle. In (d)–(f), the north pole (when polar angle is zero) is ${|0\rangle }_{\mathrm{\Delta }}$ and the south pole (when polar angle is $\pi$) is ${|1\rangle }_{\mathrm{\Delta }}$. (g)–(i) Bloch vector evolution in the “atomic” interaction picture corresponding to (a), (b), and (c). In (g)–(i), the north pole (when polar angle is zero) is ${|g\rangle }_{{\omega }_0}$ and the south pole (when polar angle is $\pi$) is ${|e\rangle }_{{\omega }_0}$. The thick red lines in (b), (c), (e), (f), (h), and (i) indicate the $-{\tau }_h<t<{\tau }_h$ region (see text). The horizontal dashed lines in (d)–(i) indicate the polar angles of the initial and final states, which show that the amounts of change in the polar angle are the same between the detuning and atomic interaction pictures.

Figure 2

(a) Final states on the Bloch sphere in the “atomic” interaction picture, spanned by the resonant chirped pulse with a temporal hole (${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2$ case), for an initial state ${\left(\right|g\rangle }_{{\omega }_0}+{|e\rangle }_{{\omega }_0})/\sqrt{2}$. The carrier phase $\phi$ and the equivalent TL pulse area $\mathcal{A}$, defined in Eq. (16), were varied, while $\tau =5.9$ ps, $\alpha =1.25$ rad/${\mathrm{ps}}^2, {\tau }_h=0.5\tau$, and $k=0.95$ were kept constant. (b) Detuning $\delta f=2\alpha \delta t/2\pi$, which is associated with the time shift $\delta t$ in Eq. (15), was used as an additional control parameter to show that (the second) $Z$- and $Y$rotation angles, ${\mathrm{\Phi }}_2$ and $\mathrm{\Theta }$, are fully spanned, where $\tau =2.95$ ps, $\alpha =2.5$ rad/${\mathrm{ps}}^2, {\tau }_h=0.5\tau$, and $k=0.7$.

Figure 3

Schematic of the experimental setup: Laser pulses were from the femtosecond laser and programed to operate arbitrary single-qubit rotations of cold-atom qubits in a magneto-optical trap. The inset shows the energy level structure of the atomic qubit along with ionization states. The arrows labeled with numbers illustrate the experimental pulse sequence.

Figure 4

(a) Measured excitation probability $P_e(\mathrm{\Theta },\phi ,{\mathrm{\Phi }}_1)$ of atoms initially in $|{\psi }_{\mathrm{init}}{\rangle =\left(\right|g\rangle }_{{\omega }_0}+{|e\rangle }_{{\omega }_0})/\sqrt{2}$, probed as a function of the equivalent TL pulse area $\mathcal{A}$ and the carrier phase $\phi$, where the chirped pulse with a temporal hole is defined in Eq. (2) with $k=0.7$ and ${\tau }_h=0.4\tau$. (b) The corresponding TDSE calculation. (c)–(h) Bloch vector dynamics (in the “detuning” interaction picture) at selected points: (c) ($\phi ,\mathcal{A})=(0, \pi$), (d) ($\pi /2,\pi$), (e) ($\pi ,\pi$), (f) ($3\pi /2,\pi$), (g) ($3\pi /2,\pi /2$), and (h) ($3\pi /2,3\pi /2$). (i) Comparison between the experimental results and calculation along the three dashed lines in (a) and (b).