Download Testing Heisenberg`s Uncertainty Principle with

Testing Heisenberg’s Uncertainty Principle with
Polarized Single Photons
Sofia Svensson
[email protected]
under the direction of
Prof. Mohamed Bourennane
Quantum Information & Quantum Optics
Department of Physics
Stockholm University
Research Academy for Young Scientists
July 10, 2013
Abstract
One of the fundamentals of quantum mechanics is Heisenberg’s uncertainty principle. The
principle states that two observables cannot be measured accurately at the same time
if their corresponding operators do not commute. The aim of the study was to test the
uncertainty principle with the help of polarized single photons dispatched from a helium
neon laser at a wavelength of 632.8 nm and an experimental setup similar to the one used
in the Stern-Gerlach experiment. The results obtained were within the standard deviation
and consistent with the theoretical results derived from the principle, thereby indicating
the principle’s validity.
Contents
1 Introduction
1
2 Theory
2
2.1
The Stern-Gerlach Experiment . . . . . . . . . . . . . . . . . . . . . . . .
2
2.2
Polarization of Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Heisenberg’s Uncertainty Principle . . . . . . . . . . . . . . . . . . . . .
8
2.4
Quantum Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3 Method
9
3.1
Measurement 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.2
Measurement 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.3
Measurement 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.4
Measurement 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4 Results
12
5 Discussion
14
Acknowledgements
15
A Results
17
1
Introduction
While more than a century old, quantum mechanics still evokes immerse interest in its
promises for future computers, cryptation and materials. An experiment carried out in
1922 by two scientists named Otto Stern and Walther Gerlach, together with a series of
other experiments, have given us insight into one of the fundamental concepts of quantum
mechanics: superpositions. Superposition describes the phenomena, that until a particle
in measured, it is said to be in all possible states at the same time. However, as soon as
the particle is measured the superposition collapses and the particle is thrown into one of
these states, called an eigenstate. The polarization of photons is an example. In the wavedescription of a photon, polarization can be visualized as the way the wave is rotated. A
photon possesses horizontal |Hi or vertical |V i polarization, but until its polarization is
measured, these two states are said to be in a superposition, described by
|Ψi = α |Hi + β |V i
(1)
where |α2 | is the probability of finding the photon in state |Hi and |β 2 | in state |V i [1].
The horizontal and the vertical polarization jointly define a basis denoted by ẑ, which
can take on the values |Hi or |V i. However, the polarization can be described in an
additional basis as well, the x̂ basis, which is shifted 45
◦
in positive direction, see Figure
1.
Figure 1: A graphical representation of the bases ẑ and x̂.
Just like the particle’s superposition consists of |Hi and |V i in ẑ basis, the two new
1
states |H+i and |V +i will describe the particle’s superposition in the x̂ basis as
|Ψi = |H+i + µ |V +i
(2)
where ||2 describes the probability of finding the photon in state |H+i, and |µ|2 is state
|V +i [1].
Heisenberg’s uncertainty principle states that two observables of a particle cannot
be measured accurately at the same time. The principle is often illustrated by the two
observables momentum and position, and says that by improving the accuracy of a measurement of the momentum, one need to sacrifice precision in the measurement of the
particle’s position. If one takes the uncertainty principle to its extreme and measures the
momentum exactly, then all the information gained about the particle’s position will be
lost [2]. In the same way Heisenberg’s uncertainty principle predicts that if one measures
the polarization in the x̂ exactly, all information gained about the polarization in ẑ will
be lost.
The aim of this study is to test Heisenberg’s uncertainty principle by means of changing the polarization of photons. This will be done by measuring the polarization states in
the two described bases and prove that the two measurements have interfered with each
other.
2
2.1
Theory
The Stern-Gerlach Experiment
A well-known experiment in quantum mechanics is the Stern-Gerlach experiment. In
this paper however, it will only be treated as an explanatory model to provide a deeper
understanding of the laws of quantum mechanics. In the experiment a beam of particles
was sent, in the z-direction, through an inhomogeneous magnetic field. This was done
