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Science

Tinker-Free Interferometry

Published Dec 13, 2007• 0 citations
Publisher

Abstract

The interference of waves, be they photons, atoms, or electrons, and the measured phase shift induced as the waves take a different route around the arms of an interferometer are widely used in applications ranging from global positioning systems and gravity wave detectors down to the more mundane characterization of optical materials. The sensitivity of the interferometer in measuring the phase shift is affected by noise contributions to the signal. To combat this noise, piezoelectric actuators are typically used to adjust the system to an optimum measuring point for maximum sensitivity. This approach has drawbacks, however, for certain applications where the determination of arbitrary phase shifts is important. Pezzé et al. introduce a method and protocol of photon counting at the output paths to determine the phase. They show that the classical, or thermal, noise can be eliminated without bias (i.e., no tinkering with the phase shift is necessary). At this quantum limit, only quantum uncertainty affects the precision of the phase measurement. — ISO

DOI
10.1126/science.318.5857.1699b

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The interference of waves, be they photons, atoms, or electrons, and the measured phase shift induced as the waves take a different route around the arms of an interferometer are widely used in applications ranging from global positioning systems and gravity wave detectors down to the more mundane characterization of optical materials. The sensitivity of the interferometer in measuring the phase shift is affected by noise contributions to the signal. To combat this noise, piezoelectric actuators are typically used to adjust the system to an optimum measuring point for maximum sensitivity. This approach has drawbacks, however, for certain applications where the determination of arbitrary phase shifts is important. Pezzé et al. introduce a method and protocol of photon counting at the output paths to determine the phase. They show that the classical, or thermal, noise can be eliminated without bias (i.e., no tinkering with the phase shift is necessary). At this quantum limit, only quantum uncertainty affects the precision of the phase measurement. — ISO
In a conventional interferometer one shines light only into one interferometer input port while the second input port is unexcited. This allows vacuum states to enter the unexcited port, leading to imperfect correlation between both the amplitude and the phase fluctuations of the light in the upper and lower interferometer arms. This, in turn, limits the interferometer sensitivity regardless of its operating panel (its arm-length difference, which we in the following will translate into a phase shift θ). The sensitivity of an ideal conventional interferometer is 1 / N. That is, a phase shift of 1 / N radians can be detected with a signal-to-noise ratio of unity by using N photons. Since N typically is a large number, it is tempting to try to find some scheme approaching the fundamental (Heisenberg) limit of phase-shift detection, 1/N. One way of doing that is to inject squeezed vacuum into the unexcited input port of the interferometer.1
Summary form only given. Interferometry is the basis of many high-precision measurements. The ultimate limit to the precision is due to quantum effects. This limit is most easily explored for a Mach-Zehnder interferometer The outputs of this device can be measured to yield an estimate /spl phi/ of the phase difference /spl phi/ between the two arms of the interferometer. It is well known that this can achieve the standard quantum limit for phase variance when an N-photon number state enters one input port. Several authors have proposed ways of reducing the phase variance to the Heisenberg limit. In this paper we show that there is a physical measurement scheme using photodetectors and feedback that is almost as good as the optimal measurement. The fundamental idea is that, in addition to the unknown phase we wish to measure, /spl phi/, in one arm of the interferometer, we introduce a known phase shift, /spl Phi/, into the other arm of the interferometer. After each photodetection we adjust this introduced phase shift in order to minimize the expected uncertainty of our best phase estimate /spl phi/ after the next photodetection. This feedback process is uniquely defined using Bayesian statistics, and can be solved exactly for moderate photon numbers.
One important problem in quantum optics is to resolve an extremely small change of phase shift. The complementarity between photon number and phase sets an ultimate limit, the so-called Heisenberg limit, on the phase measurement sensitivity. The precise phase estimation has many technological applications, such as optical gyroscopes, gravitational wave detection, quantum imaging and sensing. In this thesis I show that the utilization of the parity measurement in the optical interferometry is actually applicable to a wide range of quantum entangled input states. Comparison of the performance of the various quantum states then can be made within such a unified output measurement scheme. Based on such a universal detection scheme, we present a comparison of the phase sensitivity reduction for various quantum states of light in the presence of photon loss. I also provide a simple condition that could be used to check whether an arbitrary state can achieve the Heisenberg limit independently of the detection scheme. It implies that the fidelity between the two output states with zero phase and minimal detectable phase applied respectively should significantly different from unity as the minimal phase shift scaling as 1/N, whereas the average number of input photons N goes to infinity. Next I give several measures to characterize the which-way information in the interference experiment. We define a new distinguishability associated with the fidelity between two density matrices to measure the which-way information. We demonstrate that the changes of mutual entropy as well as the entanglement of formation of the whole system, i.e. the physical system plus the which-way detector can also be used to describe the which-way information. With such quantities, we show that as the fringe visibility of the interference pattern gets larger, the less which-way information is obtained. Finally, I show that coherent light coupled with photon number resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power. This scheme would have applications to laser radar, given the difficulty in making entangled states of light, as well as their susceptibility to atmospheric absorption.
When a photon interferes with itself while traversing a Mach-Zehnder interferometer, the output port where it emerges is influenced by the phase difference between the interferometer arms. This allows for highly precise estimation of the path length difference (delay) but is extremely sensitive to phase noise. By contrast, a delay between the arms of the two-photon Hong-Ou-Mandel interferometer directly affects the relative indistinguishability of the photon pair, affecting the rate of recorded coincidences. This likewise allows for delay estimation, notably less precise but with the advantage of being less sensitive to perturbations of the photons' phase. Focusing on two-photon input states, we here investigate to what degree of noise Mach-Zehnder interferometry retains its edge over Hong-Ou-Mandel interferometry. We also explore the competing benefits of different two-photon inputs for a Mach-Zehnder interferometer and under what parameter regimes each input performs best.
Bragg diffraction of matter waves is an established technique used in the most accurate quantum sensors. It is also the method of choice to operate large-momentum-transfer, high-sensitivity atom interferometers. It suffers, however, from an intrinsic multi-path character. Optimal control theory (OCT) has recently led to an improved robustness of atom interferometers to a range of challenging environmental effects such as vibrations or platform accelerations. In this theoretical work, we apply OCT protocols to control the Bragg diffraction phase shifts, thereby enhancing the metrological accuracy of the interferometer. We show a minimization of the diffraction phase for realistic conditions of finite temperature of the incoming wavepacket in a multi-path, high-order Bragg interferometer in a Mach–Zehnder configuration. We study input states with different momentum widths and find that our approach mitigates diffraction phases below the microradian level in the case of 1% of the photon recoil, thereby eliminating one of the leading systematic effects in atom interferometry.
The best performance of a Mach-Zehnder interferometer is achieved with the input state $\ensuremath{\mid}{N}_{T}∕2+1⟩\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩+\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩\ensuremath{\mid}{N}_{T}∕2+1⟩$, NT{N}_{T}NT​ being the total number of atoms or photons. This gives (i) a phase-shift error confidence ${C}_{68%}=2.67∕{N}_{T}$ with (ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling $\ensuremath{\sim}1∕{N}_{T}$ but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.