Random search for a dark resonance

A pair of resonant laser fields can drive a three-level system into a dark state where it ceases to absorb and emit radiation due to destructive interference. We propose a scheme to search for this resonance by randomly changing the frequency of one of the fields each time a fluorescence photon is detected. The longer the system is probed, the more likely the frequency is close to resonance and the system populates the dark state. Due to the correspondingly long waiting times between detection events, the evolution is nonergodic and the precision of the frequency estimate does not follow from the conventional Cramér-Rao bound of parameter estimation. Instead, a Lévy statistical analysis yields the scaling of the estimation error with time for precision probing of this kind.

Figure 1

(a) $\mathrm{\Lambda }$-type system driven by laser fields with Rabi frequency $\mathrm{\Omega }$. The $|1\rangle \leftrightarrow |2\rangle$ coupling laser is kept on resonance while the $|0\rangle \leftrightarrow |1\rangle$ coupling laser is detuned by an amount $\delta =\omega -{\omega }_{20}$, where ${\omega }_{20}$ is the atomic resonance frequency. Both emission channels are monitored by photodetectors, and upon detection in either channel $\delta$ is shifted randomly on a uniform interval with $\delta \in [-{\delta }_{\text{max}},{\delta }_{\text{max}}]$. (b) Quantum Monte Carlo simulated trajectory for the detuning $\delta$ as a function of time $t$. The simulation is made with $\mathrm{\Omega }=0.1\mathrm{\Gamma }/\sqrt{2}$ and ${\delta }_{\text{max}}=0.1\mathrm{\Gamma }$, where ${\mathrm{\Gamma }}^{-1}$ is the excited-state lifetime.

Figure 2

Effective frequency-dependent photoemission rate from the dark state $|{\psi }_{-}\rangle$ shown for $\mathrm{\Omega }/\mathrm{\Gamma }=0.1\mathrm{\Gamma }/\sqrt{2}$. The full line shows the exact rate and the dashed the line shows our simplified model Eq. (1). Characteristic detunings (see the main text) are annotated. The rate is an even function of $\delta$. The light shaded area is the trapping region and the dark shaded area marks the frequency range not included in the stochastic scan.

Figure 3

Top: Proportion of trapped trajectories Eq. (3) (with ${\delta }_{\text{PDS}}=0.01\mathrm{\Gamma }$) as a function of time. The dashed line depicts a quantum jump simulation of $20000$ trajectories with the same parameters as in Fig. 1. It matches the statistical model (full line) for (very) large times. Bottom: Distribution of the detuning $\delta$ after a long time $T=6\times {10}^6{\mathrm{\Gamma }}^{-1}$. The dots show simulated data, the full line the theoretical result of our statistical analysis, and the shaded area marks the fraction with $\left|\delta \right|\le {\delta }_T$. The inset shows how the characteristic width ${\delta }_T$ of the distribution scales as $T^{-1/2}$ and matches the model for times larger than $\sim {10}^5{\mathrm{\Gamma }}^{-1}$.

Figure 4

(a) Dependence on the detuning $\delta$ of the photocount variance per time $V\left(\delta \right)$ divided by the rate $\tilde{R}\left(\delta \right)$. The dashed line marks the Poissonian case in which $V\left(\delta \right)=\tilde{R}\left(\delta \right)$. (b) Information measures for estimating ${\omega }_{20}$ in the $\mathrm{\Lambda }$ system by a systematic scan and the random search protocol, respectively. In both cases, the search is restricted to an interval $[-{\delta }_{\text{max}},{\delta }_{\text{max}}]$ around the dark resonance. The shaded area is the region where ${\delta }_{\text{max}}<{\delta }_{\text{Q}}$ and our statistical model of the recycling process requires modifications. Results are shown for $\mathrm{\Omega }=0.1\mathrm{\Gamma }/\sqrt{2}$.

Figure 5

Frequency-dependent effective emission rates from the bright $|{\psi }_{+}\rangle$ and dark $|{\psi }_{-}\rangle$ state superpositions of the two ground states of a $\mathrm{\Lambda }$-type system. The rates are even functions of $\delta$, and results are shown for $\mathrm{\Omega }=0.1\mathrm{\Gamma }/\sqrt{2}$.

Figure 6

The form factor $G\left(q\right)$ [Eq. (E5)] of $\mathcal{P}(\delta ,T)$ is compared to a Lorentzian with the same tails ($\propto 1/2q^2$), and a Gaussian with the same FWHM (2.13). All distributions are normalized to an area ${\pi }^{3/2}/2$.