in order to test Bohr’s hypothesis that the direction of the angular momentum of atoms
2
is quantized. Originally, silver atoms were heated in a oven and then sent through a
magnetic field as shown in Figure 2.
Figure
2:
An
illustration
of
the
Stern-Gerlach
experiment.
http://commons.wikimedia.org/wiki/File:Stern-Gerlach_experiment.PNG
Source:
According to classical mechanics, if the direction of the angular momentum was not
quantized, one would observe a small distribution of particles coming out of the SG apparatus (Stern-Gerlach apparatus). Instead, the apparatus split the silver beam into two
different components, showing that particles does possess an intrinsic angular momentum
that only takes certain quantized values. In this experiment, the electrons of the silver
atoms possesses a property called spin which can either be up or down, Sz + and Sz −,
respectively.
As described in the introduction, a particle exists in a superposition before it is measured and the same principle applies for the electron. Before entering the magnetic field,
the electron was in a superposition consisting of Sz + and Sz −. While being in the magnetic field, the electron was measured and thrown into one of the two possible eigenstates,
yielding Sz + and Sz −. Note that the measurement was performed by the magnetic field
in the z-direction, giving the eigenstates in the z-direction.
Let’s now consider a sequential Stern-Gerlach experiment where a beam of particles
is sent through three Stern-Gerlach apparatuses (Figure 3).
Every SG apparatus has a magnetic field in the z-direction. Coming out of the first
apparatus, the beam is split into the two components Sz + and Sz −. Now, letting only the
3
Figure 3: A schematic of a sequential Stern-Gerlach experiment. SGz stands for a SGapparatus with a magnetic field in the z- direction, and SGx stands for a SG-apparatus
with a magnetic field in the x-direction.
Sz + component succeed through the rest of the apparatuses, it will go straight through.
However, in the next measurement, the magnetic field in the second apparatus is exchanged to a magnetic field in the x-direction. After succeeding through the first apparatus, the beam of particles is split. Again, only Sz + component is subjected through the
second apparatus. This time, the beam is split into two components, Sx + and Sx −, the
eigenstates of spin in the x-direction. The Sx − component is the blocked, thus only letting
the Sx + through. In the third apparatus, the Sx + is measured in the z-direction, and
intuitively only the Sz + component would be measured since Sz − was blocked. However,
that is not the case. The Sx + is split into the two components Sz + and Sz − [1]. Hence,
the superposition that first entered the experiment is unaltered, and this demonstrate
that by measure a property exactly, one loses the information of the other property.
Mathematically, each state is described by a linear combination constructed by the
two eigenstates in the other basis as shown below
1
1
|Sx +i = √ |Sx +i + √ |Sz −i
2
2
(3)
1
1
|Sx −i = − √ |Sz +i + √ |Sz −i
2
2
(4)
4
Similarly, |Siz + and |Siz − are described by
2.2
1
1
|Sz +i = √ |Sx +i + √ |Sx −i
2
2
(5)
1
1
|Sz −i = √ |Sx +i − √ |Sx −i
2
2
(6)
Polarization of Photons
An analogy to the intrinsic property spin is the polarization of photons. Since the polarization is just another example of an intrinsic property a particle could exhibit, one
can repeat the Stern-Gerlach experiment with the use of photons. Like the spin of the
electron, one can write a polarization state of one basis as a superposition constituting
the eigenstates of the other basis. The eigenstates obtained in the ẑ basis is given by
1
1
|Hi = √ |H+i + √ |V +i
2
2
(7)
1
1
|V i = √ |H+i − √ |V +i
2
2
(8)
while polarization states in the x̂ basis is described by
1
1
|H+i = √ |Hi + √ |V i
2
2
(9)
1
1
|V +i = √ |Hi − √ |V i
2
2
(10)
Every SG-apparatus is replaced by three half wave plates (HWP) and a polarizing
beam splitter (PBS) in between, see Figure 4. This setup will have the same effect on a
photon as the SG-apparatus had on the electrons since it measures the polarization of
the photon which thereby collapses into one of the eigenstates (in Section 3, these will
5
be denoted by A, B and C). Below follows a mathematical description of the setup in
order to explain the expectation value which will make us able to predict the paths of the
photons throughout the measurements.
PBS
HWP
HWP
θ
θ
θ
HWP
Figure 4: A representation of a setup corresponding to a SG-apparatus.
The HWP rotates the polarization of the transmitted light and thereby shifts between
the two bases. The HWP is described mathematically by the operator


