Protected State Enhanced Quantum Metrology with Interacting Two-Level Ensembles

Ramsey interferometry is routinely used in quantum metrology for the most sensitive measurements of optical clock frequencies. Spontaneous decay to the electromagnetic vacuum ultimately limits the interrogation time and thus sets a lower bound to the optimal frequency sensitivity. In dense ensembles of two-level systems, the presence of collective effects such as superradiance and dipole-dipole interaction tends to decrease the sensitivity even further. We show that by a redesign of the Ramsey-pulse sequence to include different rotations of individual spins that effectively fold the collective state onto a state close to the center of the Bloch sphere, partial protection from collective decoherence is possible. This allows a significant improvement in the sensitivity limit of a clock transition detection scheme over the conventional Ramsey method for interacting systems and even for noninteracting decaying atoms.

Figure 1

State protective Ramsey sequence. The ensemble of $N$ spins starts with all spins down in a collective coherent pure spin state on the surface of the collective Bloch sphere (radius $N/2$). Individual $\pi /2$ pulses are followed by phase encoding operations of angles ${\phi }_j^{(m)}=(2\pi m/N)(j-1)$ where $j=1,\dots N$ and $m=1,\dots ,[N/2]$, which brings the total spin close to the center of the Bloch sphere (the third-fifth steps are shown on small Bloch spheres of radius $1/2$). After time $\tau$, the phase encoding operation is reversed and $\pi /2$ pulses prepare the ensemble (now in a mixed state shown on the large collective Bloch sphere) for the detection of the population difference signal.

Figure 2

Two-atom metrology. (a) Normalized mutual decay rate and dipole-dipole frequency shift for a pair of atoms as a function of $r/\lambda$. For positive (negative) $\gamma$, the asymmetric state is subradiant (superradiant) (indicated by dashed/non-dashed regions). (b) Level scheme in the dressed basis showing the two independent decay channels with modified rates ${\gamma }_S$ and ${\gamma }_A$. (c) Optimal sensitivity as a function of $\tau \Gamma$. The atom separation is $r/\lambda =0.3$ corresponding to $\gamma \approx 0.41\Gamma$ and $\Omega \approx 0.29\Gamma$. The minimum for the asymmetric addressing is reached around $\tau \simeq 2/{\gamma }_A$.

Figure 3

Numerical investigations. (a) and (b) Numerical results for the square and a five atom chain. (c) Results of diagonalization for an ideal system of five equally mutually coupled emitters. The states are ordered with an increasing effective decay rate. The occupancy is shown in the histograms for SRT vs ART sequences.