Heisenberg Limit Superradiant Superresolving Metrology

We propose a superradiant metrology technique to achieve the Heisenberg limit superresolving displacement measurement by encoding multiple light momenta into a three-level atomic ensemble. We use $2N$ coherent pulses to prepare a single excitation superradiant state in a superposition of two timed Dicke states that are $4N$ light momenta apart in momentum space. The phase difference between these two states induced by a uniform displacement of the atomic ensemble has $1/4N$ sensitivity. Experiments are proposed in crystals and in ultracold atoms.

Figure 1

(a) Three-level configuration for photon momentum encoding. (b) Two counterpropagating modes collectively couple the transition between $|b⟩$ and $|a⟩$. (c) Timed Dicke state transport with $\pi$ pulses of ${\mathbf{k}}_1$ (black) and ${\mathbf{k}}_2$ (blue). They are applied alternatively to the atomic ensemble for $N$ times each to transfer the state from $|b_0⟩$ to $|b_{-2N{\mathbf{k}}_1}⟩$. (d) The schematic spatial phase oscillation of $|a_{-{\mathbf{k}}_1}⟩$ and $|b_{-20{\mathbf{k}}_1}⟩$.

Figure 2

Pictorial scheme of the displacement metrology with momentum Schrödinger cat state based on Ramsey interferometry. The white circles represent the collective ground state $|c_1,c_2,\dots ,c_{N_a}⟩$. The red and blue circles represent the superradiant sates $|b_{\mathbf{k}}⟩$ and $|a_{\mathbf{k}}⟩$. The single black arrow represent the single-photon Raman transition between the ground state and $|b_0⟩$ or $|a_0⟩$. The double black arrow represents the $\pi /2$-pulse $U_{ba}(\pi /2)$. The red and blue arrows represent the $\pi$ pulses $U_1$ and $U_2$ with the arrow’s direction indicating the transition direction.

Figure 3

Superradiant metrology with Raman transitions. (a) Atomic levels of Raman transitions. $|a⟩$, $|b⟩$, and $|c⟩$ are now the three Zeeman ground state sublevels. The atomic ensembles are initially prepared in state $|c⟩$. A single photon Raman transition (green arrows) prepares the atomic ensemble in a timed Dicke state of $|b⟩$. Then the Raman pulses drive the transition between $|b⟩$ and $|a⟩$. (b) Raman $\pi$ pulses $U_1$ and $U_2$. They are composed by a long pulse driving $d\leftrightarrow a$ transition and a short pulse driving $b\leftrightarrow d$ transition. The short pulse goes through the atomic ensemble when the atoms are uniformly covered by the long pulse.

Figure 4

Numerical simulation of $P$ for Gaussian noisy $\pi$ pulses with area variation $\mathrm{\Delta }S=0.1$ and phase variation $\mathrm{\Delta }\varphi =0.01$. $N=16$ (black squares) and 32 (red triangles).