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Dephasing in an Electronic Mach-Zehnder Interferometer,” Phys Dephasing and Quantum Noise in an electronic Mach-Zehnder Interferometer Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) der Fakultät für Physik der Universität Regensburg vorgelegt von Andreas Helzel aus Kelheim Dezember 2012 ii Promotionsgesuch eingereicht am: 22.11.2012 Die Arbeit wurde angeleitet von: Prof. Dr. Christoph Strunk Prüfungsausschuss: Prof. Dr. G. Bali (Vorsitzender) Prof. Dr. Ch. Strunk (1. Gutachter) Dr. F. Pierre (2. Gutachter) Prof. Dr. F. Gießibl (weiterer Prüfer) iii In Gedenken an Maria Höpfl. Sie war die erste Taxifahrerin von Kelheim und wurde 100 Jahre alt. Contents 1. Introduction1 2. Basics 5 2.1. The two dimensional electron gas....................5 2.2. The quantum Hall effect - Quantized Landau levels...........8 2.3. Transport in the quantum Hall regime.................. 10 2.3.1. Quantum Hall edge states and Landauer-Büttiker formalism.. 10 2.3.2. Compressible and incompressible strips............. 14 2.3.3. Luttinger liquid in the QH regime at filling factor 2....... 16 2.4. Non-equilibrium fluctuations of a QPC.................. 24 2.5. Aharonov-Bohm Interferometry..................... 28 2.6. The electronic Mach-Zehnder interferometer............... 30 3. Measurement techniques 37 3.1. Cryostat and devices........................... 37 3.2. Measurement approach.......................... 39 4. Sample fabrication and characterization 43 4.1. Fabrication................................ 43 4.1.1. Material.............................. 43 4.1.2. Lithography............................ 44 4.1.3. Gold air bridges.......................... 45 4.1.4. Sample Design.......................... 46 4.2. Characterization.............................. 47 4.2.1. Filling factor........................... 47 4.2.2. Quantum point contacts..................... 49 4.2.3. Gate setting............................ 50 5. Characteristics of a MZI 51 5.1. Basic QPC adjustments and zero bias visibility............. 51 5.2. Decoherence at finite temperatures.................... 58 5.3. Finite bias visibility............................ 61 5.3.1. Filling factor one - single side lobes............... 61 v vi Contents 5.3.2. Filling factor two......................... 63 5.4. Coherence controlled by filling factor.................. 76 6. Noise-induced phase transition 83 6.1. Description of the experiment...................... 83 6.1.1. Characterization of the samples................. 83 6.1.2. Experiments on noise detection with a Mach-Zehnder interfer- ometer.............................. 86 6.2. Theoretical model of a noise-induced phase transition.......... 89 6.3. Discussion................................. 97 7. Summary 103 A. Recipes I B. Important energy scales VII BibliographyIX Acknowledgement XVII List of Figures 2.1. Formation of a 2DEG in a GaAs/AlGaAs heterostructure........6 2.2. Quantum Hall density of states......................9 2.3. Quantum Hall edge states......................... 11 2.4. Quantized Hall resistance......................... 12 2.5. Reconstruction of quantum Hall edge.................. 15 2.6. 1D lattice and dispersion relation..................... 17 2.7. Bosonization of a 1D lattice....................... 18 2.8. Separation of charge field and spin field................. 20 2.9. Plasmon excitations at ff = 2 ....................... 23 2.10. Distributions of random variables..................... 25 2.11. Partition noise in a 1D channel...................... 27 2.12. Aharonov-Bohm effect.......................... 29 2.13. A which-path experiment in an AB interferometer............ 30 2.14. Optical Mach-Zehnder interferometer.................. 31 2.15. Electronic Mach-Zehnder interferometer................. 32 2.16. Basic bias dependence of MZI visibility................. 34 3.1. Cryostat insert............................... 38 3.2. Three point current measurement..................... 39 3.3. Measurement setup............................ 40 4.1. Optical masks............................... 44 4.2. Masks for electron beam lithography................... 45 4.3. Air bridge fabrication........................... 46 4.4. Finished sample of a MZI......................... 46 4.5. Determination of the filling factor.................... 47 4.6. QPC characteristics............................ 48 4.7. Localized state in a QPC......................... 49 4.8. Additional gates.............................. 50 5.1. Edge channel schematic and sample image................ 52 5.2. Aharonov-Bohm oscillations in a MZI.................. 54 5.3. Visibility versus QPC transmission.................... 55 vii viii List of Figures 5.4. Coherent transport through a localized state............... 56 5.5. Dependence of visibility on temperature................. 