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Quantum PhysicsMach-Zehnder
Quantum PhysicsMach-Zehnder
Quantum PhysicsMach-Zehnder InterferometerInfo
Quantum PhysicsMach-Zehnder InterferometerInfo
P Quantum Particle
Two possible states:1 or 0 (polarization, spin, …)
Detection of the state by a beam splitter
State 1
State 0
P
P
Beam splitter
Beam splitter
State 1
State 0
Beam splitter
P
P
Illustrates the two possible instatesby two different inpath A and B
A
B
Quantum PhysicsMach-Zehnder InterferometerDouble Beam Splitter
Quantum PhysicsMach-Zehnder InterferometerDouble Beam Splitter
PS1
S4
S2
S3
A
B
X
Y
P ParticleS1 – S4 Half-silvered mirrorS2 – S3 Fully silvered mirror
p12
p13 p24
p34
A particle P is coming in path A or B.
At the half-silvered mirror S1
it’s 50/50 percent chancethat the particle will go through the mirrorand travel the path p12
or be reflected and travel the path p13.
The mirrors S2 and S3 are fully silveredso the particle is reflectedand travel the path p24 or p34.
At the half-silvered mirror S4
it’s 50/50 percent chancethat the particle will go through the mirroror be reflected and travel the path X or Y.
Quantum PhysicsMach-Zehnder InterferometerClassical Particle
Quantum PhysicsMach-Zehnder InterferometerClassical 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 usthat 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.
X
Y
p12
p13 p24
p34
P
A
B
A
p12
p13
X
X
Y
Y
0.5
0.5
0.25
0.25
0.25
0.25
0.5
0.5
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Result
Quantum PhysicsMach-Zehnder InterferometerQuantum 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.
X
Y
P
A
B This result is very surprisingcompared to classical physics.
How to explain this?
P
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Measurement
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Measurement
An experiment with a quantum particle P.
Now we have a measuring instrumentto detect which path the quantum particleis 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 detectwhich path the quantum particle is travelling,then the detection ’disturbs’ the quantum effectin such a way that now we will have an equalnumber of particles travelling in path X or Y.
P
A
B
X
Y
Measuring instrument
How to explain this?
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Two orthonormal states
Quantum PhysicsMach-Zehnder InterferometerQuantum 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
BP
ijji uu
uuuu
uuuu
uu
00
110 0
1
001
11
010 1
0
101
1
02
0
1
1221
2211
1
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path A - State after 1 beam splitter
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe particle either in the state u1 or state u2.The Hadamard matrix (operator)is shown in the figure.
5.0
5.0
1
05.0
0
15.0
5.05.0 21
2
11
2
11
uu
uvuucvi
iii
ii
11
115.0H
11 uHv
5.0
5.0
0
1
11
115.011 uHv
OperatorHadamard matrix
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path A - State after 1 beam splitter - Reality / Mathematical Space
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path A - State after 1 beam splitter - Reality / Mathematical Space
P
P
P
0
11u
1u
2u
H
1u
5.0
5.0
0
1
11
115.011 uHv
5.0
5.0
1
05.0
0
15.0
5.05.0 21
2
11
2
11
uu
uvuucvi
iii
ii
Reality Mathematical Space
2u
1u
5.0
5.0
1v
0
11u
1v
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path A - State after 2 beam splitters
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe particle either in the state u1 or state u2.The Hadamard matrix (operator)is shown in the figure.
2
11
1
0
5.0
5.0
11
115.0
u
vHw
2
12
11
1
0
0
1
02
205.0
0
1
11
115.0
11
115.0
u
uHvHw
11
115.0H
11 uHv OperatorHadamard matrix
11 vHw
P
P
A
B
X
Y
P
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path A - State after 2 beam splitters - Reality / Mathematical Space
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path A - State after 2 beam splitters - Reality / Mathematical Space
2u
H
1u
Reality Mathematical Space
2u
1u
5.0
5.0
1v
5.0
5.011 uHv
1v
1w
1w 2
11
1
0
5.0
5.0
11
115.0 u
vHw
2
12
11
1
0
0
1
22
005.0
0
1
11
115.0
11
115.0
u
uHvHw
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path B - State after 1 beam splitter
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe particle either in the state u1 or state u2.The Hadamard matrix (operator)is shown in the figure.
5.0
5.0
1
05.0
0
15.0
5.05.0 21
2
12
2
12
uu
uvuucvi
iii
ii
11
115.0H
22 uHv
5.0
5.0
1
0
11
115.022 uHv
OperatorHadamard matrix
A
B
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path B - State after 1 beam splitter - Reality / Mathematical Space
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path B - State after 1 beam splitter - Reality / Mathematical Space
P
P
P
0
11u
2u2u
H
1u
5.0
5.0
1
0
11
115.022 uHv
5.0
5.0
1
05.0
0
15.0
5.05.0 21
2
11
2
12
uu
uvuucvi
iii
ii
Reality Mathematical Space
2u
1u
5.0
5.0
2v
1
02u
2v
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Path B - State after 2 beam splitters
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe particle either in the state u1 or state u2.The Hadamard matrix (operator)is shown in the figure.
