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arXiv:1101.0199 · Physical Review Letters

Amplifying Single-Photon Nonlinearity Using Weak Measurements

Amir Feizpour, Xingxing Xing, Aephraim M. Steinberg

Published Sep 23, 2011• 264 citations
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Abstract

We show that weak measurement can be used to "amplify" optical nonlinearities at the single-photon level, such that the effect of one properly postselected photon on a classical beam may be as large as that of many unpostselected photons. We find that "weak-value amplification" offers a marked improvement in the signal-to-noise ratio in the presence of technical noise with long correlation times. Unlike previous weak-measurement experiments, our proposed scheme has no classical equivalent.

DOI
10.1103/physrevlett.107.133603

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In 1988, Aharonov, Albert, and Vaidman introduced a new paradigm of quantum measurement in a paper which had the unwieldy but provocative title "How the result of a measurement of a component of the spin of a spin-1=2 particle can turn out to be 100." This paradigm, so-called "weak measurement," has since been the subject of widespread theoretical and experimental attention, both for the perspective it offers on quantum reality and for possible applications to precision measurement. Yet almost all of the weak-measurement experiments carried out so far could be alternatively understood in terms of the classical (electro-magnetic wave) theory of optics. Here we present a truly quantum version, the first in which a measurement apparatus deterministically entangles two distinct optical beams, enabling us to experimentally ask a question directly analogous to that of the original proposal: "In a two-arm interferometer containing one photon in total, can the result of a measurement of the photon number in one arm turn out to be greater than 1?" Specifically, we show that a single photon, when properly post-selected, can have an effect equal to that of eight photons: that is, in a system where a single photon has been calibrated to write a nonlinear phase shift of ϕ_0 on a probe beam, we measure phase shifts as large as 8ϕ_0 for appropriately post-selected single photons. This is the first deterministic weak-value experiment in optics which defies classical explanation, and constitutes a realization of our proposal for weak-value amplification (WVA) of the small optical nonlinearity at the single-photon level. It opens up a new regime for the study of entanglement of optical beams, as well as further investigations of the power of WVA for the measurement of small quantities.

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Weak value amplification (WVA) is a measurement technique in which the effect of a pre- and post-selected system on a weakly interacting probe is magnified. In this thesis, I present the first experimental observation of WVA of a single photon. We observed that a signal photon - sent through a polarization interferometer and post-selected by photodetection in the almost-dark port - can act like eight photons. The effect of this single photon is measured as a nonlinear phase shift on a separate laser beam. The interaction between the two is mediated by a sample of laser- cooled 85Rb atoms. Electromagnetically induced transparency (EIT) is used to enhance the nonlinearity and overcome resonant absorption. I believe this work to be the first demonstration of WVA where a deterministic interaction is used to entangle two distinct optical systems. In WVA, the amplification is contingent on discarding a large portion of the original data set. While amplification increases measurement sensitivity, discarding data worsens it. Questioning whether these competing effects conspire to improve or diminish measurement accuracy has resulted recently in controversy. I address this question by calculating the maximum amount of information achievable with the WVA technique. By comparing this information to that achievable by the standard technique, where no post-selection is employed, I show that the WVA technique can be advantageous under a certain class of noise models. Finally, I propose a way to optimally apply the WVA technique.

