Quantum Physics
Mach-Zehnder
Quantum Physics
Mach-Zehnder Interferometer
Info
State 1
State 0
P
P
Beam splitter
Beam splitter
A
State 1
P Quantum Particle
Two possible states:
1 or 0
(polarization, spin, …)
P
Detection of the state by a beam splitter
Beam splitter
B
P
State 0
Illustrates the two possible instates
by two different inpath A and B
Quantum Physics
Mach-Zehnder Interferometer
Double Beam Splitter
P
S 1 – S4
S 2 – S3
Y
p34
X
S3
A particle P is coming in path A or B.
S4
p13
p24
S1
P
S2
A
B
p12
Particle
Half-silvered mirror
Fully silvered mirror
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
it’s 50/50 percent chance
that the particle will go through the mirror
or be reflected and travel the path X or Y.
Quantum Physics
Mach-Zehnder Interferometer
Classical Particle
An experiment with a classical particle P.
Y
At the moment we have the following situation:
p34
19 particles have travelled the path p12 – p24.
18 particles have travelled the path p13 – p34.
X
P
p13
17 particles have travelled the path X.
19 particles have travelled the path Y.
p24
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.
A
p12
B
p12
0.25
X
0.5
0.25
Y
0.5
0.25
X
0.25
Y
0.5
A
0.5
p13
Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - Result
An experiment with a quantum particle P.
Y
At the moment we have the following situation:
P
X
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.
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.
Y
P
X
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:
A
B
Measuring instrument
How to explain this?
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.
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.
P A
B
1
u1   
0 
0 
u2   
1
1
u1 u1  1 0   1
0 
0 
u1 u2  1 0   0
1
0 
u2 u2  0 1   1
1
1
u2 u1  0 1   0
0 
ui u j   ij
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.
Operator
Hadamard matrix
v1  H u1
1  1
H  0.5 

1 1 
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.
2
2
i 1
i 1
v1   ci ui   ui v1 ui
 0.5 u1  0.5 u2
1  1 1  0.5 
v1  H u1  0.5 





1
1
0
0
.
5

  

1
0  0.5 
 0.5    0.5    

0 
1  0.5 
Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - Path A - State after 1 beam splitter - Reality / Mathematical Space
Reality
Mathematical Space
1
u1   
0
u2
u1
1
u1   
0
u1
P
H
u2
P
0.5
v1
P
1  1 1  0.5 
v1  H u1  0.5 





1
1
0
0
.
5

  

v1
0.5 u1
2
2
i 1
i 1
v1   ci ui   ui v1 ui
 0.5 u1  0.5 u2
1
0  0.5 
 0.5    0.5    

0
1
0
.
5
 
  

Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - Path A - State after 2 beam splitters
An experiment with a quantum particle P.
w1  H v1
Y
P
P
X
Operator
Hadamard matrix
v1  H u1
P
1  1
H  0.5 

1 1 
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.
w1  H v1
A
B
1  1
1  1 1
w1  H v1  H 2 u1  0.5 
0
.
5

1 1  0
1 1 

 
0  2 1 0
 0.5
  0   1   u 2
2
0

   
1  1  0.5 
 0 .5 


1
1
0
.
5



0 
    u2
1 
Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - Path A - State after 2 beam splitters - Reality / Mathematical Space
Reality
Mathematical Space
u2
v1
0.5
v1
0.5
u1
H
w1
 0 .5 
v1  H u1  

0
.
5


w1  H v1
1  1  0.5  0
 0.5 

     u2

1 1   0.5  1
u2
w1
u1
1  1
1  1 1
w1  H v1  H 2 u1  0.5 
0
.
5

1 1  0
1 1 

 
0 0  1 0
 0.5
  0   1   u 2
2

2

   
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.
Operator
Hadamard matrix
v2  H u2
1  1
H  0.5 

1 1 
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.
2
2
i 1
i 1
v2   ci ui   ui v2 ui
A
B
  0.5 u1  0.5 u2
v2  H u 2
1  1 0  0.5 
 0.5 





1
1
1
0
.
5

  

1
0  0.5 
  0.5    0.5    

0 
1  0.5 
Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - Path B - State after 1 beam splitter - Reality / Mathematical Space
Reality
Mathematical Space
1
u1   
0
u2
0
u2   
1
u2
u1
P
H
1  1 0  0.5 
v2  H u2  0.5 





1
1
1
0
.
5

  

u2
P
v2
v2
P
 0.5
2
2
i 1
i 1
v2   ci ui   ui v1 ui
0.5
u1
  0.5 u1  0.5 u2
1
0  0.5 
  0.5    0.5    

