Experimental Recovery of a Qubit from Partial Collapse

We describe and implement a method to restore the state of a single qubit, in principle perfectly, after it has partially collapsed. The method resembles the classical Hahn spin echo but works on a wider class of relaxation processes, in which the quantum state partially leaves the computational Hilbert space. It is not guaranteed to work every time, but successful outcomes are heralded. We demonstrate, using a single trapped ion, a better performance from this recovery method than can be obtained employing projection and postselection alone. The demonstration features a novel qubit implementation that permits both partial collapse and coherent manipulations with high fidelity.

Figure 1

(a) Relevant levels and transitions in ${}^{40}\mathrm{Ca}^{+}$. The Zeeman substructure of $3d^2{D}_{5/2}$ is shown without and then with the light shifts ${\Delta }_m$ introduced by an intense, circularly polarized 854 nm laser. Blue data (fit with thin lines) show spin precession (Rabi flopping) as a function of (b) time and (c) excitation frequency in the unshifted $D_{5/2}$ manifold. Red data (fit with thick lines) demonstrate the isolation of the two-state qubit due to the applied light shift (ac-Stark effect). The lines are numerical fits to Bloch equations. As expected, the isolated qubit spin-precession rate exceeds that of the unshifted $D_{5/2}$ manifold by a factor of $\sqrt{5}$ [17].

Figure 2

Experimental sequence for testing ${\mathcal{R}}_p^{\prime }$. The “shelve” laser is at 393 nm; the others are as described in the text. Postselection filters away attempts when 397 nm fluorescence is found in detection intervals $A$ or $B$. Scattering during $D$ recools the ion to near the Doppler limit.

Figure 3

Observed states on the Bloch sphere (a)–(d) after partial collapse, including an odd number of $\pi$ pulses (see Fig. 2), and (e)–(h) after recovery for four initial states ${\rho }_i$. Points are tomography results for 10 partial measurement strengths $p\in (0.01,\dots ,0.94)$; lines guide the eye. The evident spiraling is an azimuthal phase that can be compensated by spin echo; the increasing polar angle seen in (b)–(d) is the partial-collapse process. After recovery, most data cluster around the correct input state. $|0⟩$ at $p>0.7$ performs less well, owing to the finite branching ratio $ϵ$ in ${\mathcal{D}}_p$; in this case, ${\rho }_f$ moves down the Bloch sphere’s axis, reaching its center at $p\simeq 0.97$. (i)–(l) Measured recovery process infidelity; vertical and horizontal error bars indicate the shot noise and uncertainty in $p$, respectively. Solid lines are predictions of ${\mathcal{R}}_p^{\prime }$ with no free parameters. Dashed lines depict the infidelity of a simpler projection and postselection ${\mathcal{M}}_p$. Note: for states $|0⟩$ and $|1⟩$, $1-F_{\mathcal{M}}=0$ is off scale.