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iv R. GROSS AND A. MARX CONTENTS2.2.2 The Fraunhofer Diffraction Pattern . . . . . . . . . . . . . . . . . . . . . . . . . 542.2.3 Determination of the Maximum Josephson Current Density . . . . . . . . . . . 582.2.4 Additional Topic:Direct Imaging of the Supercurrent Distribution . . . . . . . . . . . . . . . . . . 622.2.5 Additional Topic:Short Josephson Junctions: Energy Considerations . . . . . . . . . . . . . . . . 632.2.6 The Motion of Josephson Vortices . . . . . . . . . . . . . . . . . . . . . . . . . 652.3 Long Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.3.1 The Stationary Sine-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . 682.3.2 The Josephson Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.3.3 Junction Types and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 732.3.4 Additional Topic:Josephson Current Density Distribution and Maximum Josephson Current . . . . 792.3.5 The Pendulum Analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 JJs: The Voltage State 893.1 The Basic Equation of the Lumped Josephson Junction . . . . . . . . . . . . . . . . . . 903.1.1 The Normal Current: Junction Resistance . . . . . . . . . . . . . . . . . . . . . 903.1.2 The Displacement Current: Junction Capacitance . . . . . . . . . . . . . . . . . 923.1.3 Characteristic Times and Frequencies . . . . . . . . . . . . . . . . . . . . . . . 933.1.4 The Fluctuation Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.1.5 The Basic Junction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.2 The Resistively and Capacitively Shunted Junction Model . . . . . . . . . . . . . . . . . 973.2.1 Underdamped and Overdamped Josephson Junctions . . . . . . . . . . . . . . . 1003.3 Response to Driving Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.3.1 Response to a dc Current Source . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.3.2 Response to a dc Voltage Source . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.3.3 Response to ac Driving Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.3.4 Photon-Assisted Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.4 Additional Topic:Effect of Thermal Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.4.1 Underdamped Junctions: Reduction of I c by Premature Switching . . . . . . . . 1173.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory . . . . . . . . . . . . 1183.5 Secondary Quantum Macroscopic Effects . . . . . . . . . . . . . . . . . . . . . . . . . 1223.5.1 Quantum Consequences of the Small Junction Capacitance . . . . . . . . . . . . 122© <strong>Walther</strong>-Meißner-<strong>Institut</strong>
CONTENTS APPLIED SUPERCONDUCTIVITY v3.5.2 Limiting Cases: The Phase and Charge Regime . . . . . . . . . . . . . . . . . . 1253.5.3 Coulomb and Flux Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.5.4 Coherent Charge and Phase States . . . . . . . . . . . . . . . . . . . . . . . . . 1303.5.5 Quantum Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.5.6 Macroscopic Quantum Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 1333.6 Voltage State of Extended Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . 1393.6.1 Negligible Screening Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.6.2 The Time Dependent Sine-Gordon Equation . . . . . . . . . . . . . . . . . . . . 1403.6.3 Solutions of the Time Dependent Sine-Gordon Equation . . . . . . . . . . . . . 1413.6.4 Additional Topic:Resonance Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144II Applications of the Josephson Effect 1534 SQUIDs 1574.1 The dc-SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.1.1 The Zero Voltage State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.1.2 The Voltage State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.1.3 Operation and Performance of dc-SQUIDs . . . . . . . . . . . . . . . . . . . . 1684.1.4 Practical dc-SQUIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1724.1.5 Read-Out Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.2 Additional Topic:The rf-SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.2.1 The Zero Voltage State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.2.2 Operation and Performance of rf-SQUIDs . . . . . . . . . . . . . . . . . . . . . 1824.2.3 Practical rf-SQUIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.3 Additional Topic:Other SQUID Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.3.1 The DROS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.3.2 The SQIF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.3.3 Cartwheel SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.4 Instruments Based on SQUIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.4.1 Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.4.2 Gradiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1944.4.3 Susceptometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1962005
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