Experimental Characterization of Quantum Dynamics Through Many-Body Interactions

We report on the implementation of a quantum process tomography technique known as direct characterization of quantum dynamics applied on coherent and incoherent single-qubit processes in a system of trapped ${}^{40}\mathrm{Ca}^{+}$ ions. Using quantum correlations with an ancilla qubit, direct characterization of quantum dynamics reduces substantially the number of experimental configurations required for a full quantum process tomography and all diagonal elements of the process matrix can be estimated with a single setting. With this technique, the system’s relaxation times $T_1$ and $T_2$ were measured with a single experimental configuration. We further show the first, complete characterization of single-qubit processes using a single generalized measurement realized through multibody correlations with three ancilla qubits.

Figure 1

Procedure to characterize a single-qubit process with DCQD and a GM. In DCQD (a) each experimental configuration consists of the preparation of one of four input states ${\rho }_j$ entangled between the system ion $S$ and the ancilla ion $A$. The process $E$ is applied on $S$ followed by a BSM on the output state $E({\rho }_j)$, which consists of a single MS operation followed by a projection onto the computational basis. (b) Generalized measurement via many body interactions (see text).

Figure 2

Experimental results of DCQD for unitary and decoherence processes. (a), (b) Results of the measured superoperator $\chi$ for the rotation operations $U(\pi ,0)$ in (a) and $U(\pi ,\pi /2)$ in (b). Ideally, the target processes have only nonzero elements at positions indicated by the gray (orange)-bordered bars. (c), (d) Bloch sphere representation of the ideal (c) and measured (d) amplitude damping process with 60% probability [22]. (e), (f) Bloch sphere representation of the ideal (e) and measured (f) phase damping process with 60% probability [22]. Bloch sphere axes in black evolve into the spheroid primed axes in blue. A slight imperfection due to residual light on the ancilla ion can be observed as a rotation of the spheroids in the measured decohering processes.

Figure 3

Simultaneous measurement of phase decoherence (a) and the spontaneous decay (b) of a two-qubit system. The DCRT technique (green dots) is compared to a Ramsey-contrast measurement (red diamonds) and a spontaneous-decay measurement (blue triangles) (see text). The measurement using the DCRT method in (a) was carried out on the entangled two-qubit system [$\exp (-4t/T_2^{\mathrm{DCRT}})$ scaling] whereas the red diamonds were measured on a single qubit with the Ramsey-contrast technique [$\exp (-t/T_2^{\mathrm{trad}})$ scaling]. The shaded areas correspond to the envelope of the curves with the decay times $T_{1,2}^{\mathrm{DCRT},\mathrm{trad}}\pm \Delta T_{1,2}^{\mathrm{DCRT},\mathrm{trad}}$, considering the statistical errors $\Delta T_{1,2}^{\mathrm{DCRT},\mathrm{trad}}$. The relaxation time measurements, using the DCRT method and, in comparison, the traditional Ramsey-contrast and spontaneous-decay measurement, yield $T_2^{\mathrm{DCRT}}=18.8(5)\mathrm{ms}$, $T_2^{\mathrm{trad}}=19.4(8)\mathrm{ms}$, $T_1^{\mathrm{DCRT}}=1130(47)\mathrm{ms}$, and $T_1^{\mathrm{trad}}=1160(30)\mathrm{ms}$.