Yutaka Shikano Visualizing a Quantum State Self‐Introduction http://qm.ims.ac.jp 東京工業大学（大岡山キャンパス） 学部４年間、修士課程、博士課程と通ったところです。 宇宙物理学理論研究室というところにいました。 マサチューセッツ工科大学 博士課程の際に、機械工学科に留学していました。（１年半） チャップマン大学 ロサンゼルス（Disney Landの近く）。ここで客員助教授をやっています。 ２０１２年２月から。 任期５年あるのであ れば、まずＭ１・Ｂ４ に戻ってみよう。 現在、２年半経ちまし た。残り２年半。 6 現メンバー ＰＤ１人 ビジターたくさん 技術支援員 ２人 最近やっている仕事一覧 Macro Math Phys Micro 最近やっている仕事？ Macro Brownian motion Thermodynamics Nitrogen Vacancy Center in Diamond Fractal & Localization Math Quantum Walks Photophysics in Semiconductor Micro Phys Optomechanical System 物理学（量子力学）の基礎を 測定理論を通じて理解したい。 観ることによって 理解は深まる。 観る/ 魅せる物理 Quantum State Magnet can be taken as the “spin” Magnetic dipole = “classical spin”. Problem Stern‐Gerlach Experiment From Wikipedia http://en.wikipedia.org/wiki/File:Quantum_spin_and_the_Stern‐Gerlach_experiment.ogv Classical spin Measurement Magnetic field is applied. Stern‐Gerlach Experiment From Wikipedia http://en.wikipedia.org/wiki/File:Quantum_spin_and_the_Stern‐Gerlach_experiment.ogv Quantum spin ≠ Magnetic dipole of magnet Quantum Spin = Quantum State = Qubit (Quantum Bit) Density Operator = Quantum State Quantum state is written in the vector form. matrix As the vector can be characterized by the complete orthogonal basis, the matrix can also be. Two‐by‐two matrix is given by with Example: with The quantum state can be depicted in the sphere of the three‐dimensional ball!! = Bloch sphere representation Quantum Spin Bloch sphere representation Stern‐Gerlach Experiment From Wikipedia http://en.wikipedia.org/wiki/File:Quantum_spin_and_the_Stern‐Gerlach_experiment.ogv Strong Measurement Setting #1 Strong Measurement Setting #1 Setting #2 Strong Measurement Setting #1 Setting #2 Setting #3 Strong measurement must lead to the change of the quantum state. Strong measurement can only tell us the probability of physical quantity. Born Rule for Density Operator Probability of quantum measurement is given by Born rule : Projection operator When the quantum state is measured by the direction of Since the quantum state can be transformed to the density operator form, Problem Quantum State Tomography State Tomography for Qubit X direction Y direction Z direction Bloch sphere Expectation value for density operator In general, the expectation value is Why? When we know the expectation value of the Pauli operators, we can evaluate the quantum state; mapping to the Bloch sphere. State Tomography for Qubit X direction Y direction Z direction Bloch sphere We need the post‐ information processing. Is it possible to demonstrate the quantum state without the post‐ information processing? Polarized State Tomography Stereographic projection point w/ Hirokazu Kobayashi, Koji Nonaka (Kochi Tech) Definition of Weak Values Def: Weak values of observable A pre-selected state post-selected state To measure the weak value… Weak Measurement (Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)) Weak measurement (Review) The expectation value shifts on position and momentum should be needed. = Post‐information processing is needed. Our Approach Optical Propagation From the Maxwell equation, Paraxial approximation If u is cylindrically symmetric, Review of Lagurre‐Gauss mode l = 0 ‐> Gaussian mode Holoeye LC 2012 How to create it? • Spatially phase modulation • Reflect the conical mirror (See Optics Express 20, 14064 (2012).) Rough explanation Two‐dimensional Imaging of the weak value (H. Kobayashi, G. Puentes, and YS, Phys. Rev. A 86, 053805 (2012)) • Interaction Hamiltonian Up to the first order calculation We can evaluate the weak values from the imaging data. • General calculations (up to second‐order calculation) were done. Optical vortex ∝ Streographic Projection Point Stereographic projection point w/ Hirokazu Kobayashi, Koji Nonaka (Kochi Tech) How to implement it? Experiment by Hirokazu Kobayashi (Kochi Tech) Polarized State = Qubit From Wikipedia http://www.radartutorial.eu/06.antenn as/pic/zirkulanim.gif Polarized state is used. Qubit Polarized State Remapping to Poincare Sphere Summary and Future works • We proposed the weak measurement with LG modes. • We experimentally demonstrated the visualization of a quantum state from the imaging data under the specific configuration via the above proposed weak measurement. Phys. Rev. A 89, 053816 (2014)
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