•Jun 1, 2016•1 citation
We show that weak measurement can be used to "amplify" optical nonlinearities at the single-photon level, such that the effect of one properly postselected photon on a classical beam may be as large as that of many unpostselected photons. We find that "weak-value amplification" offers a marked improvement in the signal-to-noise ratio in the presence of technical noise with long correlation times. Unlike previous weak-measurement experiments, our proposed scheme has no classical equivalent.
Abstract Most studies for postselected weak measurement in optomechanical system focus on using a single photon as a measured system. However, we find that using weak coherent light instead of a single photon can also amplify the mirror’s position displacement of one photon. In the weak value amplification (WVA) regime, the weak value of one photon can lie outside the eigenvalue spectrum, proportional to the difference of the mirror’s position displacement between the successful and failed postselection and its successful postselection probability is dependent of the mean photon number of the coherent light and improved by adjusting it accordingly. Outside the WVA regime, the amplification limit can reach the level of the vacuum fluctuations, and when the mean photon number and the optomechanical coupling parameter are selected appropriately, its successful postselection probability becomes higher, which is beneficial to observe the maximum amplification value under the current experimental conditions. This result breaks the constraint that it is difficult to detect outside the WVA regime. This opens up a new regime for the study of a single photon nonlinearity in optomechanical system.
In 1988, Aharonov, Albert, and Vaidman introduced a new paradigm of quantum measurement in a paper which had the unwieldy but provocative title "How the result of a measurement of a component of the spin of a spin-1=2 particle can turn out to be 100." This paradigm, so-called "weak measurement," has since been the subject of widespread theoretical and experimental attention, both for the perspective it offers on quantum reality and for possible applications to precision measurement. Yet almost all of the weak-measurement experiments carried out so far could be alternatively understood in terms of the classical (electro-magnetic wave) theory of optics. Here we present a truly quantum version, the first in which a measurement apparatus deterministically entangles two distinct optical beams, enabling us to experimentally ask a question directly analogous to that of the original proposal: "In a two-arm interferometer containing one photon in total, can the result of a measurement of the photon number in one arm turn out to be greater than 1?" Specifically, we show that a single photon, when properly post-selected, can have an effect equal to that of eight photons: that is, in a system where a single photon has been calibrated to write a nonlinear phase shift of ϕ_0 on a probe beam, we measure phase shifts as large as 8ϕ_0 for appropriately post-selected single photons. This is the first deterministic weak-value experiment in optics which defies classical explanation, and constitutes a realization of our proposal for weak-value amplification (WVA) of the small optical nonlinearity at the single-photon level. It opens up a new regime for the study of entanglement of optical beams, as well as further investigations of the power of WVA for the measurement of small quantities.
In a quantum-noise limited system, weak-value amplification using postselection normally does not produce more sensitive measurements than standard methods for ideal detectors: the increased weak value is compensated by the reduced power due to the small postselection probability. Here, we experimentally demonstrate recycled weak-value measurements using a pulsed light source and optical switch to enable nearly deterministic weak-value amplification of a mirror tilt. Using photon counting detectors, we demonstrate a signal improvement by a factor of 4.4±0.2 and a signal-to-noise ratio improvement of 2.10±0.06, compared to a single-pass weak-value experiment, and also compared to a conventional direct measurement of the tilt. The signal-to-noise ratio improvement could reach around six for the parameters of this experiment, assuming lower loss elements.
In a weak measurement with postselection, a measurement value, called the weak value, can be amplified beyond the eigenvalues of an observable. However, there are some controversies whether weak-value amplification is practically useful in increasing the sensitivity of the measurement, in which fundamental quantum noise dominates. In this paper, we investigate the sensitivity limit of an optical interferometer when weak-value amplification is implemented, properly accounting for quantum shot noise and radiation-pressure noise. To do so, we formulate weak-value amplification in the Heisenberg picture instead of in the Schr\"odinger picture, which is conventionally used. This formulation enables us to understand intuitively what happens when the measurement outcome is postselected and the weak value is amplified. As a result, we find that the sensitivity limit is given by the standard quantum limit that is the same as in a standard interferometry. We also discuss a way to circumvent the standard quantum limit.
Weak value amplification (WVA) is a measurement technique in which the effect of a pre- and post-selected system on a weakly interacting probe is magnified. In this thesis, I present the first experimental observation of WVA of a single photon. We observed that a signal photon - sent through a polarization interferometer and post-selected by photodetection in the almost-dark port - can act like eight photons. The effect of this single photon is measured as a nonlinear phase shift on a separate laser beam. The interaction between the two is mediated by a sample of laser- cooled 85Rb atoms. Electromagnetically induced transparency (EIT) is used to enhance the nonlinearity and overcome resonant absorption. I believe this work to be the first demonstration of WVA where a deterministic interaction is used to entangle two distinct optical systems. In WVA, the amplification is contingent on discarding a large portion of the original data set. While amplification increases measurement sensitivity, discarding data worsens it. Questioning whether these competing effects conspire to improve or diminish measurement accuracy has resulted recently in controversy. I address this question by calculating the maximum amount of information achievable with the WVA technique. By comparing this information to that achievable by the standard technique, where no post-selection is employed, I show that the WVA technique can be advantageous under a certain class of noise models. Finally, I propose a way to optimally apply the WVA technique.