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Photonics

Weak Value Amplification of Photons in Optical Nonlinear Medium, Opto-Mechanical, and Spin-Mechanical Systems

Sergio Carrasco, Miguel Orszag

Published Mar 23, 2024• 2 citations
Publisher

Abstract

A measurement of an observable A performed on a quantum system that is initially prepared in a state ρi, followed by a probabilistic procedure that leaves the system in a final state ρf, a process often referred as state postselection (or filtering process), can yield, on average, anomalous measurement results, i.e., values that may exceed the eigenvalue range of the observable being measured or be complex numbers. There is, therefore, an amplification effect of the average measurement result, i.e., the effect of the system on the measurement device is increased. When the coupling between the system and the measurement device satisfies some weakness conditions, the amplification effect occurs due to the weak value of the operator A. In this article, the amplification effect due to the postselection process is reviewed, and theoretical proposals and experiments published in the recent literature on the field are commented on. The emphasis is made on interactions occurring in optical nonlinear media and opto-mechanical and spin-mechanical systems, in which the amplification of number operators takes place.

DOI
10.3390/photonics11040291

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A measurement of an observable A performed on a quantum system that is initially prepared in a state ρi, followed by a probabilistic procedure that leaves the system in a final state ρf, a process often referred as state postselection (or filtering process), can yield, on average, anomalous measurement results, i.e., values that may exceed the eigenvalue range of the observable being measured or be complex numbers. There is, therefore, an amplification effect of the average measurement result, i.e., the effect of the system on the measurement device is increased. When the coupling between the system and the measurement device satisfies some weakness conditions, the amplification effect occurs due to the weak value of the operator A. In this article, the amplification effect due to the postselection process is reviewed, and theoretical proposals and experiments published in the recent literature on the field are commented on. The emphasis is made on interactions occurring in optical nonlinear media and opto-mechanical and spin-mechanical systems, in which the amplification of number operators takes place.
An experimental proposal is presented in which dark port postselection together with weak measurements are used to enlarge the radiation pressure effect of a single photon on a mechanical oscillator placed in the middle of a Fabry-P\'erot cavity and initialized in the ground state. By preparing and postselecting the photon (the system) in two quasiorthogonal states, the weak value of the radiation force operator can lie outside the eigenvalue spectrum, producing a large shift on the wave function of the mechanical oscillator (the measuring device) in the position representation. Consequently, the effect of a single photon on the average position of the oscillator can be amplified as compared to the effect caused by a photon without postselection, i.e., only preselected. The strong measurement scenario is also analyzed. In this case, a higher amplification effect is achieved and the mean position of the oscillator can reach the level of the zero-point fluctuations, but the postselection probabilities are smaller. Finally, the situation in which the oscillator starts in a thermal equilibrium state is studied. In this case, the amplification of the average position remains the same, i.e., proportional to the weak value, but the signal-to-noise ratio is reduced by a factor that depends on the average number of thermal phonons.
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.
Abstract To describe the pre‐ and postselected quantum ensembles, a complex quantity called the weak value of an operator is used. The weak value is highly controversial due to the fact that it is not bounded by the possible eigenvalues of the corresponding operator. Nevertheless, the obtaining of the anomalous weak value is regarded as a powerful technique in the quantum interferometry nowadays. Here it is shown that the postselection of a quantum system recovers a completely hidden interference effect in the measurement apparatus. Studying the interference pattern shows the optimal settings for the amplification and the parameter estimation. It also proves that the weak value is not an element of reality. Using single photons, it is investigated how a postselected photon can impart a π phase shift (the peak of the amplification) to a photon interacting weakly with it in a nonlinear optical medium. The increasing of the degree of the entanglement lies behind the effectiveness of the postselection in the parameter estimation. In particular, arranging to postselect on pure entangled states can optimize the signal‐to‐noise ratio, allowing to achieve high‐sensitive measurements using low input power.
Abstract An interferometric arrangement is proposed in which the technique of weak value amplification is implemented in order to enlarge the effect of a single photon on the quadratures of a movable mirror of an optical cavity. The photon interacts weakly with the mirror via radiation pressure and is postselected in the dark port of the interferometer. The real and imaginary parts of weak values of angular momentum type photonic operators produce an amplification of the mirror quadratures, which is large as compared to the scenario in which all photons are taken into consideration, i.e. when no postselection is performed. The effect is studied both for a mirror initialized in a thermal and coherent states. For a thermal state, the weak value amplification effect is boosted with the number of particles of the mirror, which occurs due to the imaginary part of the weak values.
The weak value of an operator is the average result of measuring that operator on a quantum system given specified initial and final states. The overlap between these states appears in the denominator of the weak value, and thus if that overlap is small, the weak value can be arbitrarily large. In 1988, Albert, Aharonov, and Vaidman proposed a weak Stern-Gerlach experiment which would be able to measure a so-called anomalous weak value, a measurement result which lies out- side of the eigenvalue spectrum of the operator being measured. Though anomalous weak values have been measured in numerous optical systems, here we present the first such realization of this experiment as originally proposed. By placing a Bose-Einstein condensate of 87Rb in a magnetic gradient and performing an unlikely postselection, we measure an effective magnification of a weak magnetic momentum kick (~ 8 μμμm/s) by a factor of 14. We also demonstrate that increasing the momentum kick decreases the "weakness" of the measurement and reduces the amplification.
Postselected weak measurement has been widely used in experiments to observe weak effects in various physical systems. However, it is still unclear how large the amplification ability of a weak measurement can be and what determines the limit of this ability, which is fundamental to understanding and applying weak measurements. The limitation of the conventional weak-value formalism for this problem is the divergence of weak values when the pre- and postselections are nearly orthogonal. In this paper, we study this problem by a variational approach for a general Hamiltonian ${H}_{\mathrm{int}}=gA\ensuremath{\bigotimes}\ensuremath{\Omega}\ensuremath{\delta}(t\ensuremath{-}{t}_{0}),\phantom{\rule{0.16em}{0ex}}g\ensuremath{\ll}1$. We derive a general asymptotic solution and show that the amplification limit is essentially independent of ggg and is determined by only the initial state of the detector and the number of distinct eigenvalues of AAA or $\ensuremath{\Omega}$. An example of spin-12\frac{1}{2}21​ particles with a pair of Stern-Gerlach devices is given to illustrate the results. The limiting case of continuous-variable systems is also investigated to demonstrate the influence of system dimension on the amplification limit.