1. 1 Steven Anlage Basics of RF Superconductivity and QC-Related Microwave Issues Short Course Tutorial Superconducting Quantum Computing 2014 Applied Superconductivity Conference Charlotte, North Carolina USA Steven M. Anlage Center for Nanophysics and Advanced Materials Physics Department University of Maryland College Park, MD 20742-4111 USA anlage@umd.edu

2. 2 Steven Anlage Objective To give a basic introduction to superconductivity, superconducting electrodynamics, and microwave measurements as background for the Short Course Tutorial “Superconducting Quantum Computation” Digital Electronics (10 sessions) Nanowire / Kinetic Inductance / Single-photon Detectors (10 sessions) Transition-Edge Sensors / Bolometers (8 sessions) SQUIDs / NanoSQUIDs / SQUIFs (6 sessions) Superconducting Qubits (5 sessions) Microwave / THz Applications (4 sessions) Mixers (1 session) Electronics Sessions at ASC 2014:

3. 3 Steven Anlage Outline Hallmarks of Superconductivity Superconducting Transmission Lines Network Analysis Superconducting Microwave Resonators for QC Microwave Losses Microwave Modeling and Simulation QC-Related Microwave Technology Essentials of Superconductivity Fundamentals of Microwave Measurements

4. 4 Steven Anlage Please Ask Questions!

5. 5 Steven Anlage I. Hallmarks of Superconductivity The Three Hallmarks of Superconductivity Zero Resistance Complete* Diamagnetism Macroscopic Quantum Effects Superconductors in a Magnetic Field Vortices and Dissipation I. Hallmarks of Superconductivity

6. 6 Steven Anlage The Three Hallmarks of Superconductivity Zero Resistance I V DC Resistance Temperature TcTc 0 Complete Diamagnetism Magnetic Induction Temperature TcTc 0 T>T c T<T c Macroscopic Quantum Effects Flux  Flux quantization  = n  0 Josephson Effects B I. Hallmarks of Superconductivity [BCS Theory]BCS Theory [Mesoscopic Superconductivity]Mesoscopic Superconductivity

7. 7 Steven Anlage Zero Resistance R = 0 only at  = 0 (DC) R > 0 for  > 0 and T > 0 E Quasiparticles Cooper Pairs  0 The Kamerlingh Onnes resistance measurement of mercury. At 4.15K the resistance suddenly dropped to zero Energy Gap I. Hallmarks of Superconductivity

8. 8 Steven Anlage Perfect Diamagnetism Magnetic Fields and Superconductors are not generally compatible The Meissner Effect The Yamanashi MLX01 MagLev test vehicle achieved a speed of 361 mph (581 kph) in 20032003 Super- conductor T>T c T<T c Spontaneous exclusion of magnetic flux T (T) (0) TcTc vacuum superconductor is independent of frequency (  < 2  ħ ) magnetic penetration depth  surface screening currents I. Hallmarks of Superconductivity

9. 9 Steven Anlage Macroscopic Quantum Effects Superconductor is described by a single Macroscopic Quantum Wavefunction Consequences: Magnetic flux is quantized in units of  0 = h/2e (= 2.07 x 10 -15 Tm 2 ) R = 0 allows persistent currents Current I flows to maintain  = n  0 in loop n = integer, h = Planck’s const., 2e = Cooper pair charge [DETAILS]DETAILS Flux  I superconductor Sachdev and Zhang, Science Magnetic vortices have quantized flux vortex lattice B screening currents x B(x) |  (x)| 0  Type II vortex core A vortex Line cut I. Hallmarks of Superconductivity

10. 10 Steven Anlage Definition of “Flux” surface I. Hallmarks of Superconductivity

11. 11 Steven Anlage Macroscopic Quantum Effects Continued Josephson Effects (Tunneling of Cooper Pairs) I DC Josephson Effect (V DC =0) AC Josephson Effect (V DC  0) Quantum VCO: V DC 1 2 (Tunnel barrier) Gauge-invariant phase difference: I. Hallmarks of Superconductivity [DC SQUID Detail]Detail[RF SQUID Detail]Detail

