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Quantum Interferometry with Multiports: Entangled Photons in Optical Fibers

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Quantum Interferometry with Multiports: Entangled Photons in Optical Fibers Quantum Interferometry with Multiports: Entangled Photons in Optical Fibers Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften eingereicht von Mag. phil. Mag. rer. nat. Michael Hunter Alexander Reck im Juli 1996 durchgef¨uhrt am Institut f¨ur Experimentalphysik, Naturwissenschaftliche Fakult¨at der Leopold-Franzens Universit¨at Innsbruck unter der Leitung von o. Univ. Prof. Dr. Anton Zeilinger Diese Arbeit wurde vom FWF im Rahmen des Schwerpunkts Quantenoptik (S06502) unterst¨utzt. Contents Abstract 5 1 Entanglement and Bell’s inequalities 7 1.1 Introduction .............................. 7 1.2 Derivation of Bell’s inequality for dichotomic variables ...... 10 1.3 Tests of Bell’s inequality ....................... 14 1.3.1 First experiments ....................... 14 1.3.2 Entanglement and the parametric downconversion source .16 1.3.3 New experiments with entangled photons .......... 17 1.4 Bell inequalities for three-valued observables ............ 18 2 Theory of linear multiports 23 2.1 The beam splitter ........................... 24 2.2 Multiports ............................... 25 2.2.1 From experiment to matrix ................. 26 2.2.2 From matrix to experiment ................. 27 2.3 Symmetric multiports ......................... 32 2.3.1 Single-photon eigenstates of a symmetric multiport .... 33 2.3.2 Two-photon eigenstates of a symmetric multiport ..... 33 2.4 Multiports and quantum computation ................ 35 3 Optical fibers 39 3.1 Single mode fibers ........................... 39 3.1.1 Optical parameters of fused silica .............. 40 3.1.2 Material dispersion and interferometry ........... 43 3.2 Components of fiber-optical systems ................. 46 3.3 Coupled waveguides as multiports ................. 47 4 Experimental characterization of fiber multiports 49 4.1 A three-path Mach-Zehnder interferometer using all-fiber tritters .49 4.1.1 Experimental setup ...................... 49 4.1.2 Theoretical description .................... 52 4.1.3 Experimental results ..................... 55 1 2 4.2 Two-photon interferences in optical fiber multiports ........ 56 4.2.1 Introduction .......................... 56 4.2.2 Theoretical description .................... 56 4.2.3 Experimental results ..................... 58 5 Two-photon three-path interference 63 5.1 Energy entanglement in three-path interferometers ......... 63 5.2 Multipath interferences ........................ 64 5.3 Experiment .............................. 70 5.3.1 The experimental setup ................... 70 5.3.2 Polarization and path length adjustment .......... 79 5.3.3 Detection system ....................... 84 5.3.4 Calibration of phase settings ................. 88 5.3.5 Data acquisition ........................ 90 5.3.6 Temperature drifts ...................... 91 5.4 Experimental results ......................... 93 5.4.1 Description of data ...................... 93 5.4.2 Data analysis ......................... 100 5.5 Interpretation ............................. 105 5.5.1 Bell’s inequalities ....................... 105 5.5.2 Classical picture vs. quantum picture ............ 106 6 Conclusions and Outlook 111 A Devices 115 A.1 Parametric downconversion crystals ................. 115 A.1.1 KDP and LiIO3 ........................ 115 A.1.2 Beta Barium Borate (BBO) ................. 116 A.2 Optical fibers and tritters ...................... 117 A.3 Lasers ................................. 119 A.3.1 Argon-Ion laser ........................ 119 A.3.2 HeNe laser ........................... 120 A.3.3 Diode laser .......................... 121 A.4 Single-photon detectors ........................ 121 A.5 Control and detection electronics .................. 122 A.6 Miscellanea .............................. 123 A.6.1 Electro-mechanic shutter ................... 123 A.6.2 Parallel printer port as programmable TTL output .... 123 A.6.3 Temperature sensors ..................... 125 B Reprint from Phys. Rev. Lett. 73, 58–61 (1994) 127 C A computer program for the design of unitary interferometers 133 3 C.1 Mathematica notebook ........................ 133 C.2 Mathematica package ......................... 136 D Wave packets in multiports 141 D.1 A single photon in a multiport .................... 141 D.2 Photon pairs in multiports ...................... 143 D.3 Material dispersion and interference ................. 