0
1
0
.
5
 
  

Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - Path B - State after 2 beam splitters
An experiment with a quantum particle P.
w2  H v2
Y
P
P
X
Operator
Hadamard matrix
v2  H u2
P
1  1
H  0.5 

1 1 
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.
w2  H v2
A
B
1  1
1  1 0
w2  H v2  H 2 u2  0.5 
0
.
5

1 1  1
1 1 

 
0  2 0  1
 0.5
 1   0    u1
2
0

   
1  1  0.5 
 0 .5 


1
1
0
.
5



 1
     u1
0
Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - State B - State after 2 beam splitters - Reality / Mathematical Space
Reality
Mathematical Space
u2
v2
v2
 0.5 
v2  H u 2  

0
.
5


0.5
 0.5
u1
H
w2
w2  H v2
1  1  0.5   1
 0.5 

      u1

1 1   0.5   0 
u2
w1
u1
1  1
1  1 0
w2  H v2  H 2 u2  0.5 
0
.
5

1 1  1
1 1 

 
 0 0  0   0 
 0.5
 1   1   u2
2

2

   
Quantum Physics
1  1
H  0.5 

1 1 
Mach-Zehnder Interferometer
Quantum Particle - Hadamard Operator
Beam splitter 1
Beam splitter 2
u2
u2
0.5
u1
 0.5 
v1  

0
.
5


H
u2
H
u1
0.5 u1
u1
u2
u2
u2
u2
0.5
 0.5 
v2  

0
.
5


H
u1
w1  u2
 0.5
H w2   u1
u1
Hadamard operator rotates the state vector 450 counterclockwise
u1
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
P A
p12
B
Detector
1
u1   
0 
0 
u2   
1
1
u1 u1  1 0   1
0 
0 
u1 u2  1 0   0
1
0 
u2 u2  0 1   1
1
1
u2 u1  0 1   0
0 
ui u j   ij
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.
Operator
Hadamard matrix
p13
v1  H u1
A
1  1
H  0.5 

1 1 
2
2
i 1
i 1
v1   ci ui   ui v1 ui
p12
B
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).
Detector
1  1 1  0.5 
v1  H u1  0.5 





1
1
0
0
.
5

  

 0.5 u1  0.5 u2
1
0  0.5 
 0.5    0.5    

0 
1  0.5 
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.
p34
p13
A
p24
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.
p12
B
Detector
0
P v1  u2   
1
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.
p34
p13
A
0
u2   
1
p24
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.
p12
B
Detector
0
P v1  u2   
1
Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - State A - Particle detected in path p12 passing beam splitter S4
Y
w2  H u2
p34
 0.5 


 0.5 
X
0
u2   
1
p13
A
p24
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.
2
2
i 1
i 1
w2   ci ui   ui w2 ui
p12
B
An experiment with a quantum particle P.
Detector
1  1 0  0.5 
w2  H u2  0.5 





1
1
1
0
.
5

  

  0.5 u1  0.5 u2
1
0  0.5 
  0.5    0.5    

0 
1  0.5 
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.
p34
p13
A
p24
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).
p12
B
Detector
1
P v1  u1   
0
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.
P
1
u1   
0
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.
A
B
Detector
1
P v1  u1   
0
Quantum Physics
Mach-Zehnder Interferometer
Quantum Particle - State A - Detector in path p12, no detection there - Passing beam splitter S4
Y
1
u1   
0
 0.5 
w1  H u1  

0
.
5


X
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.
2
2
i 1
i 1
w1   ci ui   ui w1 ui
A
B
Detector
1  1 1  0.5 
w1  H u1  0.5 





1
1
0
0
.
5

  

 0.5 u1  0.5 u2
1
0  0.5 
 0.5    0.5    

0 
1  0.5 
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.
p13
Operator
Hadamard matrix
v1  H u1
v2  H u2
A
1  1
H  0.5 

1 1 
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.
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).
1  1 1  0.5 
v1  H u1  0.5 





1
1
0
0
.
5

  

1  1 0  0.5 
v2  H u2  0.5 

 1  
1
1

    0.5 
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
w1  H v1  u2
w2  H v2   u1
Y
Y
P
P
P
X
v1  H u1
 0.5 


0
.
5


X
S4
v2  H u 2
P
 0.5 


 0.5 
S1
S1
A
P
P
B
S4
P
Operator
Hadamard matrix
1  1
H  0.5 

1 1 
A
P
B
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.