12. 12 Steven Anlage Abrikosov Vortex Lattice Superconductors in a Magnetic Field The Vortex State T  H c1 (0) TcTc Meissner State H c2 (0) B = 0, R = 0 B  0, R  0 Normal State Type II SC vortex Lorentz Force Moving vortices create a longitudinal voltage V>0 I Vortices also experience a viscous drag force: I. Hallmarks of Superconductivity [Phase diagram Details]Details

13. 13 Steven Anlage II. Superconducting Transmission Lines Property I: Low-Loss Two-Fluid Model Surface Impedance and Complex Conductivity [Details]Details BCS Electrodynamics [Details]Details Property II: Low-Dispersion Kinetic Inductance Josephson Inductance II. Superconducting Transmission Lines

14. 14 Steven Anlage Why are Superconductors so Useful at High Frequencies? Low Losses: Filters have low insertion loss  Better S/N, filters can be made small NMR/MRI SC RF pickup coils  x10 improvement in speed of spectrometer High Q  Filters with steep skirts, good out-of-band rejection Low Dispersion: SC transmission lines can carry short pulses with little distortion RSFQ logic pulses – 2 ps long, ~1 mV in amplitude: II. Superconducting Transmission Lines [RSFQ Details]Details

15. 15 Steven Anlage Electrodynamics of Superconductors in the Meissner State (Two-Fluid Model) E Quasiparticles (Normal Fluid) Cooper Pairs (Super Fluid)  0 LsLs nn  Superfluid channel Normal Fluid channel Energy Gap  =  n – i  2 J = J s + J n JsJs JnJn AC Current-carrying superconductor J =  E  n = n n e 2  /m  2 = n s e 2 /m  n n = number of QPs n s = number of SC electrons  = QP momentum relaxation time m = carrier mass  = frequency T 0 n n n (T) n s (T) TcTc J n = n s (T) + n n (T) II. Superconducting Transmission Lines

16. 16 Steven Anlage Surface Impedance x -z y J E H Surface Resistance R s : Measure of Ohmic power dissipation RsRs conductor Local Limit Surface Reactance X s : Measure of stored energy per period XsXs X s =  L s =  L geo L kinetic II. Superconducting Transmission Lines [London Eqs. Details1, Details2]Details1Details2

17. 17 Steven Anlage Two-Fluid Surface Impedance Because R s ~  2 : The advantage of SC over Cu diminishes with increasing frequency HTS: R s crossover at f ~ 100 GHz at 77 K LsLs nn Superfluid channel Normal Fluid channel M. Hein, Wuppertal R s ~  2 R n ~  1/2 II. Superconducting Transmission Lines

18. 18 Steven Anlage Kinetic Inductance Flux integral surface A measure of energy stored in magnetic fields both outside and inside the conductor A measure of energy stored in dissipation-less currents inside the superconductor (valid when << w, see Orlando+Delin) For a current-carrying strip conductor: In the limit of t <<  or w << : … and this can get very large for low-carrier density metals (e.g. TiN or Mo 1-x Ge x ), or near T c II. Superconducting Transmission Lines

19. 19 Steven Anlage J. Baselmans, JLTP (2012) Microwave Kinetic Inductance Detectors (MKIDs) Kinetic Inductance (Continued) Microstrip HTS transmission line resonator (B. W. Langley, RSI, 1991) (C. Kurter, PRB, 2013) II. Superconducting Transmission Lines

20. 20 Steven Anlage Josephson Inductance Josephson Inductance is large, tunable and nonlinear Resistively and Capacitively Shunted Junction (RCSJ) Model The Josephson inductance can be tuned e.g. when the JJ is incorporated into a loop and flux  is applied rf SQUID Here is a non-rigorous derivation of L JJ Start with the dc Josephson relation: Take the time derivative and use the ac Josephson relation: Solving for voltage as: Yields: is the gauge-invariant phase difference across the junction II. Superconducting Transmission Lines

21. 21 Steven Anlage Single-rf-SQUID Resonance Tuning with DC Magnetic Flux Comparison to Model RF power = -80 dBm, @6.5K Maximum Tuning: 80 THz/Gauss @ 12 GHz, 6.5 K Total Tunability: 56% rf SQUID II. Superconducting Transmission Lines M. Trepanier, PRX 2013 Red represents resonance dip RCSJ model

