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 Photophysics in Semiconductor Math Fractal & Localization Nitrogen Vacancy Center in Diamond Phys Quantum Walks Micro 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 / 魅せる物理 観ることによって理解は深まる
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 postinformation processing. Is it possible to demonstrate the quantum state without the postinformation 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)