從1932年以來，穿隧效應要花費的時間就一直是懸而未決的問題，這期間 有各式各樣關於穿隧時間的模型被提出，其中有些定義甚至預測物質波在穿隧 時的速度能超越真空光速。到了2005年，Y. Couder 的團隊將盛裝了矽油的容 器放在震動台上，發現在略低於產生法拉第波的臨界加速度時，在矽油表面挑 起的矽油滴可以長時間存活，以此兼具粒子(矽油滴)和波動(油滴撞擊油面 產生的漣漪)二象性的純古典系統模擬出了一系列的類量子現象，包含狹縫的 干涉與繞射，量子化的軌道，二維量子柵欄(quantum corral)，以及2009年的 穿隧機率 P 與障礙物寬度 D 有 P ∝ e−αD 關係。此論文以矽油滴類穿隧效應的 實驗作為延伸，測量油滴從進入到離開障礙物所花費的時間。結果發現此穿隧 時間 t，在障礙物狹窄時，與寬度 D 成正比，但是對於較廣的障礙物，會呈指 數正相關，有下面的關係 t ∝ DeβD，其中 α ≠ β。我們發現這個行為迥異於文 獻眾多的穿隧時間定義，並提出可能的物理解釋。

Tunneling time has been a pending issue since 1932. There exists various competing models, including the dwell time and phase time that suffer from the defect of violating the special relativity. In 2005 Y. Couder et al. found that a droplet bouncing on a vertically vibrated bath can be propelled by the surface wave it generates and becomes a “walker" that moves at a constant speed. Although purely classical, this walker has been reported to exhibit many quantum-like characteristics, such as diffraction and interference, quantized orbits, quantum corral, and tunneling probability. This thesis sets out to marry these two developments by extending the setup of walker to tackle the stagnant issue of finding a tunnel time that consists with the special relativity. This constraint is guaranteed because the walker can be constantly monitored while "tunneling". Unlike all existing candidates, our measured tunnel time is linear initially, but rises exponentially as the barrier widens. A theoretical model was proposed that is capable of explaining both behaviors. However, the quantity of our data was limited by the difficulties in tuning the temperature, effective acceleration, and amount of liquid to realize the tunnel action. Future work is thus required to come up with the best recipe for tuning the parameters; otherwise, the robustness of our tunnel time and its other quantum-like properties may be called into question.

[1] Yves Couder, Emmanuel Fort, C.-H. Gautier, and Arezki Boudaoud, From Bouncing to Floating: Noncoalescence of Drops on a Fluid Bath, Phys. Rev. Lett. 94, 177801 (2005).
[2] Yves Couder, Suzie Protière, Emmanuel Fort, and Arezki Boudaoud, Walking and orbiting droplets, Nature 437, 208 (2005).
[3] Suzie Protière, Arezki Boudaoud, and Yves Couder, Particle-wave association on a fluid interface, J. Fluid Mech. 554, 85 (2006).
[4] Yves Couder and Emmanuel Fort, Single-Particle Diffraction and Interference at a Macroscopic Scale, Phys. Rev. Lett. 97, 154101 (2006).
[5] Antonin Eddi, Emmanuel Fort, Frederic Moisy, and Yves Couder, Unpredictable Tunneling of a Classical Wave-Particle Association, Phys. Rev. Lett. 102, 240401 (2009).
[6] Daniel M. Harris, Julien Moukhtar, Emmanuel Fort, Yves Couder, and John W. M. Bush, Wavelike statistics from pilot-wave dynamics in a circular corral, Phys. Rev. E 88, 011001(R) (2013).
[7] John W. Bush, Quantum mechanics writ large, PNAS, 107, 17455 (2010).
[8] Antonin Eddi, Julien Moukhtar, Emmanuel Fort, and Yves Couder, Level
Splitting at Macroscopic Scale, Phys. Rev. Lett. 108, 264503 (2012).
[9] David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I, Phys. Rev. 85, 166 (1952).
[10] David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II, Phys. Rev. 85, 180 (1952).
[11] Niels Bohr, The Quantum Postulate and the Recent Development of Atomic Theory, Nature, 121, 580 (1928).
[12] John G. Cramer, The transactional interpretation of quantum mechanics, Rev. Mod. Phys. 58, 647 (1986).
[13] L. A. MacColl, Note on the Transmission and Reflection of Wave Packets by Potential Barriers, Phys. Rev. 40, 621 (1932).
[14] Markus Büttiker and Rolf Landauer, Traversal Time for Tunneling, Phys. Rev. Lett. 49, 23 (1982).
[15] Principles of quantum mechanics. Ramamurti Shankar, Kluwer Academic Plenum Publishers (1994).
[16] Felix T. Smith, Lifetime Matrix in Collision Theory, Phys. Rev. 119, 2098 (1960).
[17] V. Bargmann, Louis Michel, and V. L. Telegdi, Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field, Phys. Rev. Lett. 2, 435 (1959).
[18] Herbert G. Winful, Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox, Physics Reports 436, 1 (2006).
[19] Herbert G. Winful, Delay time and the Hartman effect in quantum tunneling, Phys. Rev. Lett. 91, 260401 (2003)
[20] Markus Büttiker, Larmor precession and the traversal time for tunneling, Phys. Rev. B 27, 6178 (1983).