22. 22 Steven Anlage

23. 23 Steven Anlage III. Network Analysis Network vs. Spectrum Analysis Scattering (S) Parameters Quality Factor Q Cavity Perturbation Theory [Details]Details III. Network Analysis

24. 24 Steven Anlage 24 Network vs. Spectrum Analysis Agilent – Back to Basics Seminar III. Network Analysis

25. 25 Steven Anlage Network Analysis Assumes linearity! 2-port system frequency (f) f0f0 |S 21 (f)| 2 resonator transmission P. K. Day, Nature, 2003 |S 21 | (dB) input (1) output (2) B 12 III. Network Analysis Resonant Cavity Co-Planar Waveguide (CPW) Resonator

26. 26 Steven Anlage Transmission Lines Transmission lines carry microwave signals from one point to another They are important because the wavelength is much smaller than the length of typical T-lines used in the lab You have to look at them as distributed circuits, rather than lumped circuits The wave equations V III. Network Analysis

27. 27 Steven Anlage Transmission Lines Wave Speed Take the ratio of the voltage and current waves at any given point in the transmission line: = Z 0 The characteristic impedance Z 0 of the T-line Reflections from a terminated transmission line ZLZL Z0Z0 Reflection coefficient Some interesting special cases: Open Circuit Z L = ∞,  = 1 e i0 Short Circuit Z L = 0,  = 1 e i  Perfect Load Z L = Z 0,  = 0 e i  III. Network Analysis

28. 28 Steven Anlage 28 Transmission Lines and Their Characteristic Impedances Agilent – Back to Basics Seminar Attenuation is lowest at 77  50  Standard Power handling capacity peaks at 30  Normalized Values Characteristic Impedance for coaxial cable (  [Transmission Line Detail]Detail

29. 29 Steven Anlage N-Port Description of an Arbitrary System N – Port System N-Port System Described equally well by: ► Voltages and Currents, or ► Incoming and Outgoing Waves Z matrixS matrix V 1, I 1 V N, I N S = Scattering Matrix Z = Impedance Matrix Z 0 = (diagonal) Characteristic Impedance Matrix III. Network Analysis Z 0,1 Z 0,N

30. 30 Steven Anlage Linear vs. Nonlinear Behavior Agilent – Back to Basics Seminar Device Under Test III. Network Analysis

31. 31 Steven Anlage Quality Factor Two important quantities characterise a resonator: The resonance frequency f 0 and the quality factor Q Where U is the energy stored in the cavity volume and P c /  0 is the energy lost per RF period by the induced surface currents 31 Frequency Offset from Resonance f – f 0 0 ff ff  f Stored Energy U Some typical Q-values: SRF accelerator cavity Q ~ 10 11 3D qubit cavity Q ~ 10 8 III. Network Analysis [Q of a shunt-coupled resonator Detail]Detail

32. 32 Steven Anlage Scattering Parameter of Resonators III. Network Analysis

33. 33 Steven Anlage IV. Superconducting Microwave Resonators for QC Thin Film Resonators Co-planar Waveguide Lumped-Element SQUID-based Bulk Resonators Coupling to Resonators IV. Superconducting Microwave Resonators for QC

34. 34 Steven Anlage Resonators … the building block of superconducting applications … Microwave surface impedance measurements Cavity Quantum Electrodynamics of Qubits Superconducting RF Accelerators Metamaterials (  eff < 0 ‘atoms’) etc. P out co-planar waveguide (CPW) resonator P in frequency f0f0 |S 21 (f)| 2 resonator transmission Port 1 Port 2 Transmitted Power IV. Superconducting Microwave Resonators for QC CPW Field Structure

35. 35 Steven Anlage http://www.phys.unsw.edu.au Animation link The Inductor-Capacitor Circuit Resonator frequency (f) f0f0 Re[Z] Im[Z] Impedance of a Resonator Im[Z] = 0 on resonance IV. Superconducting Microwave Resonators for QC

36. 36 Steven Anlage Lumped-Element LC-Resonator Z. Kim P out IV. Superconducting Microwave Resonators for QC

