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Journal of Optics B Quantum and Semiclassical Optics

Imaging of faint phase objects at the Heisenberg limit with quadrature-squeezed light

I. V. Sokolov

Published Apr 1, 2000• 7 citations
Publisher

Abstract

We evaluate the limiting imaging sensitivity of a two-dimensional faint phase object, placed within an interferometer. We consider the case when both input ports of the interferometer are illuminated by spatially multimode quadrature-squeezed light, and a differenced detection scheme where spatial resolution is used. This provides the Heisenberg-limited measurement sensitivity of the spatial phase distribution in the object. The optimum choise of space-time scales of the system and the resolving power of the measurement in space and time are discussed.

DOI
10.1088/1464-4266/2/2/320

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We evaluate the limiting imaging sensitivity of a two-dimensional faint phase object, placed within an interferometer. We consider the case when both input ports of the interferometer are illuminated by spatially multimode quadrature-squeezed light, and a differenced detection scheme where spatial resolution is used. This provides the Heisenberg-limited measurement sensitivity of the spatial phase distribution in the object. The optimum choise of space-time scales of the system and the resolving power of the measurement in space and time are discussed.
Summary form only given. To achieve super-resolution one must take into account a priori information about localization of the object. Such an a priori information exists in the case of an object of finite size. Its Fourier transform located in the focal plane of the lens is an entire analytic function, and analytic continuation of the object spectrum outside the spatial-frequency band allows for unlimited resolution. However such a super-resolution is extremely sensitive to the noise in the detected image. The ultimate limit of super-resolution is determined by the quantum fluctuations of light in the detection plane. We present the quantum theory of super-resolution for a one-dimensional coherent optical imaging scheme and discuss possibilities of improving the resolution limit by using multimode squeezed light. For quantum-mechanical description of this scheme we take into account the vacuum fluctuations of the electromagnetic field in the input plane outside the object. Interference of these vacuum fluctuations with the field amplitude of the object in the detection plane is the origin of the standard quantum limit or shot noise of photodetection in the image. To go beyond this limit we suggest to illuminate the object with multimode squeezed light having the spot size much larger than the object area. Then the vacuum fluctuations outside the object can be effectively suppressed resulting in sub-shot-noise detection of the image.
The ultimate precision of phase estimation is limited by the Heisenberg scaling Δϕ0=K/NΔϕ_0 = K/NΔϕ0​=K/N, where K∼1K\sim1K∼1 is a numerical prefactor and NNN is the mean number of photons interacting with the phase shifting object(s). However, achieving this fundamental limit often comes at the cost of an extremely narrow high-sensitive range, rendering schemes impractical. We analyze the precision limits of phase measurements in single- and two-arm optical interferometers with input Gaussian states. We consider two detection methods: conventional homodyne measurement and non-Gaussian threshold detection that saturates the quantum Cramér-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity Δϕ0Δϕ_0Δϕ0​ and the width δϕδϕδϕ of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that δϕδϕδϕ strongly depends on KKK. We derive an approximate analytic expression that describes this trade-off. We show also that the two-arm interferometer with antisymmetrically squeezed inputs exhibits exceptional performance, simultaneously achieving Heisenberg-limited sensitivity and a broad high-sensitivity range δϕ=π/2δϕ=π/2δϕ=π/2.
We study in detail a system of two interferometers aimed at detecting extremely faint phase fluctuations. This system can represent a breakthrough for detecting a faint correlated signal that would remain otherwise undetectable even using the most sensitive individual interferometric devices, as in the case of so-called holographic noise. The signature of this kind of noise emerges as a correlation between the output signals of the interferometers. On the other hand, when holographic noise is absent one expects uncorrelated signals since the time-averaged fluctuations due to shot noise and other independent contributions vanish (though limiting the overall sensitivity). We show how injecting quantum light in the free ports of the interferometers can reduce the photon noise of the system beyond the shot noise, enhancing the resolution in the phase-correlation estimation. We analyze the use of both the two-mode squeezed vacuum and two independent squeezed states. Our results confirm the benefit of using squeezed beams together with strong coherent beams in interferometry. We also investigate the possible use of the two-mode squeezed vacuum, discovering interesting and unexplored areas of application of bipartite entanglement, in particular the possibility of reaching in principle a surprising uncertainty reduction.
We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.
We show that the phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1/{N}_{T}$, where NT{N}_{T}NT​ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter rrr and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.