cos 2θ sin 2θ 
R̂ = 

sin 2θ − cos 2θ
(11)
and when set to basis ẑ, θ = 0 ◦ , and when set to x̂, θ = 22.5 ◦ .
The PBS divides the incident beam into two output beams and can mathematically
be described as the quantum operator


1 0 
P̂θ = 

0 −1
(12)
The matrix representing the measurement in the ẑ basis is derived by
σz = R̂0 P̂ R̂0
similarly, the matrix representing the measurement in x̂ basis is derived by
6
(13)
σx = R̂22.5 P̂ R̂22.5
(14)
Though the expressions look alike, note that θ does not take on the same values when
set to measure ẑ and x̂. σz measures the polarization in the z-direction. Hence the photon
will always be thrown into one of the eigenstates |Hi or |V i. When measured in σx , it
will always be thrown into |H+i or Vˆ+. Thus, one can say that the states are eigenstates
to the respective operator. The two operators σz and σx are represented by


1 0 
σz = 

0 −1
(15)


0 1
σx = 

1 0
(16)
By describing the two different measurements with matrices one can calculate the
expectation value of the polarization. The expectation value for these two bases is a
number between 1 and -1, and it describes the distribution between two states constituting
a basis. If the expectation value of σz is 1, then 100% of the photons emerging from the
measurement is in the |Hi state, and if it is -1, then the polarization of the photons is
|V i. However, if the expectation value is 0, then all the particles are in a superposition
consisting of the two states |Hi and |V i, and the chance of finding the photon in each
state is 50% respectively. The same applies for x̂ and the two states |H+i and |V +i. [4]
For example, if |Hi is subjected through a measurement in the ẑ basis, the expectation
value is calculated as

 
 
1 0  1
1
hH| σz |Hi = 1 0 
  = 1 0   = 1
0 −1 0
0
(17)
The expectation value is equal to 1, meaning that with a certainty of 100%, the
photons are |Hi polarized. However, if |Hi polarized photons are subjected through a
7
measurement in the x̂ base, then the expectation value is calculated as

 
 
0 1 1
0
hH| σx |Hi = 1 0 
  = 1 0   = 0
1 0
0
1
(18)
Since the expectation value is equal to 0, the photons exists in superpositions of the
two states and the probability of finding the photon in respective state will be 50% [4].
2.3
Heisenberg’s Uncertainty Principle
The mathematical description of the inherent relation between two properties, which
makes it impossible to measure both exactly at the same time, is denoted commutator.
The commutator of two quantum mechanical operators  and B̂ is defined by
[Â, B̂] = ÂB̂ − B̂ Â
(19)
If the two operators commute, i.e. [Â, B̂] = 0, then it is possible to measure the
properties represented by  and B̂ simultaneously with complete accuracy. However, if
the two operators do not commute, the measurement of one property will change the result
obtained in a previous measurement of the complementary property. This introduces an
uncertainty relationship which is formally described by Heisenberg’s uncertainty principle
[2].
The theory predicts that σz and σx do not commute, since they do not satisfy Eq.19.