37. 37 Steven Anlage Resonators (continued) T = 79 K P = - 10 dBm f = 5.285 GHz YBCO/LaAlO 3 CPW Resonator Excited in Fundamental Mode Imaged by Laser Scanning Microscopy* W strip = 500  m 1 x 8 mm scan *A. P. Zhuravel, et al., J. Appl. Phys. 108, 033920 (2010) G. Ciovati, et al., Rev. Sci. Instrum. 83, 034704 (2012) IV. Superconducting Microwave Resonators for QC [Trans. Line Resonator Detail]Detail

38. 38 Steven Anlage IV. Superconducting Microwave Resonators for QC Three-Dimensional Resonator Inductively-coupled cylindrical cavity with sapphire “hot-finger” TE 011 mode Electric fields Rahul Gogna, UMD M. Reagor, APL 102, 192604 (2013) Al cylindrical cavity TE 011 mode Q = 6 x 10 8 T = 20 mK

39. 39 Steven Anlage Coupling to Resonators InductiveCapacitive Lumped Inductor Lumped Capacitor P. Bertet SPEC, CEA Saclay IV. Superconducting Microwave Resonators for QC

40. 40 Steven Anlage V. Microwave Losses Microscopic Sources of Loss 2-Level Systems (TLS) in Dielectrics Flux Motion What Limits the Q of Resonators? V. Microwave Losses

41. 41 Steven Anlage V. Microwave Losses Microwave Losses / 2-Level Systems (TLS) in Dielectrics M. Hein, APL 80, 1007 (2002) Effective Loss @ 2.3 GHz (  ) -60 dBm -20 dBm TLS in MgO substrates YBCO film on MgO YBCO/MgO T = 5K Nb/MgO Energy Low Power High Power Two-Level System (TLS) Classic reference: photon

42. 42 Steven Anlage Microwave Losses / Flux Motion Single-vortex response (Gittleman-Rosenblum model) Equation of motion for vortex in a rigid lattice (vortex-vortex force is constant) Effective mass of vortex Vortex viscosity ~  n Ignore vortex inertia vortex Pinning potential Frequency f / f 0 Dissipated Power P / P 0 Gittleman, PRL (1966) V. Microwave Losses

43. 43 Steven Anlage What Limits the Q of Resonators? Assumption: Loss mechanisms add linearly Power dissipated in load impedance(s) V. Microwave Losses

44. 44 Steven Anlage VI. Microwave Modeling and Simulation Computational Electromagnetics (CEM) Finite Element Approach (FEM) Finite Difference Time Domain (FDTD) Solvers Eigenmode Driven Transient Time-Domain Examples of Use VI. Microwave Modeling and Simulation

45. 45 Steven Anlage Finite-Element Method (FEM): Create a finite-element mesh (triangles and tetrahedra), expand the fields in a series of basis functions on the mesh, then solve a matrix equation that minimizes a variational functional corresponds to the solutions of Maxwell’s equations subject to the boundary conditions. Finite Difference Time Domain (FDTD): Directly approximate the differential operators on a grid staggered in time and space. E and H computed on a regular grid and advanced in time. Computational EM: FEM and FDTD The Maxwell curl equations MethodAdvantagesDisadvantagesExamples FEM Conformal meshes model curved surfaces well. Handles dispersive materials. Good for finding eigenmodes. Can be linked to other FEM solvers (thermal, mechanical, etc.) Does not handle nonlinear materials easily. Meshes can get very large and limit the computation. Solvers are often proprietary. High Frequency Structure Simulator (HFSS) and other frequency-domain solvers, including the COMSOL RF module. FDTD Handles nonlinearity and wideband signals well. Less limited by mesh size than FEM – better for electrically-large structures. Easy to parallelize and solve with GPUs. Not good for high-Q devices or dispersive materials. Staircased grid does not model curved surfaces well. CST Microwave Studio, XFDTD

46. 46 Steven Anlage Example FDTD grid with ‘Yee’ cells Examples FEM triangular / tetrahedral meshes Dave Morris, Agilent 5990-97595990-9759 Example CEM Mesh and Grid VI. Microwave Modeling and Simulation Grid approximation for a sphere

