Testing Heisenberg's Uncertainty Principle with Polarized Single

Testing Heisenberg’s Uncertainty Principle with Polarized Single Photons

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 wave- description 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

Figure 1: A graphical representation of the bases zˆ and xˆ.

Just like the particle’s superposition consists of |Hi and |V i in zˆ basis, the two new

1 states |H+i and |V +i will describe the particle’s superposition in the xˆ 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 xˆ exactly, all information gained about the polarization in zˆ 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 Theory

2.1 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. Source: http://commons.wikimedia.org/wiki/File:Stern-Gerlach_experiment.PNG

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 SG- apparatus 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 ex- changed 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 (3) 2 2

1 1 |Sx−i = −√ |Sz+i + √ |Sz−i (4) 2 2

4 Similarly, |Siz + and |Siz − are described by

1 1 |Sz+i = √ |Sx+i + √ |Sx−i (5) 2 2

1 1 |Sz−i = √ |Sx+i − √ |Sx−i (6) 2 2

2.2 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 zˆ basis is given by

1 1 |Hi = √ |H+i + √ |V +i (7) 2 2

1 1 |V i = √ |H+i − √ |V +i (8) 2 2

while polarization states in the xˆ basis is described by

1 1 |H+i = √ |Hi + √ |V i (9) 2 2

1 1 |V +i = √ |Hi − √ |V i (10) 2 2

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.

θ θ

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 ˆ   R =   (11) sin 2θ − cos 2θ and when set to basis zˆ, θ = 0 ◦, and when set to xˆ, θ = 22.5 ◦. The PBS divides the incident beam into two output beams and can mathematically be described as the quantum operator

  1 0 ˆ   Pθ =   (12) 0 −1

The matrix representing the measurement in the zˆ basis is derived by

ˆ ˆ ˆ σz = R0P R0 (13) similarly, the matrix representing the measurement in xˆ basis is derived by

6 ˆ ˆ ˆ σx = R22.5P R22.5 (14)

Though the expressions look alike, note that θ does not take on the same values when set to measure zˆ and xˆ. 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 =   (15) 0 −1

  0 1   σx =   (16) 1 0

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 xˆ and the two states |H+i and |V +i. [4] For example, if |Hi is subjected through a measurement in the zˆ basis, the expectation value is calculated as

      1 0 1 1       hH| σz |Hi = 1 0     = 1 0   = 1 (17) 0 −1 0 0

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 xˆ base, then the expectation value is calculated as

      0 1 1 0       hH| σx |Hi = 1 0     = 1 0   = 0 (18) 1 0 0 1

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 Aˆ and Bˆ is deﬁned by

[A,ˆ Bˆ] = AˆBˆ − BˆAˆ (19)

If the two operators commute, i.e. [A,ˆ Bˆ] = 0, then it is possible to measure the properties represented by Aˆ and Bˆ 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 (20) −1 0

2.4 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 zˆ and 22.5 ◦ in order to measure in xˆ.

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, zˆzˆ, zˆxˆ, xˆzˆ and xˆxˆ.

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 zˆ, 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 |β2| if the photon was measured in the zˆ basis or |2| and |µ2| if the photon was measured in the xˆ 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 A Probability D1 Probability D2 |Hi z 0.004 0.996 |Hi x 0.435 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 zˆ 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 xˆ, 56.5% was |H+i and 43.5% was measured as |V +i. In the second measurement, when measuring in zˆzˆ, nearly all of the photons were

12 Input A B P. D1 P. D3 P. D4 |Hi z z 0.003 0.002 0.994 |Hi z x 0.003 0.417 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 zˆxˆ, the detected photons were rather equally distributed between D3 and D4, which detected |H+i and |V +i.

Input A B C P. D1 P. D3 P. D5 P. D6 |Hi x z x 0.452 0.279 0.115 0.154 |Hi z x z 0.003 0.47 0.221 0.306

Table 3: The results from the third measurement.

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 A Probability D1 Probability D2 |Hi z 0 1 |Hi x 0.5 0.5

Table 4: The theoretically expected results in measurement 1.

Input A B P. D1 P. D3 P. D4 |Hi z z 0 0 1 |Hi z x 0 0.5 0.5

Table 5: The theoretically expected results in measurement 2.

Input A B C P. D1 P. D3 P. D5 P. D6 |Hi x z x 0.5 0.25 0.125 0.125 |Hi z x z 0 0.5 0.25 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