Quantum-limited metrology with product states

We study the performance of initial product states of $n$-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear $k$-body $\left(k\ll n\right)$ Hamiltonian. We obtain the theoretical lower bound on the uncertainty in the estimate of the parameter. For arbitrary initial states, the lower bound scales as $1/n^k$, and for initial product states, it scales as $1/n^{k-1/2}$. We show that the latter scaling can be achieved using simple, separable measurements. We analyze in detail the case of a quadratic Hamiltonian $\left(k=2\right)$, implementable with Bose-Einstein condensates. We formulate a simple model, based on the evolution of angular-momentum coherent states, which explains the $O\left(n^{-3/2}\right)$ scaling for $k=2$; the model shows that the entanglement generated by the quadratic Hamiltonian does not play a role in the enhanced sensitivity scaling. We show that phase decoherence does not affect the $O\left(n^{-3/2}\right)$ sensitivity scaling for initial product states.

Figure 1

(Color online) Sensitivity (solid red lines) vs $\varphi$ $\left(-\pi /8\le \varphi \le \pi /8\right)$ using an optimal initial state at angle $\beta =\pi /4$: (a) $J_x$ measurements; (b) $J_y$ measurements. The total angular momentum $J$ of the probe is 200, corresponding to $n=400$. The lower bound on the sensitivity, $1/\sqrt{2}{J}^{3/2}$, is plotted as the dotted (green) line. The sensitivity is characterized by rapidly oscillating fringes and a decay of sensitivity away from the best sensitivities near $\varphi =0$. The sensitivity patterns repeat with periodicity $\pi$; only one-quarter of a period is plotted because the sensitivity worsens even more outside the plotted region. Part (a) also shows the sensitivity for $J_x$ measurements when $\beta =\pi /2$ (dashed blue line); notice the absence of fringes in this case and the substantially degraded sensitivity.

Figure 2

(Color online) Central few fringes of the measurement precision for $\beta =\pi /4$ and $J=2500$ $\left(n=5000\right)$: (a) $J_x$ measurements; (b) $J_y$ measurements. The solid (red) lines are the exact sensitivities, the dashed (blue) lines are the sensitivities given by the uniform-fringe approximation of Eqs.(5.22, 5.23), $\delta {\varphi }_x\simeq \left(1/2J^{3/2}\right)\sqrt{1+\left[1/{\text{sin}}^2\left(2J\varphi \text{cos}\beta \right)\right]}$, $\delta {\varphi }_y\simeq \left(1/2J^{3/2}\right)\sqrt{1+\left[1/{\text{cos}}^2\left(2J\varphi \text{cos}\beta \right)\right]}$, and the dotted (green) lines are the Gaussian-envelope approximation of Eqs. (5.17). The uniform-fringe approximation locates the fringes precisely, but misses entirely the degradation in sensitivity as one moves away from the central fringes and also fails to characterize accurately the shape of the fringes. The Gaussian-envelope approximation improves on this performance by capturing the degradation of sensitivity quite well, but still fails on the fringe shapes. Even the central fringe for $J_y$ measurements is noticeably flatter than in the two approximations. To get the best sensitivity, one should operate right on the central fringe, at $\varphi =q\pi$, for $J_y$ measurements and on one of the two central fringes, centered at $\varphi =q\pi \pm \pi /4J\text{cos}\beta$ for $J_x$ measurements. Notice that $J_x$ measurements achieve nearly optimal sensitivity at points near the outside of these two central fringes.

Figure 3

(Color online) Scaling exponent $\xi$ for $J_x$ measurements. The dotted (red) line is for $J={10}^3$, the dashed (green) line for $J={10}^5$, and the solid (blue) line for $J={10}^7$.

Figure 4

(Color online) Scaling exponent $\xi$ for $J_y$ measurements. The dotted (red) line is for $J={10}^3$, the dashed (green) line for $J={10}^5$, and the solid (blue) line for $J={10}^7$.