Tracking the Dynamics of an Ideal Quantum Measurement

The existence of ideal quantum measurements is one of the fundamental predictions of quantum mechanics. In theory, an ideal measurement projects a quantum state onto the eigenbasis of the measurement observable, while preserving coherences between eigenstates that have the same eigenvalue. The question arises whether there are processes in nature that correspond to such ideal quantum measurements and how such processes are dynamically implemented in nature. Here we address this question and present experimental results monitoring the dynamics of a naturally occurring measurement process: the coupling of a trapped ion qutrit to the photon environment. By taking tomographic snapshots during the detection process, we show that the process develops in agreement with the model of an ideal quantum measurement with an average fidelity of 94%.

Figure 1

Experimental realization of the process tomography of a Lüders measurement. (a) Level scheme of ${{}^{88}\mathrm{Sr}}^{+}$. A magnetic field of 0.36 mT applied along the trap axis splits the $m_J=-5/2$ and $m_J=-3/2$ Zeeman sublevels of the metastable states $4D_{5/2}$ by $2\pi \times 6.0\mathrm{MHz}$. Together with the $5S_{1/2}{m}_J=-1/2$ ground state, these states form the qutrit basis states $|0⟩$, $|1⟩$, and $|2⟩$. The qutrit transitions are driven by a narrow-linewidth laser at 674 nm (linewidth $<2\pi \times 600\mathrm{Hz}$) tuned to either the $|0⟩\leftrightarrow |1⟩$ or $|0⟩\leftrightarrow |2⟩$ transition. A 422 nm laser driving the $|0⟩\leftrightarrow |e⟩\equiv 5P_{1/2}$ transition is used to induce the Lüders measurement process and the fluorescence detection during state tomography. To prevent loss of population from the qutrit subspace during the measurement process, 1092 nm laser light and circularly polarized 422 nm laser light are used to repump population from $4D_{3/2}$ and $5S_{1/2}{m}_J=+1/2$ to $|0⟩$. (b) Experimental sequence for the process tomography. First, the ion is prepared in one of nine initial states (see Supplemental Material [12]) followed by a measurement that implements snapshots of an ideal measurement process, and finally, state tomography is performed using a qutrit rotation and fluorescence detection.

Figure 2

Choi matrices ${\chi }_{\exp }$ reconstructed from the experimental data. The colors indicate the phase, $g_0$ is calculated according to our model, see Eq. (7). (a)–(d) The coupling strength of $|0⟩$ to the photon environment is increased and coherences involving state $|0⟩$ are destroyed. (d) Similar to the ideal Lüders process, and from (d) $\to$ (a) the process becomes more similar to the trivial process $\rho \mapsto \rho$.