On signal amplification via weak measurement

On signal amplification via weak measurement Yutaka Shikano Citation: AIP Conference Proceedings 1633, 84 (2014); doi: 10.1063/1.4903102 View online: http://dx.doi.org/10.1063/1.4903102 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1633?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The Q Weak p Experiment A Search For Physics Beyond The Standard Model Via A Measurement Of The Proton’s Weak Charge AIP Conf. Proc. 1265, 322 (2010); 10.1063/1.3480194 Signal amplification in a nanomechanical Duffing resonator via stochastic resonance Appl. Phys. Lett. 90, 013508 (2007); 10.1063/1.2430689 Coherent Signal Amplification in a Nanomechanical Oscillator via Stochastic Resonance AIP Conf. Proc. 850, 1675 (2006); 10.1063/1.2355353 Thermoacoustic amplification of photoacoustic signal Rev. Sci. Instrum. 67, 2317 (1996); 10.1063/1.1146939 Surface amplification of photoionization signal J. Chem. Phys. 79, 4608 (1983); 10.1063/1.446377

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On Signal Amplification via Weak Measurement Yutaka Shikano Institute for Molecular Science, Okazaki 444-8585, Japan Schmid College of Science and Technology, Chapman University, Orange, CA 92866, USA Abstract. One of the practical applications of the weak measurement proposed by Aharonov and his colleague is the signal amplification, which is called the weak-value amplification. We show the probe wavefunction to maximize the shift basically derived in [Y. Susa, Y. Shikano and A. Hosoya, Phys. Rev. A 85, 052110 (2012).] while this mode is not the propagation mode in light. Keywords: Signal Amplification, Weak Measurement, Non-Gaussian Distribution PACS: 03.65.Ta, 42.50.Dv, 42.50.Gy

The weak measurement was proposed in the context of the time-symmetric quantum measurement without collapsing the quantum state [1]. The weak value as the measurement outcome of the weak measurement can exceed the eigenvalue. By this fact, the signal can be amplified. This is called the weak-value amplification. To study the invisible region under the standard technique, there are several studies on the weak-value amplification as seen in the review [2, 3, 4]. Here, the following question arises. How can the signal maximize? To solve this problem, the probe wavefunction should be changed from the Gaussian distribution, which is originally used [1]. In this paper, we mathematically recapitulate the optimal probe wavefunction shown in Ref. [5] and discuss the properties of this probe wavefunction. Let us assume the following setup as the interaction Hamiltonian coupled between the system and the probe is given by Hint = gAˆ ⊗ pˆδ (t − t0 ),

(1)

where g is the coupling constant. Here, for the simplicity, we assume the instantaneous interaction at t0 . To study the maximization of the shift of the expectation value in the position space, we apply the variational principle under the Lagrangian ) [ ∫ ] (∫ ∗ 2 ∗ ′ ˆ f −λ d p∣ξi (p)∣ − 1 − Im μ dkξi (p)ξi (p) , (2) L[ξi (p), ξi (p), λ , μ ] := ⟨q⟩ where λ and μ are the Lagrange multipliers. It is noted that this is added the gauge fixing term to answer the commentary paper [6] from the original paper [5]. It is also noted that we can only find the stationary solution on the maximization of the shift of the expectation value in the position space with fixing the coupling constant g and the weak value Aw := ⟨ f ∣A∣i⟩/⟨ f ∣i⟩. Also, we have to set the initial expectation value of the position zero. We have already derived the optimal probe wavefunction as the following theorem [5]. Theorem 1. Let the momentum space be the compact support: −π /2g ≤ p ≤ π /2g. Also, the periodic boundary condition is assumed at p = ±π /2g with U(1) degree of

Eleventh International Conference on Quantum Communication, Measurement and Computation (QCMC) AIP Conf. Proc. 1633, 84-86 (2014); doi: 10.1063/1.4903102 © 2014 AIP Publishing LLC 978-0-7354-1272-9/$30.00

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freedom. Under the assumption of Aˆ 2 = 1 and ReAw ∕= 0, the optimal wavefunction is given by [ ] √ g(∣Aw ∣2 +1) g∣ReAw ∣ exp −i 2ReAw p (3) ξi (p) = π cos gp − iAw sin gp in the case of μ = 0. Then, the shift is given by ˆ i = ⟨q⟩ ˆ f= Δ⟨q⟩ ˆ := ⟨q⟩ ˆ f − ⟨q⟩

g(∣Aw ∣2 + 1) . 2ReAw

(4)

Otherwise, μ ∕= 0, there is no stationary solution. The proof of this theorem is seen in Refs. [5, 7]. It is remarked that the probe wavefunction in the position space has the discrete value only since the probe wavefunction is only defined in the region −π /2g ≤ p ≤ π /2g with the periodic boundary condition without U(1) degree of freedom, which was used on setting the initial expectation value of the position. However, there is the following problem to demonstrate this optimal probe wavefunction in experiment. We have not yet considered the concrete experimental setup of this situation. Especially in light, this probe wavefunction does not satisfy the solution of the optical equation. Therefore, this should be immediately broken after the measurement interaction since this is not the propagation mode. Within the propagation modes, for examples, Hermite-Gaussian modes and Laguerre-Gaussian modes, we have to find the optimal probe wavefunction from the practical viewpoint. As the first trial, the weak measurement in Laguerre-Gaussian mode, ( 2 ) x + y2 ∣l∣ , (5) ξLG (x) = C{x + i ⋅ sgn(l)y} exp − 4σ 2 where l is the azimuthal index, σ is the variance in the case of l = 0, and C is the normalization constant, was considered in Refs. [8, 9]. However, we have not yet calculated the all-order effects of this mode. This is the challenging task of the weak-value amplification. As alluded before, we need the concrete experimental setup to apply somewhat. For example, the research and development project in the gravitational wave detector. Within this, we have to find the useful case on the weak-value amplification. As far as we know, the case that the effect is theoretically predicted but seems to be experimentally detected like the spin Hall effect of light [10] is very useful to experimentally verify the new physical theory. The author (Y.S.) thanks collaboration with Yuki Susa and Akio Hosoya. Also, Y.S. thanks useful discussion with Antonio Di Lorenzo.

REFERENCES 1. 2.

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). Y. Aharonov and L. Vaidman, in Time in Quantum Mechanics, Vol. 1, edited by J. G. Muga, R. Sala Mayato, and I. L. Egusquiza (Springer, Berlin Heidelberg, 2008) p. 399.

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3.

Y. Aharonov and J. Tollaksen, in Visions of Discovery: New Light on Physics Cosmology and Consciousness, edited by R. Y. Chiao, M. L. Cohen, A. J. Leggett, W. D. Phillips, and C. L. Harper, Jr. (Cambridge University Press, Cambridge, 2011), p. 105. 4. Y. Shikano, in Measurements in Quantum Mechanics, edited by M. R. Pahlavani (InTech, 2012) p. 75, arXiv:1110.5055. 5. Y. Susa, Y. Shikano and A. Hosoya, Phys. Rev. A 85, 052110 (2012). 6. A. Di Lorenzo, arXiv:1210.3274 to appear in Phys. Rev. A. 7. Y. Susa, Y. Shikano and A. Hosoya, submitted to Phys. Rev. A. 8. G. Puentes, N. Hermosa, and J. P. Torres, Phys. Rev. Lett. 109, 040401 (2012). 9. H. Kobayashi, G. Puentes, and Y. Shikano, Phys. Rev. A 86, 053805 (2012). 10. O. Hosten and P. Kwiat, Science 319, 787 (2008).

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