Yutaka Shikano. Visualizing a Quantum State

Transcription

1 Yutaka Shikano Visualizing a Quantum State

2 Self Introduction

3 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

4 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 )

5 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

6 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

7 現メンバー PD1 人ビジターたくさん技術支援員 2 人

8 最近やっている仕事一覧 Macro Math Phys Micro

9 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

10 物理学 ( 量子力学 ) の基礎を 測定理論を通じて理解したい

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12 観る / 魅せる物理 観ることによって理解は深まる

13 Quantum State

14 Magnet can be taken as the spin Magnetic dipole = classical spin.

15 Problem

16 Stern Gerlach Experiment From Wikipedia Gerlach_experiment.ogv

17 Classical spin Measurement Magnetic field is applied.

18 Stern Gerlach Experiment From Wikipedia Gerlach_experiment.ogv

19 Quantum spin Magnetic dipole of magnet

20 Quantum Spin = Quantum State = Qubit (Quantum Bit)

21 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

22 Example: with The quantum state can be depicted in the sphere of the three dimensional ball!! = Bloch sphere representation

23 Quantum Spin Bloch sphere representation

24 Stern Gerlach Experiment From Wikipedia Gerlach_experiment.ogv

25 Strong Measurement Setting #1

26 Strong Measurement Setting #1 Setting #2

27 Strong Measurement Setting #1 Setting #2 Setting #3

28 Strong measurement must lead to the change of the quantum state. Strong measurement can only tell us the probability of physical quantity.

29 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,

30 Problem

31 Quantum State Tomography

32 State Tomography for Qubit X direction Y direction Z direction Bloch sphere

33 Expectation value for density operator In general, the expectation value is

34 Why? When we know the expectation value of the Pauli operators, we can evaluate the quantum state; mapping to the Bloch sphere.

35 State Tomography for Qubit X direction Y direction Z direction Bloch sphere

36 We need the postinformation processing. Is it possible to demonstrate the quantum state without the postinformation processing?

37 Polarized State Tomography Stereographic projection point w/ Hirokazu Kobayashi, Koji Nonaka (Kochi Tech)

38 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))

39 Weak measurement (Review)

40 The expectation value shifts on position and momentum should be needed. = Post information processing is needed.

41 Our Approach

42 Optical Propagation From the Maxwell equation, Paraxial approximation If u is cylindrically symmetric,

43 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, (2012).)

44 Rough explanation

45 Two dimensional Imaging of the weak value (H. Kobayashi, G. Puentes, and YS, Phys. Rev. A 86, (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.

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47 Optical vortex Streographic Projection Point Stereographic projection point w/ Hirokazu Kobayashi, Koji Nonaka (Kochi Tech)

48 How to implement it? Experiment by Hirokazu Kobayashi (Kochi Tech)

49 Polarized State = Qubit From Wikipedia as/pic/zirkulanim.gif

50 Polarized state is used.

51 Qubit Polarized State

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60 Remapping to Poincare Sphere

61 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, (2014)