47. 47 Steven Anlage CEM Solvers Eigenmode: Closed system, finds the eigen-frequencies and Q values Driven: The system has one or more ‘ports’ connected to infinity by a transmission line or free-space propagating mode. Calculate the Scattering (S) Parameters. Driven anharmonic billiard VI. Microwave Modeling and Simulation Gaussian wave-packet excitation with antenna array Rahul Gogna, UMD Bo Xiao, UMD Harita Tenneti, UMD Propagation simulates time-evolution

48. 48 Steven Anlage Domain wall Loop antenna (3 turns) Transient (FDTD): The system has one or more ‘ports’ connected to infinity by a transmission line or free-space propagating mode. Calculate the transient signals. CEM Solvers (Continued) VI. Microwave Modeling and Simulation Bo Xiao, UMD

49. 49 Steven Anlage Uses Finding unwanted modes or parasitic channels through a structure Understanding and optimizing coupling Evaluating and minimizing radiation losses in CPW and microstrip Current + field profiles / distributions Computational Electromagnetics TE 311 VI. Microwave Modeling and Simulation

50. 50 Steven Anlage VII. QC-Related Microwave Technology Isolating the Qubit from External Radiation Filtering Attenuation / Screening Cryogenic Microwave Hardware Passive Devices (attenuator, isolator, circulator, directional coupler) Active Devices (Amplifiers) Microwave Calibration VII. QC-Related Microwave Technology

51. 51 Steven Anlage VII. QC-Related Microwave Technology A. J. Przybysz, Ph.D. Thesis, UMD (2010) Grounding the leads prevents electro-static discharge in to the device Isolating the Qubit from External Radiation Cu patch box helps to thermalize the leads Low-Pass LC-filter (f c = 10 MHz) Cu Powder Filter (strong suppression of microwaves) The experiments take place in a shielded room with filtered AC lines and electrical feedthroughs

52. 52 Steven Anlage Typical SC QC Experimental set-up LNA T = 25 mK f LO IF dc block Mixer IF amp Bias Tee Resonator CPB Isolators RF 4 K300K f pump for qubit VgVg Directional coupler (HP87301D, 1-40 GHz) 300K LNA 3 dB Attenuator PiPi PoPo Pulse Isolator 4-8 GHz LO Attenuators f probe for resonator Digital oscilloscope Amplitude and phase (slide courtesy Z. Kim, LPS) VII. QC-Related Microwave Technology

53. 53 Steven Anlage Passive Cryogenic Microwave Hardware Much of microwave technology is designed to cleanly separate the “left-going” from the “right-going waves! Directional coupler Input Output Coupled Signal (-x dB) x = 6, 10, 20, etc. Bias Tee DC input RF input RF+DC output Isolator RF input Output “Left-Going” reflected signals are absorbed by the matched loa d VII. QC-Related Microwave Technology

54. 54 Steven Anlage Active Cryogenic Microwave Hardware Low-Noise Amplifier (e.g. Low-Noise Factory LNF-LNC6_20B) LNA 13 mW dissipated power A measure of noise added to signal Mixer RF LO IF Mixer (e.g. Marki Microwave T3-0838) The ‘RF’ signal is mixed with a Local Oscillator signal to produce an Intermediate Frequency signal at the difference frequency VII. QC-Related Microwave Technology

55. 55 Steven Anlage Cryogenic Microwave Calibration J.-H. Yeh, RSI 84, 034706 (2013); L. Ranzani, RSI 84, 034704 (2013) VII. QC-Related Microwave Technology

56. 56 Steven Anlage References and Further Reading Z. Y. Shen, “High-Temperature Superconducting Microwave Circuits,” Artech House, Boston, 1994. M. J. Lancaster, “Passive Microwave Device Applications,” Cambridge University Press, Cambridge, 1997. M. A. Hein, “HTS Thin Films at Microwave Frequencies,” Springer Tracts of Modern Physics 155, Springer, Berlin, 1999. “Microwave Superconductivity,” NATO- ASI series, ed. by H. Weinstock and M. Nisenoff, Kluwer, 2001. T. VanDuzer and C. W. Turner, “Principles of Superconductive Devices and Circuits,” Elsevier, 1981. T. P. Orlando and K. A. Delin, “Fundamentals of Applied Superconductivity,” Addison-Wesley, 1991. R. E. Matick, “Transmission Lines for Digital and Communication Networks,” IEEE Press, 1995; Chapter 6. Alan M. Portis, “Electrodynamcis of High-Temperature Superconductors,” World Scientific, Singapore, 1993.

