Pulsed Quantum-State Reconstruction of Dark Systems

We propose a novel strategy to reconstruct the quantum state of dark systems, i.e., degrees of freedom that are not directly accessible for measurement or control. Our scheme relies on the quantum control of a two-level probe that exerts a state-dependent potential on the dark system. Using a sequence of control pulses applied to the probe makes it possible to tailor the information one can obtain and, for example, allows us to reconstruct the density operator of a dark spin as well as the Wigner characteristic function of a harmonic oscillator. Because of the symmetry of the applied pulse sequence, this scheme is robust against slow noise on the probe. The proof-of-principle experiments are readily feasible in solid-state spins and trapped ions.

Figure 1

Measurement of $⟨{\sigma }_y⟩$ of a dark $\mathrm{spin}\text{−}1/2$ system. (a) Dependence of $|\sin \varphi n_y|$ on the pulse-sequence parameters $\tau$ and $N$ for $a_z=0.015{\omega }_0$ and $a_x=0.08{\omega }_0$. (b) Vertical cut along the free evolution time ${\tau }_1=2\pi /({\omega }_1+{\omega }_0)$, indicated by red circles in (a), resulting in an optimal pulse-cycle number $N=\pi /4v_x=10$. (c) Horizontal cut along $N=10$, indicated by blue lines in (a).

Figure 2

State reconstruction of a dark harmonic oscillator. (a) Reachable points $\xi$ in reciprocal phase space for different numbers of pulse cycles, $N=1$ and $N=5$ in the first two panels and all points for $N=(1,\dots ,10)$ in the right panel. (b) First 20 contour lines (blue) sampled from the characteristic function of a squeezed vacuum state $S(\lambda )|0⟩$, with $\lambda =\log (1/2)$ and $g/\nu =3/40$. Red curves are obtained using $\chi (-\xi )=\chi (\xi {)}^{*}$. (c) Density matrices reconstructed from an interpolated characteristic function obtained from $N=(1,\dots ,20)$ for $g/\nu =3/40$. Left panel: Squeezed vacuum from (b). Right panel: Coherent state $|\eta ⟩=D(\eta )|0⟩$ with $\eta =1$. Gray inner bars represent the exact values for comparison.