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.
The weak-value (WV) measurement proposed by Aharonov, Albert, and Vaidman (AAV) has attracted a great deal of interest in connection with quantum metrology. In this paper, we extend the analysis beyond the AAV limit and obtain a few main results. (i) We obtain a nonperturbative result for the signal-to-noise ratio (SNR). In contrast to the AAV prediction, we find that the SNR asymptotically gets worse when the AAV WV
Aw becomes large, i.e., in the case $g|{A}_{w}{|}^{2}\ensuremath{\gg}1$, where
g is the measurement strength. (ii) With the increase of
g (but also small), we find that the SNR is comparable to the result under the AAV limit, while both can reach---actually the former can slightly exceed---the SNR of the standard measurement. However, with a further increase of
g, the WV technique will become less efficient than the standard measurement, despite the fact that the postselection probability is increased. (iii) We find that the Fisher information can characterize the estimate precision qualitatively well as the SNR, yet the difference will become more prominent with the increase of
g. (iv) We carry out analytic expressions of the SNR in the presence of technical noises and illustrate the particular advantage of the imaginary WV measurement. The nonperturbative result of the SNR manifests a favorable range of the noise strength and allows an optimal determination.
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.
Interpretations of an experiment on the backaction in a weak measurement in [Iinuma et al., New J. Phys. 13, 033041 (2011)] are revisited. We show two different but essentially equivalent interpretations for this experiment along the original scenario of weak measurements proposed by Aharonov, Albert, and Vaidman [Phys. Rev. Lett. 60, 1351 (1988)]. To do this, we introduce the notion of extended weak values, which is associated not only with the states of the system but also the state of the measuring device. We also evaluate fluctuations in this experiment, and found that an optimal measurement strength exists for a fixed polarization angle prepared as an initial state, at which fluctuations in measurement results vanish. The consistency of this evaluation with the experimental results is discussed.
We extend the idea of weak measurements to the general case, provide a complete treatment, and obtain results for both the regime when the preselected and postselected states (PPS) are almost orthogonal and the regime when they are exactly orthogonal. We surprisingly find that for a fixed interaction strength, there may exist a maximum signal amplification and a corresponding optimum overlap of PPS to achieve it. For weak measurements in the orthogonal regime, we find interesting quantities that play the same role that weak values play in the nonorthogonal regime.
We report on the use of an interferometric weak value technique to amplify small transverse deflections of an optical beam. Our techniques consists of coupling the beam’s transverse degrees of freedom to the which-path states of a Sagnac interferometer. The theory for the amplification, including signal to noise ratio considerations, is presented along with the experimental results which are in good agreement. Of particular interest, we measured the angular deflection of a mirror down to 560 femtoradians and showed the signal to noise ratio to be amplified by a factor of 150.
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
g and is determined by only the initial state of the detector and the number of distinct eigenvalues of
A or $\ensuremath{\Omega}$. An example of spin-
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.