57. 57 Steven Anlage Graduate course on Superconductivity (Anlage) http://www.physics.umd.edu/courses/Phys798I/anlage/AnlageFall12/index.html Superconductivity Links Gallery of Abrikosov Vortex Lattices http://www.fys.uio.no/super/vortex/ Wikipedia article on Superconductivity http://en.wikipedia.org/wiki/Superconductivity YouTube videos of Superconductivity (a classic film by Prof. Alfred Leitner) http://www.youtube.com/watch?v=nLWUtUZvOP8

58. 58 Steven Anlage

59. 59 Steven Anlage Please Ask Questions!

60. 60 Steven Anlage

61. 61 Steven Anlage Details and Backup Slides

62. 62 Steven Anlage What are the Limits of Superconductivity? Phase Diagram Superconducting State Normal State Ginzburg-Landau free energy density Applied Magnetic FieldCurrents Temperature Dependence TcTc  0 H c2 JcJc [Return]Return

63. 63 Steven Anlage BCS Theory of Superconductivity http://www.chemsoc.org/exemplarchem/entries/igrant/hightctheory_noflash.html First electron polarizes the lattice Second electron is attracted to the concentration of positive charges left behind by the first electron S v + + + + S v + + + + s-wave ( = 0) pairing Spin singlet pair Cooper Pair  Debye is the characteristic phonon (lattice vibration) frequency N(E F ) is the electronic density of states at the Fermi Energy V is the attractive electron-electron interaction An energy 2  (T) is required to break a Cooper pair into two quasiparticles (roughly speaking) Cooper pair ‘size’: A many-electron quantum wavefunction  made up of Cooper pairs is constructed with these properties: Bardeen-Cooper-Schrieffer (BCS) [Return]Return

64. 64 Steven Anlage 0 0 1()1() 2()2() 1()1()  2 (  ) ~ 1/      i   Superconductor Electrodynamics “binding energy” of Cooper pair ( 100 GHz ~ few THz) T = 0 ideal s-wave Surface Impedance (  > 0) Normal State Superconducting State (  < 2  ) Penetration depth (0) ~ 20 – 200 nm  n s e   m  Finite-temperature: X s (T) =  L =   (T) → ∞ as T →T c (and  ps (T) → 0) Narrow wire or thin film of thickness t : L(T) =   (T) coth(t/ (T)) →  0 2 (T)/t Kinetic Inductance Superfluid density 2 ~ m/n s ~ 1/  ps 2 T n s (T) TcTc 0 0 Normal State (T > T c ) (Drude Model) 1/  [Return]Return

65. 65 Steven Anlage BCS Microwave Electrodynamics Low Microwave Dissipation Full energy gap → R s can be made arbitrarily small for T < T c /3 in a fully-gapped SC R s,residual ~ 10 -9  at 1.5 GHz in Nb M. Hein, Wuppertal Filled Fermi Sea ss kxkx kyky HTS materials have nodes in the energy gap. This leads to power-law behavior of (T) and R s (T) and residual losses R s,residual ~ 10 -5  at 10 GHz in YBa 2 Cu 3 O 7-  Filled Fermi Sea dd node kxkx kyky [Return]Return

66. 66 Steven Anlage The London Equations Newton’s 2 nd Law for a charge carrier Superconductor: 1/   0 1 st London Equation  = momentum relaxation time J s = n s e v s 1 st London Eq. and (Faraday) yield: London surmise 2 nd London Equation These equations yield the Meissner screening vacuum superconductor L L ~ 20 – 200 nm L is frequency independent (  < 2  ħ ) [Return]Return

67. 67 Steven Anlage The London Equations continued Normal metalSuperconductor B is the source of zero frequency J s, spontaneous flux exclusion E is the source of J n E=0: J s goes on forever Lenz’s Law 1 st London Equation  E is required to maintain an ac current in a SC Cooper pair has finite inertia  QPs are accelerated and dissipation occurs [Return]Return

