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Testing Heisenberg's Uncertainty Principle with Polarized Single 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 wave- description of a photon, polarization can be visualized as the way the wave is rotated. A photon possesses horizontal jHi or vertical jV i polarization, but until its polarization is measured, these two states are said to be in a superposition, described by jΨi = α jHi + β jV i (1) where jα2j is the probability of finding the photon in state jHi and jβ2j in state jV i [1]. The horizontal and the vertical polarization jointly define a basis denoted by z^, which can take on the values jHi or jV i. However, the polarization can be described in an additional basis as well, the x^ basis, which is shifted 45 ◦ in positive direction, see Figure 1. Figure 1: A graphical representation of the bases z^ and x^. Just like the particle’s superposition consists of jHi and jV i in z^ basis, the two new 1 states jH+i and jV +i will describe the particle’s superposition in the x^ basis as jΨi = jH+i + µ jV +i (2) where jj2 describes the probability of finding the photon in state jH+i, and jµj2 is state jV +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 mea- surement 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 x^ exactly, all information gained about the polarization in z^ will be lost. The aim of this study is to test Heisenberg’s uncertainty principle by means of chang- ing 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 ap- paratus (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 mea- sured 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 mag- netic 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 appara- tus, 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 jSx+i = p jSx+i + p jSz−i (3) 2 2 1 1 jSx−i = −p jSz+i + p jSz−i (4) 2 2 4 Similarly, jSiz + and jSiz − are described by 1 1 jSz+i = p jSx+i + p jSx−i (5) 2 2 1 1 jSz−i = p jSx+i − p jSx−i (6) 2 2 2.2 Polarization of Photons An analogy to the intrinsic property spin is the polarization of photons. Since the po- larization 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 z^ basis is given by 1 1 jHi = p jH+i + p jV +i (7) 2 2 1 1 jV i = p jH+i − p jV +i (8) 2 2 while polarization states in the x^ basis is described by 1 1 jH+i = p jHi + p jV i (9) 2 2 1 1 jV +i = p jHi − p jV 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. 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 2 3 cos 2θ sin 2θ ^ 6 7 R = 4 5 (11) sin 2θ − cos 2θ and when set to basis z^, θ = 0 ◦, and when set to x^, θ = 22:5 ◦.
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