 0 1
[σz , σx ] = 2 
 6= 0
−1 0
2.4
(20)
Quantum Cryptography
The quantum properties of photons have been of practical use when it comes to quantum information processing. The idea is to use the laws of quantum mechanics in order
to transfer and manipulate data. Below follows a short description of one of the major
8
branches of quantum information processing: quantum cryptography, in order to demonstrate the importance of Heisenberg’s uncertainty principle.[2]
The purpose of quantum cryptography is to provide a secure way of transmitting a
key with the knowledge that it has not been intercepted along the way. It is fulfilled by
utilizing a postulate of quantum mechanics: the only possible result of a measurement of a
state is one of the eigenstate to the corresponding operator [4]. Meaning, if someone tries
to eavesdrop and measures the state of a photon, the state will change and the photon
will not contain the same quantum information as before, and thus introduce errors.
By analyzing the errors of the results, the two parties who wished to share information
can determine whether or not someone has eavesdropped, and thereby conclude if the key
is secure. The method is called quantum key distribution [2].
3
Method
In order to experimentally test whether the two operators σz and σx commute, four measurement were performed with different setups. Each measurement was repeated with the
four different states of input |Hi , |V i , |H+i and |V +i, respectively. The photons were
counted by a single-photon detector.
The experimental setup consisted of four parts, a preparation part and the three components A, B and C, see Figure 5.
The preparation part consisted of a helium-neon (He-Ne) laser which dispatched photons at a wavelength of 632.8 nm which were sent to a polarizer via an iris, a single
mode fiber (SMF) and an attenuator. The iris decreased the fluctuations due to reflection
back to the laser, while the SMF and the attenuator were utilized to absorb most of the
photons, letting only a few to be subjected. The polarizer prepared the different input
states, |Hi,|V i, |H+i and |V +i. Thereafter, the components A, B and C followed, which
are described in Section 2.2.
9
Figure 5: A schematic of the experimental setup. M stands for mirror, A stands for
attenuator, P for polarizer and D for detector.
3.1
Measurement 1
The aim was to verify that the PBS followed the theory from section 2.2, and divided the
incoming beam. The setup utilized for the measurement is shown in Figure 6.
Figure 6: Similarly to Figure 5, a schematic of the experimental setup utilized in experiment 1.
10
Each of the input states was measured twice, once in each base. The polarizer was set
to 0 ◦ to measure |Hi, 45 ◦ for |H+i, 90 ◦ for |V i and 315 ◦ to measure |V +i. The HWP
were set to 0 ◦ in order to measure in ẑ and 22.5 ◦ in order to measure in x̂.
3.2
Measurement 2
The measurement utilized the two components A and B, viewed in Figure 7, to test if
the two operators σz and σx do not commute. The four input states were measured in all
possible combinations of the bases, ẑ ẑ, ẑ x̂, x̂ẑ and x̂x̂.
Figure 7: Similarly to Figure 5, a schematic of the experimental setup utilized in experiment 2.
3.3
Measurement 3
The third measurement utilized the whole setup viewed in Figure 5 with the purpose to
test whether or not the information gained by the measurement in A was destroyed by
the measurement in B, according to Heisenberg’s uncertainty principle.
Again, the four different states were measured in all the possible combinations of the
bases. For this, only the detectors 1, 3, 5 and 6 were used since the mirrors were flipped
11