68. 68 Steven Anlage Fluxoid Quantization Superconductor is described by a single Macroscopic Quantum Wavefunction Consequences: Magnetic flux is quantized in units of  0 = h/2e (= 2.07 x 10 -15 Tm 2 ) R = 0 allows persistent currents Current I flows to maintain  = n  0 in loop n = integer, h = Planck’s const., 2e = Cooper pair charge Flux  I superconductor Flux  Circuit C Surface S B Fluxoid Quantization n = 0, ±1, ±2, … [Return]Return

69. 69 Steven Anlage Cavity Perturbation Objective: determine R s, X s (or  1,  2 ) from f 0 and Q measurements of a resonant cavity containing the sample of interest Sample at Temperature T Microwave Resonator InputOutput frequency transmission f0f0 ff f0’f0’  f’  f = f 0 ’ – f 0   (Stored Energy)  (1/2Q)   (Dissipated Energy) Quality Factor ~ microwave wavelength T1T1 T2T2 B Cavity perturbation means  f << f 0  is the sample/cavity geometry factor III. Network Analysis [Return]Return

70. 70 Steven Anlage Superconducting Quantum Interference Devices (SQUIDs) The DC SQUID A Sensitive Magnetic Flux-to-Voltage Transducer Flux IcIc IcIc Bias Current 2I c One can measure small fields (e.g. 5 aT) with low noise (e.g. 3 fT/Hz 1/2 ) Used extensively for Magnetometry (m vs. H) Magnetoencephalography (MEG) Magnetic microscopy Low-field magnetic resonance imaging Wikipedia.org [Return]Return

71. 71 Steven Anlage Superconducting Quantum Interference Devices (SQUIDs) The RF SQUID A High-Frequency Magnetic Flux-to-Voltage Transducer R. Rifkin, J. Appl. Phys. (1976) Not as sensitive as DC SQUIDs Less susceptible to noise at 77 K with HTS junctions Basis for many qubit and meta-atom designs RF SQUID Tank Circuit [Return]Return

72. 72 Steven Anlage The power absorbed in a termination is: Transmission Lines, continued Model of a realistic transmission line including loss Traveling Wave solutions with Shunt Conductance III. Network Analysis [Return]Return

73. 73 Steven Anlage How Much Power Reaches the Load? Agilent – Back to Basics Seminar III. Network Analysis [Return]Return

74. 74 Steven Anlage Waveguides Rectangular metallic waveguide H III. Network Analysis [Return]Return

75. 75 Steven Anlage Transmission Line Unit Cell Transmission Line Resonators Transmission Line Model Transmission Line Resonator Model C coupling IV. Superconducting Microwave Resonators for QC [Return]Return

76. 76 Steven Anlage Measuring the Q of a Shunt-Coupled Resonator Ch. Kaiser, Sup Sci Tech 23, 075008 (2010)Sup Sci Tech 23, 075008 (2010) [Return]Return

77. 77 Steven Anlage Rapid Single Flux Quantum Logic superconducting “classical” digital computing  When (Input + I bias ) exceeds JJ critical current I c, JJ “flips”, producing an SFQ pulse.  Area of the pulse is  0 =2.067 mV-ps  Pulse width shrinks as J C increases  SFQ logic is based on counting single flux quanta  SFQ pulses propagate along impedance-matched passive transmission line (PTL) at the speed of light in the line (~ c/3).  Multiple pulses can propagate in PTL simultaneously in both directions.  Circuits can operate at 100’s of GHz Input ~1mV ~2ps I bias JJ Courtesy Arnold Silver [Return]Return

78. 78 Steven Anlage Mesoscopic Superconductivity BCS Ground state wavefunction Coherent state with no fixed number of Cooper pairs Number-phase uncertainty (conjugate variables) Bulk Superconductor Small superconducting island Josephson tunnel barrier phase difference  Typically: R barrier > 2ħ/e 2 ~ 6 k  e 2 /2C > k B T (requires C ~ fF at 1 K) KE PE Using Q = CV and Q/2e = N = i ∂/∂  ∂/∂  the Hamiltonian is: Two important limits: E C << E J well-define phase   utilize phase eigenstates, like E C >> E J all values of  equally probable  utilize number eigenstates Classical energy [Return]Return