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"Study of Weak Measurement Amplification in Simple Optomechanical Systems"

MANOJ DAS

Published Jan 1, 2022• 0 citations
Publisher

Abstract

Abstract We have studied the amplification in displacement of the mechanical mirror in a simple optomechanical system. Here photon number acts as system and the mirror acts as the meter. In the weak coupling condition, post selection of the photon number operator enhance the value of mirror displacement. The displacement of the mirror depends on the weak value of the photon number operator and on the value of relative phase(φ) between the pre and post selected states. Amplification of the mirror can be seen for maximum coupling strength(.25Ω m ) in weak coupling criteria for optomechanical systems, Ωm being the mechanical frequency.

DOI
10.21203/rs.3.rs-1713089/v1

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Abstract We have studied the amplification in displacement of the mechanical mirror in a simple optomechanical system. Here photon number acts as system and the mirror acts as the meter. In the weak coupling condition, post selection of the photon number operator enhance the value of mirror displacement. The displacement of the mirror depends on the weak value of the photon number operator and on the value of relative phase(φ) between the pre and post selected states. Amplification of the mirror can be seen for maximum coupling strength(.25Ω m ) in weak coupling criteria for optomechanical systems, Ωm being the mechanical frequency.
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.
We revisit the scheme of single-photon weak-coupling optomechanics using postselection, proposed by Pepper, Ghobadi, Jeffrey, Simon, and Bouwmeester [Phys. Rev. Lett. 109, 023601 (2012)], by analyzing the exact solution of the dynamical evolution. Positive and negative amplification effects of the displacement of the mirror's position can be generated when the Kerr phase is considered. This effect occurs when the postselected state of the photon is orthogonal to the initial state, which cannot be explained by the usual weak measurement results. The amplification effect can be further modulated by a phase shifter, and the maximal displacement state can appear within a short evolution time.
We analyze the amplification by the Aharonov-Albert-Vaidman weak quantum measurement on a Sagnac interferometer [Dixon et al., Phys. Rev. Lett. 102, 173601 (2009)] up to all orders of the coupling strength between the measured system and the measuring device. The amplifier transforms a small tilt of a mirror into a large transverse displacement of the laser beam. The conventional analysis has shown that the measured value is proportional to the weak value, so that the amplification can be made arbitrarily large in the cost of decreasing output laser intensity. It is shown that the measured displacement and the amplification factor are in fact not proportional to the weak value and rather vanish in the limit of infinitesimal output intensity. We derive the optimal overlap of the pre- and postselected states with which the amplification become maximum. We also show that the nonlinear effects begin to arise in the performed experiments so that any improvements in the experiment, typically with an amplification greater than 100, should require the nonlinear theory in translating the observed value to the original displacement.
An interferometric experiment is proposed in order to measure the radiation force exerted by a single photon on a nano mechanical oscillator placed in the middle of a Fabry-P\'erot cavity. The force is measured by observing the shift given to the probability density function of the position of the oscillator, using weak measurements and post-selection of photons. We show that this shift is proportional to the weak value of the operator that accounts for the difference of photons between both sides of the cavity. For single-photon states, this operator has eigenvalues 1 or -1, depending on the side of the cavity in which is the photon, but the weak value can exceed by far this range. Thus, by preparing and post-selecting the photon in two nearly orthogonal states the weak value is increased and the radiation effect of a single photon turns out to be much larger than the pressure exerted by a photon without post-selection. This opens the possibility to study weak values of number operators in optomechanics and the radiation effect in pre- and post-selected ensembles.