Quantum Physics Mach-Zehnder

Quantum Physics Mach-Zehnder Interferometer Info State 1 P State 0 P Beam splitter P Quantum Particle Two possible states: 1 or 0 (polarization, spin, …) Beam splitter A State 1 P Detection of the state by a beam splitter Beam splitter B P Illustrates the two possible instates by two different inpath A and B State 0

Quantum Physics Mach-Zehnder Interferometer Double Beam Splitter P Particle S1 – S4 Half-silvered mirror S2 – S3 Fully silvered mirror Y p34 X S3 A particle P is coming in path A or B. At the half-silvered mirror S1 it’s 50/50 percent chance that the particle will go through the mirror and travel the path p12 or be reflected and travel the path p13. The mirrors S2 and S3 are fully silvered so the particle is reflected and travel the path p24 or p34. At the half-silvered mirror S4 or be reflected and travel the path X or Y. S4 p13 p24 S1 P S2 A p12 B

Quantum Physics Mach-Zehnder Interferometer Classical Particle An experiment with a classical particle P. At the moment we have the following situation: 19 particles have travelled the path p12 – p24. 18 particles have travelled the path p13 – p34. 17 particles have travelled the path X. 19 particles have travelled the path Y. Ordinary statistical theory tells us that there will be 50/50 percent of particles travelling the path p12 - p24 or p13 - p34. There will be 50/50 percent of particles travelling the path X or Y. Y p34 X P p13 p24 A p12 B 0.25 p12 X 0.5 0.5 0.25 Y 0.5 A 0.25 X 0.5 p13 0.25 Y

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Result An experiment with a quantum particle P. At the moment we have the following situation: 34 particles have been travelling out the path X. 0 particles have been travelling out the path Y. Quantum theory tells us the following: If the quantum particle is starting in the path A, then every particle will be travelling the out-path X. If the quantum particle is staring in the path B, then every particle will be travelling the out-path Y. Y P X A P B This result is very surprising compared to classical physics. How to explain this?

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Measurement An experiment with a quantum particle P. Now we have a measuring instrument to detect which path the quantum particle is travelling. At the moment we have the following situation: 22 particles have been travelling the out-path X. 22 particles have been travelling the out-path Y. Quantum theory tells us the following: If we have a measuring instrument (either in only one or both path) to detect which path the quantum particle is travelling, then the detection ’disturbs’ the quantum effect in such a way that now we will have an equal number of particles travelling in path X or Y. Y P X A B Measuring instrument How to explain this?

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Two orthonormal states An experiment with a quantum particle P. There are two possible initial states u1 and u2 for the particle P dependent of the in-path A or B. Let these two possible instates be: A: u1 = [1,0] B: u2 = [0,1] These two states are orthonormal. A P B

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Path A - State after 1 beam splitter An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in the state u1 or state u2. The Hadamard matrix (operator) is shown in the figure. Operator Hadamard matrix

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Path A - State after 1 beam splitter - Reality / Mathematical Space Reality Mathematical Space P P P

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Path A - State after 2 beam splitters An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in the state u1 or state u2. The Hadamard matrix (operator) is shown in the figure. Y P P X Operator Hadamard matrix P A B

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Path A - State after 2 beam splitters - Reality / Mathematical Space Reality Mathematical Space

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Path B - State after 1 beam splitter An experiment with a quantum particle P. The particle P starts in the state u2. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in the state u1 or state u2. The Hadamard matrix (operator) is shown in the figure. Operator Hadamard matrix A B

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Path B - State after 1 beam splitter - Reality / Mathematical Space Reality Mathematical Space P P P

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Path B - State after 2 beam splitters An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in the state u1 or state u2. The Hadamard matrix (operator) is shown in the figure. Y P P X Operator Hadamard matrix P A B

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State B - State after 2 beam splitters - Reality / Mathematical Space Reality Mathematical Space

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Hadamard Operator Beam splitter 1 Beam splitter 2 Hadamard operator rotates the state vector 450 counterclockwise

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Detector An experiment with a quantum particle P. We have one or two detectors to detect the travelling path p12 or p13 of the particle. p13 A p12 P B Detector

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State A - Approaching the detector(s) An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to detect the particle either in the path p12 or p13. The particle is approaching the detector(s). Operator Hadamard matrix p13 A p12 B Detector

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State A - Particle is detected in path p12 An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to detect the particle in the path p12 or p|3. Now the particle is detected in the path p12. The detection of the particle force the particle into one of the eigenstates (here u2) of the detection operator P. p34 p13 p24 A p12 B Detector

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State A - Particle deteced in path p12 approaching beam splitter S2 An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in the path p12 or path13. The detector has detected the particle in path p12. The particle is now in state u2 and approaches the second beam splitter S4. p34 p13 p24 A p12 B Detector

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State A - Particle detected in path p12 passing beam splitter S4 An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure The particle has been detected in path p12 and forced into state u2. After passing the second beam splitter it’s equalt probability to detect the particle in path X or path Y. Y p34 X p13 p24 A p12 B Detector

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State A - Detector in path p12, but no detection there An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in path p12 or p13. Detector in path p12, but no detection there. Anyway the detector change the state and the particle is forced into one of the eigenstate of detection operator P (here u1). p34 p13 p24 A p12 B Detector

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State A - Detector in path p12 ,no detection there - Approaching beam splitter S4 An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in path p12 or p13. Detector in path p12, but no detection there. Anyway the detector change the state and the particle is forced into one of the eigenstate of detection operator P (here u1). The particle is approaching the second beam splitter S4. P A B Detector

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State A - Detector in path p12, no detection there - Passing beam splitter S4 An experiment with a quantum particle P. The particle P starts in the state u1. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in the path p12 or p13. Detector in path p12, but no detection there. Particle is forced in into state u1 and equal probability in path X og Y after second beam splitter. Y X A B Detector

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - State A - Approaching the detector(s) An experiment with a quantum particle P. The particle P starts in the state u1 or u2. The beam splitter is represented mathematically by an operator called the Hadamard matrix. After the beam splitter (mirror) we have equal probability to measure the particle either in the state u1 or state u2. The particle is approaching the detector(s). Operator Hadamard matrix p13 A p12 B Detector There is no possibility to decide if the particle is coming from A or B using a detector in the path p12 or p13 after the first beam splitter S1. We have to let the particle be undisturbed until passing the second beam splitter S4. Equal probability for detecting particle in path p12 or p13 independent of particle in-path A or B.

Quantum Physics Mach-Zehnder Interferometer Quantum Particle - Conclusion X X S4 S4 Operator Hadamard matrix P P S1 S1 A A P B B P Let the particle be undisturbed between beam splitter S1 and S2. Detect the particle after beam splitter S2. The particle out-path is X if in-path is A. The particle out-path is Y if in-path is B.