Key for a Hidden Quantum State

Quantum trajectories are crucial to understanding the evolution of open systems. We consider an open cavity mode undergoing up and down multistate quantum jumps due to the emission and absorption of photons. We prove that among all subtrajectories, starting simultaneously from different photon number states, only one survives a long single-run evolution. A random Fock state terminating the subtrajectory becomes known for the ergodic case via the key—the processed record of the input and output photocounts, and the trajectory duration. Based on this result, we propose a robust protocol to infer the Fock state, a valuable resource for quantum applications.

Figure 1

(A) Quantum compound trajectory of the cavity field as a sum of the contributing subtrajectories, specifically, $m$ trajectories [lines (a),(b), and (c)] initialized at different Fock states $m$ synchronously. The weights of distinct trajectories indicated by the circles’ sizes change in time, and after the threshold time $t_N^{(m_0)}$ they are accumulated on one dominant $m_0$ trajectory that visits the vacuum state for the ergodic case. (B) The higher-lying ($m_0+1$) trajectory, generated by the same jump sequence as in (a), preferentially ends earlier, within the time window $(t_N^{(m_0+1)},t_N^{(m_0)})$. (C) Principle of the K4HQS protocol. (C1) The proof-of-principle experiment. One of the twin photons generated at the SPDC and detected by detector $D_{+1}$ records the photons exciting the cavity externally (a) or internally (b). Detector $D_{-1}$ records the photons leaving the cavity. So, the experiment produces the raw measurement record $K_N=\{{\xi }_k,t_k\}$ of $N$ random clicks by both detectors $D_{\xi }$. Then the cavity is locked (C2), the recorded data are processed (C3), and, finally, the cavity is opened (C4) for possible exploitation of a particular Fock state $|n_K⟩$ inferred from the record $K_N$.

Figure 2

Two related standard deviations: ${\sigma }_m^{(t)}$, of the $m$ trajectory distribution at the local maxima points ${\overline{t}}_N^{(m)}$, and ${\sigma }_t^{(m)}$, of the distribution ${\overline{\Re }}_m^{(N)}(t_N)$ at fixed $m>m_0$ and varied $t_N$, depicted as uncertainty ellipses, with axes ${\sigma }_m^{(t)}$ and $2\mathrm{\Gamma }{\sigma }_t^{(m)}$, that are centered at points $(m,{\overline{t}}_N^{(m)})$ marked as crosses. Depending on duration $t_N$ the ellipses can be time oriented or energy oriented. Inset(A): functions ${\Re }_m^{(N)}(t_N)$ for the generated forty-click $f$ key: $f_{40}=1100122112210000111234433443222111111100$, with $k_{-1}=1$, $q=0.75$, and for ${\rho }_0^{(m)}={\rho }_T^{(m)}$ [5]. The key was obtained from Eq. (2) by the Monte Carlo method. This $f$ key encodes the photon number state $n_K=1$ generated at $2\mathrm{\Gamma }{t}_N>15.7$. Inset(B): standard deviations ${\sigma }_m^{(t)}$ (1), $2\mathrm{\Gamma }{\sigma }_t^{(m)}$ (2), and their product (3) at points $t={\overline{t}}_N^{(m)}$.