down. All the possible combinations were measured to test that the principle applies to
all given states.
3.4
Measurement 4
Theoretical expectation values predict that by changing the HWP gradually, the expectation value will behave as a cosine function. The measurement utilized the same setup
as shown in Figure 7. The HWP in component A were fixed at measuring ẑ, while the
HWPs in component B were first set at 0,◦ and then adjusted to 45,◦ by switching 5,◦
every measurement. This were repeated twice, one with the input state |Hi, the other
with |H+i.
4
Results
The collected data was transformed in order to determine the probability of finding a
photon in a specific state, thus determining |α2 | and |β 2 | if the photon was measured in
the ẑ basis or |2 | and |µ2 | if the photon was measured in the x̂ basis.
In Table 1-3 and Figure 8-9, a representative set of values from the measurements
was chosen to convey the main point of the experiment. However, all results from the
measurements are found in the A Resultat.
Input
|Hi
|Hi
A
z
x
Probability D1
0.004
0.435
Probability D2
0.996
0.565
Table 1: The results obtained in the first measurement.
In Table 1, the results of the first measurement are depicted. When the horizontally
polarized photons were measured in the ẑ basis, 99, 6% was detected at D2 and were
thereby |Hi and 0.04% was detected at D1 and thus measured as being |V i. In the x̂,
56.5% was |H+i and 43.5% was measured as |V +i.
In the second measurement, when measuring in ẑ ẑ, nearly all of the photons were
12
Input
|Hi
|Hi
A
z
z
B
z
x
P. D1
0.003
0.003
P. D3
0.002
0.417
P. D4
0.994
0.58
Table 2: The results obtained in the second measurement.
detected at the fourth detector which measured |Hi, see Table 2. Thereafter, when measuring in ẑ x̂, the detected photons were rather equally distributed between D3 and D4,
which detected |H+i and |V +i.
Input
|Hi
|Hi
A
x
z
B
z
x
C
x
z
P. D1
0.452
0.003
P. D3
0.279
0.47
P. D5
0.115
0.221
P. D6
0.154
0.306
Table 3: The results from the third measurement.
The results from the third measurement, listed in Table 3, show that when |Hi was
measured in the bases x̂ẑ x̂, the |Hi was split in the first component A to 45.2% |V +i and
54.8% |H+i. In the B component, the |H+i was split into 27.9% |V i and 26.9% |Hi.|Hi
was then split in 11.5% |V +i and 15.4%|H+i in the last component.
Figure 8: The fourth measurement with Figure 9: The fourth measurement with |Hi
|H+i as input. The blue marks represent D1, as input. The blue marks represent D1, red
red marks D3 and green marks D4.
marks D3 and green marks D4.
Figures 8 and 9 represent the last measurement. Here the blue marks represent D1
and measures the probability of |V i, red marks D3 and green marks D4. If the HWP in
13
the B component was set to 0 ◦ , then D3 measured probability |V i, and D4 |Hi. When
the HWP was set to 22.5 ◦ D3 measured |V +i and D4 |H+i.
5
Discussion
Table 4, 5 and 6 show the theoretically expected results of each measurement described
above, except measurement 4, derived from the fact that, according to the uncertainty
principle, a measurement in one basis will erase the previously gained information about
the polarization in the other basis.
Input
|Hi
|Hi
A
z
x
Probability D1
0
0.5
Probability D2
1
0.5
Table 4: The theoretically expected results in measurement 1.
Input
|Hi
|Hi
A
z
z
B
z
x
P. D1
0
0
P. D3
0
0.5
P. D4
1
0.5
Table 5: The theoretically expected results in measurement 2.
Input
|Hi
|Hi
A
x
z
B
z
x
C
x
z
P. D1
0.5
0
P. D3
0.25
0.5
P. D5
0.125
0.25
P. D6
0.125
0.25
Table 6: The theoretically expected results in measurement 3.