1
22
0
1
5.0
5.0
11
115.0
u
vHw
1
22
22
0
1
1
0
02
205.0
1
0
11
115.0
11
115.0
u
uHvHw
11
115.0H
22 uHv OperatorHadamard matrix
22 vHw
P
P
A
B
X
Y
P
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State B - State after 2 beam splitters - Reality / Mathematical Space
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State B - State after 2 beam splitters - Reality / Mathematical Space
2u
H
1u
Reality Mathematical Space
2u
1u
5.0
5.0
2v
5.0
5.022 uHv
2v
2w
1w
1
22
0
1
5.0
5.0
11
115.0 u
vHw
2
22
22
1
0
1
0
22
005.0
1
0
11
115.0
11
115.0
u
uHvHw
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Hadamard Operator
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Hadamard Operator
1u
2u
H1u
2u
1u
5.0
5.0
5.0
5.01v 2u
1u
21 uw
2u2u
1u
2u
1u
5.0
5.0
5.0
5.02v
2u
1u
12 uw H
H
H
Hadamard operator rotates the state vector 450 counterclockwise
11
115.0H
Beam splitter 1 Beam splitter 2
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Detector
Quantum PhysicsMach-Zehnder InterferometerQuantum 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.
ijji uu
uuuu
uuuu
uu
00
110 0
1
001
11
010 1
0
101
1
02
0
1
1221
2211
1
A
BP
Detector
p12
p13
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Approaching the detector(s)
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to detectthe particle either in the path p12 or p13.The particle is approaching the detector(s).
5.0
5.0
1
05.0
0
15.0
5.05.0 21
2
11
2
11
uu
uvuucvi
iii
ii
5.0
5.0
0
1
11
115.011 uHv
11
115.0H
11 uHv
OperatorHadamard matrix
A
B Detector
p12
p13
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Particle is detected in path p12
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to detectthe 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.
1
021 uvP
A
B
p12
p13
Detector
p24
p34
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Particle deteced in path p12 approaching beam splitter S2
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe 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.
A
B
p12
p13
Detector
p24
p34
1
021 uvP
1
02u
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Particle detected in path p12 passing beam splitter S4
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measureThe 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.
A
B Detector
p12
p13 p24
p34
X
Y
1
02u
5.0
5.0
1
0
11
115.022 uHw
5.0
5.0
1
05.0
0
15.0
5.05.0 21
2
12
2
12
uu
uwuucwi
iii
ii
5.0
5.022 uHw
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Detector in path p12, but no detection there
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe 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).
A
B
p12
p13 p24
p34
Detector
0
111 uvP
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Detector in path p12 ,no detection there - Approaching beam splitter S4
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Detector in path p12 ,no detection there - Approaching beam splitter S4
A
B Detector
An experiment with a quantum particle P.
The particle P starts in the state u1.The beam splitter is represented mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe 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.
0
111 uvP
0
11u
P
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Detector in path p12, no detection there - Passing beam splitter S4
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe 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 Yafter second beam splitter.
5.0
5.0
1
05.0
0
15.0
5.05.0 21
2
11
2
11
uu
uwuucwi
iii
ii
5.0
5.0
0
1
11
115.011 uHw
A
B Detector
X
Y
5.0
5.011 uHw
0
11u
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - State A - Approaching the detector(s)
Quantum PhysicsMach-Zehnder InterferometerQuantum 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 mathematicallyby an operator called the Hadamard matrix.
After the beam splitter (mirror)we have equal probability to measurethe particle either in the state u1 or state u2.The particle is approaching the detector(s).
5.0
5.0
0
1
11
115.011 uHv
11
115.0H11 uHv
OperatorHadamard matrix
A
B Detector
22 uHv
5.0
5.0
1
0
11
115.022 uHv
There is no possibility to decide if the particle is coming from A or B using a detectorin the path p12 or p13 after the first beam splitter S1.We have to let the particle be undisturbeduntil passing the second beam splitter S4.
Equal probabilityfor detecting particle in path p12 or p13
independent of particle in-path A or B.
p12
p13
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Conclusion
Quantum PhysicsMach-Zehnder InterferometerQuantum Particle - Conclusion
5.0
5.0
11 uHv
Let the particle be undisturbed between beam splitter S1 and S2. Detect the particle after beam splitter S2.
P
P
A
B
X
Y
P
5.0
5.0
22 uHv
P
P
A
B
X
Y
P
211 uvHw 122 uvHw
P
P
OperatorHadamard matrix
11
115.0HS1
S4
S1
S4
The particle out-path is X if in-path is A. The particle out-path is Y if in-path is B.