59 5.6. Temperature dependence for different arm lengths............ 60 5.7. Lobe structure with single side lobes................... 62 5.8. Lobe structure with multiple side lobes.................. 64 5.9. Lobe structure for different QPC transmissions at T0 = 1 ........ 66 5.10. Plasmons in a Mach-zehnder interferometer............... 68 5.11. Lobe structure for T0 = 2 ......................... 71 5.12. Lobe structure for different QPC transmissions at T0 = 2 ........ 72 5.13. Model for lobe structure at T0 = 2 .................... 74 5.14. Two-point conductance and visibility vs. magnetic field......... 76 5.15. Maximum visibility is controlled by filling factor............ 77 5.16. Temperature dependence for different magnetic fields.......... 78 5.17. Lobe structures for different magnetic fields............... 79 5.18. Energy scales of the lobe structure.................... 80 6.1. Sample micrograph and sketch...................... 84 6.2. Lobe structure for T0 = 1 ......................... 85 6.3. Lobe structure for various T0 ....................... 87 6.4. Phase evolutions for various T0 ...................... 88 6.5. Decay of 2nd side lobe and node positions vs. T0 ............ 89 6.6. Theory: Lobe structure for various T0 .................. 92 6.7. Order parameter and node position vs. T0 ................ 94 6.8. Dephasing versus transmission...................... 95 6.9. Lobe structure from experiment and numerics.............. 96 6.10. Differences between samples A and B.................. 97 6.11. Ambiguity of Sample A.......................... 98 6.12. Lobe structure of sample A for T0 = 0:4 and theoretical curves..... 99 6.13. AB-phase evolution at T0 = 0:5 ..................... 100 B.1. Important energy scales.......................... VIII 1. Introduction “You are not thinking. You are merely being logical.” Niels Bohr to Albert Einstein during their great debate on Quantum Mechanics Dephasing, i.e. decoherence is the loss of quantum interference due to interactions of the interfering degree of freedom with the many degrees of freedom of the environment. Since no quantum mechanical system is perfectly decoupled from its environment, de- coherence and dephasing is an important subject to understand the evolution of the states of a system and to utilize them in an application as a quantum computer or in quantum cryptography. It is subject to active research only since the 1980s [1]. Before this time the problem was treated only rather “philosophically” by the known collapse of the wave function by a classical macroscopic measurement apparatus in the Copenhagen interpretation of quantum mechanics. For this reason dephasing is closely related to the measurement problem of quantum mechanics, because this describes a decoherence process as well. Even when a single state interacts with a second state, the time evolu- tion of their product state will lead to an entangled state. And this can lead to surprising effects of this quantum correlations of a state or a system with its environment. So the investigation of dephasing is attended by the recognition of the fundamental importance of entanglement of quantum systems or states. The question arises here if this interaction between a system of interest and its envi- ronment can only be explained by the randomization of the phases of the states, or if it is also important how the system influences the environment and if these effects are equivalent or must be distinguished. The first is an almost classical view that the envi- ronment acts as a perturbance on the system, while the second is more abstract, because the decoherence is due to the information the environment “measures” on the system. It was together with the advent of mesoscopic physics that these decoherence effects were studied more and more extensively in the physics community. Dephasing describ- ing a process where the behavior of a quantum mechanical particle (or system) becomes gradually classical, due to interaction with the environment, can be best investigated in systems which are at the border between the microscopic, pure quantum mechani- cal world and the macroscopic classical world. They range from few nanometers (such as large molecules [2], carbon nanotubes [3], or self assembled quantum dots [4]) up to several micrometers (as in 2DEG material with large Fermi wave lengths and large mean 1 2 CHAPTER 1. INTRODUCTION free paths). These sizes are now accessible for technological processing and this offers the possibility to control most parameters of a quantum mechanical system. For exam- ple one can realize in a two-dimensional electron gas by applying metal top gates and charging them negatively the typical textbook problem of a particle in a box, a so-called quantum dot [5]. With these gates the size
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