Quantum-limited measurement of space-time curvature with scaling beyond the conventional Heisenberg limit

We study the problem of estimating the phase shift due to the general relativistic time dilation in the interference of photons using a nonlinear Mach-Zender interferometer setup. By introducing two nonlinear Kerr materials, one in the bottom and one in the top arm, we can measure the nonlinear phase ${\varphi }_{\text{NL}}$ produced by the space-time curvature and achieve a scaling of the standard deviation with photon number ($N$) of $1/N^{\beta }$ where $\beta >1$, which exceeds the conventional Heisenberg limit of a linear interferometer ($1/N$). The nonlinear phase shift is an effect that is amplified by the intensity of the probe field. In a regime of high photon number, this effect can dominate over the linear phase shift.

Figure 1

Nonlinear interferometer of arm length $L$ in a gravitational field. Coherent light passes through a $50/50$ beamsplitter at 1. The phase from 1 to 3 at $r=r_A$ is set to ${\varphi }_{13}=0$ and the time interval that light traverses is ${\tau }_{13}=\frac{L}{c}$. The effect of the gravitational redshift cancels out and no phase shift is imposed as light traverses vertically. In the top and bottom arms, we have a nonlinear medium with $\chi$ coupling. A phase difference due to slower interaction time with the nonlinearity in the bottom arm is detected after recombining at the second beamsplitter. The time intervals $\mathrm{\Delta }{\tau }_{13}$ and $\mathrm{\Delta }{\tau }_{24}$ contain the Schwarzschild radius $r_s$. $\beta$ represents an adjustable linear phase shift.

Figure 2

Error bound of the Schwarzschild radius plotted against the number of coherent photons for various nonlinearity couplings of interferometer arm size $L=1$ cm and height $h=10$ m. The solid lines represent the exact lower bound for the best possible measurement. The dashed lines are the quadrature measurement error bounds. These lines terminate before the condition $\left|\alpha \right|\chi \tau \ll 1$ is violated. The solid black line is the case where squeezing of all photons is used to enhance the sensitivity; however, small amounts of loss ($\epsilon =1–{10}^{-6}$) means the scaling is still at the SQL. From top to bottom, the red solid lines represent the SQL limit for a linear interferometer of heights $h=10, {10}^2$, and ${10}^3$ m. [Other parameters: Number of measurements $M={10}^{10}$, the central frequency $\omega =100$ THz, and the radius $r_A=6.37\times {10}^6$ m (Earth's radius).]