Quantum Metrology: Dynamics versus Entanglement

A parameter whose coupling to a quantum probe of $n$ constituents includes all two-body interactions between the constituents can be measured with an uncertainty that scales as $1/n^{3/2}$, even when the constituents are initially unentangled. We devise a protocol that achieves the $1/n^{3/2}$ scaling without generating any entanglement among the constituents, and we suggest that the protocol might be implemented in a two-component Bose-Einstein condensate.

Figure 1

Nonlinear interferometer giving $J_z^2$ and $n{J}_z$ couplings. An incoming beam of $n$ bosons is split at a beam splitter, which puts each boson into an appropriate superposition of being in the two arms (modes). The two initial nonlinear phase shifters produce Kerr phase shifts ${\chi }_1{n}_1^2$ and ${\chi }_2{n}_2^2$. The phase shifter at the intersection of the beams produces a cross-Kerr phase shift $2{\chi }_{12}{n}_1{n}_2$. The final $50/50$ beam splitter converts the required measurement of an equatorial component of $\mathit{J}$ into a measurement of $J_z$, i.e., a counting of the difference of the numbers of particles in the two output beams. The net effect of the nonlinear phase shifters is the same as a probe Hamiltonian $\mathcal{H}$ acting for a time $t$, with $\mathcal{H}t/\hslash ={\chi }_1{n}_1^2+{\chi }_2{n}_2^2+2{\chi }_{12}{n}_1{n}_2=(\chi +{\chi }_{12})n^2/2+({\chi }_1-{\chi }_2)n{J}_z+2(\chi -{\chi }_{12})J_z^2$, where $\chi =\frac{1}{2}({\chi }_1+{\chi }_2)$ is the average Kerr phase shift. The term proportional to $n^2$ produces an overall phase shift and can be ignored. The $n{J}_z$ coupling comes from having different Kerr phase shifters in the two arms; to eliminate the $J_z^2$ interaction requires a cross-Kerr coupling ${\chi }_{12}=\chi$. Under these circumstances, we have $\mathcal{H}=\hslash \gamma n{J}_z$, with $\gamma t={\chi }_1-{\chi }_2$.