Deterministic Superresolution with Coherent States at the Shot Noise Limit

Interference of light fields plays an important role in various high-precision measurement schemes. It has been shown that superresolving phase measurements beyond the standard coherent state limit can be obtained either by using maximally entangled multiparticle states of light or using complex detection approaches. Here we show that superresolving phase measurements at the shot noise limit can be achieved without resorting to nonclassical optical states or to low-efficiency detection processes. Using robust coherent states of light, high-efficiency homodyne detection, and a deterministic binarization processing technique, we show a narrowing of the interference fringes that scales with $1/\sqrt{N}$ where $N$ is the mean number of photons of the coherent state. Experimentally we demonstrate a 12-fold narrowing at the shot noise limit.

Figure 1

Experimental setup and illustration of the principles. (a) Schematic of the experimental setup. A product of a coherent state, $|\alpha ⟩$, and a vacuum state, $|0⟩$, is transformed through an interferometer and measured with a homodyne detector described by the ideal projector $⟨p|$. The evolutions in phase space of the two states are illustrated by the insets. (b) The phase response function $⟨\Pi ⟩$ for the standard interferometer scheme (dashed curve) and for the superresolving scheme (solid curve). (c) Experimental setup. Light from a picosecond pulsed Ti:sapphire laser operating at 830 nm and with a repetition rate of 800 kHz is controllably divided into two beams using a half-wave plate (HWP) and a polarizing beam splitter (PBS) thereby creating a signal and a local oscillator (LO) beam for homodyne detection (HD). The power of the signal beam is controlled by a neutral density filter (NDF), a HWP and a PBS, and subsequently sent into a Michelson interferometer (MI), the function of which is identical to the one in (a). A piezocrystal (PZT) attached to one of the interferometer mirrors scans the phase, $\varphi$, and the resulting output is measured by means of HD.

Figure 2

Performance of the superresolving interferometer. (a) and (b) show the experimental results (dots) and the theory (solid curve) for the response function with $a=0.5$ for two different mean photon numbers $N=19$ and $N=132$, respectively. The dashed curves represent the standard Rayleigh limited strategy. The insets of (a) and (b) are the results of the sensitivity as a function of $\varphi$ for the same values of $N$. The dashed curves represent the SNL. (c) The resolution in terms of FWHM of our scheme as a function of the mean photon number (red dots) together with theory for our approach (solid curve) and for Rayleigh limited strategy (dashed line). (d) Minimum value of the sensitivity for our scheme for different mean photon numbers (red dots). The solid curve represents the theoretical curve, $\Delta {\varphi }_{\min }\approx 1.37/\sqrt{N}$, and the dashed line stands for the SNL. Error bars of (c), (d) and the insets are smaller than the data points.

Figure 3

Results for the multiple binning approach. (a) Response function with 8 fringes per period obtained by binning the measurement outcomes along 5 intervals for a coherent state with $N=139$. The central points of these intervals are located at $p=b·k$ where $k\in [-2,-1,0,1,2]$ and $b=3.17$. The average fidelity of all fringes within a period is 95%. The data (red dots) fit well with theory (blue line) and the uncertainty for each point lies inside the theoretically predicted uncertainty, represented by the shaded area. The interference fringe corresponding to the Rayleigh limited approach is shown by the dashed curve. (b) Phase sensitivity for the multiple binning approach associated with the experimental results (dots) and the theory (solid curve). Near the SNL, performance (represented by the dashed line) is obtained for several phases. (c) Plot of the number of fringes $M$ for which the visibility is larger than 0.95 (red solid trace and dots) and 0.90 (blue dashed trace and squares) as a function of the average photon number, $N$, of the coherent state. The dots and squares correspond to numerical estimates whereas the curves are theoretical fits.