The results obtained by the measurements were consistent with the theoretically expected results, within one standard derivation. Hence, it can be concluded that the measurements supports the principle of uncertainty. In measurements like this, errors can be
caused by dark counts. These were however taken into account and counted off in the
calculations.
14
Acknowledgements
First of all I want thank to my project partner, Felix Tellander, for all the help he has
provided. Without him, this project would never have succeeded. I would also like to
thank Prof. Mohamed Bourennane from the Quantum Information & Quantum Optics
group at the Department of Physics at Stockholm University for his mentorship and
expertise which has been essential for this research. Many great thanks to Ph.D student
Mohamed Nawareg, who with his patience and skills with the optical instruments have
been to an incredible assistence. A special thanks to my counselor Johannes Orstadius and
to Johan Henriksson who have guided me throughout this project and provided me with
all the support I needed. Finally huge thanks to Mikael Ingemyr and Research Academy
for Young Scientists for giving me this opportunity, and to Volvo and Teknikföretagen
for making Rays possible.
15
References
[1] Sakurai J.J. Modern Quantum Mechanics. Addison-Wesley Publishing Company;
1994.
[2] Fox M. Quantum Optics: an introduction. New York: Oxford University Press; 2006.
[3] Pedrotti F.J. Pedrotti L.S. Pedrotti L.M. Introduktion to Optics. Upper Saddle
River, New Jersey: Pearson Prentice Hall; 2007.
[4] McIntyre D.H. Quantum Mechanics, A Paradigms Approach. Sansome St: Pearson
Education; 2012
[5] Rådmark M. Photonic quantum information and experimental tests of fundations of
quantum mechanics. Stockholm; Universitetsservice US-AB; 2010
16
A
Results
Table 7: All results from measurement 1.
Input A Probability D1 Probability D2
|Hi
z
0.004
0.996
|Hi
x
0.435
0.565
|V i
z
0.998
0.002
|V i
x
0.434
0.566
|H+i z
0.433
0.567
|H+i x
0.002
0.998
|V +i z
0.431
0.569
|V +i x
0.994
0.006
Table 8: All results from
Input A B P. D1
|Hi
z z 0.003
|Hi
z x 0.003
|Hi
x z 0.453
|Hi
x x 0.418
|V i
z z 0.998
|V i
z x 0.996
|V i
x z 0.436
|V i
x x 0.414
|H+i z z 0.413
|H+i z x 0.447
|H+i x z 0.003
|H+i x x 0.002
|V +i z z 0.411
|V +i z x 0.481
|V +i x z 0.994
|V +i x x
0.99
17
measurement 2.
P. D3 P. D4
0.002
0.994
0.417
0.58
0.227
0.319
0.002
0.581
0.001
0.001
0.001
0.003
0.234
0.33
0.002
0.584
0.001
0.586
0.23
0.319
0.416
0.581
0.002
0.995
0.004
0.585
0.219
0.301
0
0.007
0.004
0.006
Table
Input
|Hi
|Hi
|Hi
|Hi
|Hi
|Hi
|Hi
|Hi
|V i
|V i
|V i
|V i
|V i
|V i
|V i
|V i
|H+i
|H+i
|H+i
|H+i
|H+i
|H+i
|H+i
|H+i
|V +i
|V +i
|V +i
|V +i
|V +i
|V +i
|V +i
|V +i
9: The theoretical results in measurement 3.
A B C P. D1 P. D3 P. D5 P. D6
x x x 0.411
0.006
0.007
0.576
x x z 0.426
0.013
0.238
0.325
x z x 0.452
0.279
0.115
0.154
x z z 0.424
0.273
0.008
0.295
z x x 0.003
0.48
0.006
0.511
z x z 0.003
0.47
0.221
0.306
z z x 0.005
0.007
0.412
0.577
z z z 0.004
0.005
0.005
0.986
x x x 0.402
0.011
0.01
0.577
x x z 0.425
0.012
0.238
0.324
x z x 0.436
0.29
0.118
0.156
x z z 0.407
0.285
0.003
0.304
z x x 0.995
0.002
0.002
0
z x z 0.994
0.002
0.002
0.001
z z x 0.993
0.003
0.002
0.002
z z z 0.994
0.003
0.002
0.001
x x x 0.005
0.012
0.013
0.969
x x z 0.003
0.018
0.411
0.568
x z x 0.003
0.506
0.208
0.284
x z z 0.005
0.471
0.011
0.513
z x x 0.319
0.325
0.004
0.352
z x z 0.446
0.26
0.125
0.17
z z x 0.507
0.004
0.206
0.283
z z z 0.411
0.004
0.004
0.582
x x x 0.996
0.016
0.015
0.003
x x z 0.965
0.016
0.013
0.006
x z x 0.971
0.009
0.011
0.009
x z z 0.967
0.011
0.014
0.007
z x x 0.401
0.284
0.012
0.298
z x x 0.431
0.273
0.125
0.171
z z x 0.425
0.015
0.23
0.33
z z z 0.393
0.011
0.009
0.587
18