144 EPhotographing Type-II downconversion 149 4 Abstract The present thesis is the result of theoretical and experimental work on the physics of optical multiports. The theoretical results show that multiport interfer- ometers can be used to realize any discrete unitary transformation operating on modes of a classical or a quantum radiation field. Tests of a Bell-type inequality for higher-dimensional entangled states are thus possible using entangled photon pairs from a parametric downconversion source. The experimental work measured the nonclassical interferences at the fiber-optical three-way beam splitters (trit- ters) and three-path fiber interferometers. The experimental results are discussed in the context of Bell’s inequalities and the physics of entanglement. The first chapter gives a brief review of entanglement and the Bell inequal- ities. Quantization and the superposition principle together form the basis of quantum physics. The physics of entangled states dramatically demonstrates the difference between the quantum and the everyday world. Multiports are the logical generalization of the beam splitter in classical and quantum optics. The second chapter deals with the theory of linear multiports and presents an algorithmic proof that any unitary operator can be built in the laboratory. Optical fibers and integrated optics, the basic components of many earth- bound telecommunication systems, will be necessary in any practical realization of quantum communication and quantum information processing. The third chapter introduces optical fibers as building blocks for the experimental realization of multiport interferometers. The fourth chapter studies the properties of fiber optical multiports in a classical multipath interferometer. A three-path interference experiment reveals 5 6 the typical features of multipath interferometry. In another experiment, entangled photon pairs from the spontaneous parametric downconversion process were used to demonstrate a purely quantum effect, the antibunching of photon pairs at the output of a multiport. Both experiments demonstrate the use of fiber multiports for coherent operation on single quanta. The main part of this work is concerned with the study of time-energy en- tanglement in two three-path interferometers built with fiber optical multiports. This pair of quantum interferometers is the first realization of an entangled three- state system. It is the first multipath experiment to show quantum interferences. A first test of a Bell inequality for a multistate system is attempted with this system. Before the final summary, the quantum and classical pictures of the ex- periment are discussed giving an outlook to new experiments. Technical details about the experiments, a Mathematica program for the design of unitary interferometers, some calculations, and photographs of type-II downconversion light have been included in the appendices. Chapter 1 Entanglement and Bell’s inequalities 1.1 Introduction In their seminal paper of 1935 “Can quantum-mechanical description of physical reality be considered complete” Einstein, Podolsky, and Rosen wrote: “If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” [Einstein35] They proceeded to show that if this apparently innocuous definition of “elements of reality” is accepted, the description of reality provided by the quantum mechan- ical wave function must be incomplete. They finally expressed the belief that a complete description could be found. One possibility to complete the description, which was not mentioned in the EPR paper, would be to supplement quantum mechanics with hidden parameters that play the role of the (hidden) positions and velocities of particles in statistical mechanics. Schr¨odinger’s analysis “The present situation in quantum mechanics” of 1935 was partially motivated by the EPR paper. He for the first time introduced 7 8 the concept of ‘entanglement’. “Maximal knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all. ...Iftwoseparatedbodies,eachbyitselfknownmaximally,entera situation in which they influence each other, and separate again, then there occurs regularly that which I have just called entanglement of our knowledge of the two bodies.”1 [Schr¨odinger35] Schr¨odinger interpreted the wave function as a description of our knowledge of the quantum system. He could not refute the EPR argument, but noticing that quantum states can be ‘entangled’ he pointed to the principal difference between the quantum world and the world of classical physics. His famous cat paradox illustrates this in a very drastic way. The particular case studied in the EPR paper consisted of two quanta which have interacted at some time. These particles are in an entangled state, i.e. the properties of the whole system are well defined, but the properties of the indi- vidual particles are not. The joint state
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