Report

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 1 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” Electronics Sessions at ASC 2014: 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) 2 Steven Anlage Outline • • • • • • • 3 Hallmarks of Superconductivity Essentials of Superconductivity Superconducting Transmission Lines Network Analysis Superconducting Microwave Resonators for QC Microwave Losses Fundamentals of Microwave Microwave Modeling and Simulation Measurements QC-Related Microwave Technology Steven Anlage Please Ask Questions! 4 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 5 Steven Anlage I. Hallmarks of Superconductivity The Three Hallmarks of Superconductivity Zero Resistance Complete Diamagnetism Macroscopic Quantum Effects Flux F V 0 Tc Temperature Magnetic Induction DC Resistance I T>Tc T<Tc B 0 Tc Temperature Flux quantization F = nF0 Josephson Effects [BCS Theory] [Mesoscopic Superconductivity] 6 Steven Anlage I. Hallmarks of Superconductivity Zero Resistance R = 0 only at w = 0 (DC) R > 0 for w > 0 and T > 0 E Quasiparticles 2D The Kamerlingh Onnes resistance measurement of mercury. At 4.15K the resistance suddenly dropped to zero Energy Gap 0 7 Cooper Pairs Steven Anlage I. Hallmarks of Superconductivity Perfect Diamagnetism Magnetic Fields and Superconductors are not generally compatible The Meissner Effect Superconductor H H B 0 H M 0 l(T) magnetic penetration depth l superconductor H H0e z / lL l T<Tc Spontaneous exclusion of magnetic flux B0 z surface screening currents l(0) l is independent of frequency (w < 2D/ħ) 8 H T>Tc H z vacuum H Tc Steven Anlage T The Yamanashi MLX01 MagLev test vehicle achieved a speed of 361 mph (581 kph) in 2003 I. Hallmarks of Superconductivity Macroscopic Quantum Effects ei Superconductor is described by a single Macroscopic Quantum Wavefunction Flux F Consequences: Magnetic flux is quantized in units of F0 = h/2e (= 2.07 x 10-15 Tm2) R = 0 allows persistent currents Current I flows to maintain F = n F0 in loop n = integer, h = Planck’s const., 2e = Cooper pair charge I superconductor [DETAILS] Magnetic vortices have quantized flux A vortex B F n F0 B(x) |(x)| Type II x 2l 0x 9 Line cut l x Steven Anlage vortex core vortex lattice screening currents Sachdev and Zhang, Science I. Hallmarks of Superconductivity Definition of “Flux” dA ⊥ Φ= surface 10 ∙ = ⊥ surface Steven Anlage I. Hallmarks of Superconductivity Macroscopic Quantum Effects Continued Josephson Effects (Tunneling of Cooper Pairs) i1 2 2 e 1 2 1 1 e A i2 I (Tunnel barrier) B A VDC Gauge-invariant phase difference: 2 2e 1 2 A dl 1 11 I I c sin DC Josephson Effect (VDC=0) d eVDC dt eVDC I I c sin t 0 AC Josephson Effect (VDC0) e* 1 MHz 483.593420 Quantum VCO: h F0 μV [DC SQUID Detail] Steven Anlage [RF SQUID Detail] I. Hallmarks of Superconductivity Superconductors in a Magnetic Field The Vortex State H Hc2(0) Normal State Lorentz Force vortex B = 0, R = 0 Meissner State Type II SC T Tc x 2l Vortices also experience a viscous drag force: FDrag vvortex Moving vortices create a longitudinal voltage I V>0 [Phase diagram Details] 12 J B 0, R 0 Abrikosov Vortex Lattice Hc1(0) ˆ0 FL J F Steven Anlage I. Hallmarks of Superconductivity II. Superconducting Transmission Lines • Property I: Low-Loss • Two-Fluid Model • Surface Impedance and Complex Conductivity [Details] • BCS Electrodynamics [Details] • Property II: Low-Dispersion • Kinetic Inductance • Josephson Inductance 13 Steven Anlage II. Superconducting Transmission Lines 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: V t dt F 0 2.07 mV ps [RSFQ Details] Superconducting Transmission Lines microstrip Kinetic Inductance Lkin Geometrical Inductance Lgeo (thickness t) J C B E ground plane propagating TEM wave attenuation a ~ 0 14 Lkin ~ l2 t v phase 1 LC L = Lkin + Lgeo is frequency independent Steven Anlage II. Superconducting Transmission Lines Electrodynamics of Superconductors in the Meissner State (Two-Fluid Model) Normal Fluid channel E Quasiparticles (Normal Fluid) 2D sn Energy Gap Ls Cooper Pairs (Super Fluid) 0 Superfluid channel AC Current-carrying superconductor J J=sE s = sn – i s2 15 Js J = Js + Jn Jn e2t/m sn = nn s2 = nse2/mw n 0 Steven Anlage ns(T) nn(T) Tc T n = ns(T) + nn(T) nn = number of QPs ns = number of SC electrons t = QP momentum relaxation time m = carrier mass w = frequency II. Superconducting Transmission Lines Surface Impedance Z s Rs iX s E J z dz y iw s Local Limit -z H E J x conductor Surface Resistance Rs: Measure of Ohmic power dissipation PDissipated 1 2 1 1 2 Re J E dV R Rss H dA ~ I Rs 2 Volume 2 2 Surface Surface Reactance Xs: Measure of stored energy per period WStored 2 1 H Im J E 2 Volume Lgeo Lkinetic 2 1 2 dV 1 X H dA ~ LI X ss 2w Surface 2 Xs = wLs = wl 16 Steven Anlage [London Eqs. Details1, Details2] II. Superconducting Transmission Lines Two-Fluid Surface Impedance Normal Fluid channel sn 1 Rs w 2 0 l3s n 2 Z s Rs iX s 0 10 X s 0wl -2 Rn ~ w1/2 10 Cu(77K) s Superfluid channel Surface resistance R (W) Ls -1 10 w2: Because Rs ~ The advantage of SC over Cu diminishes with increasing frequency -3 10 poly -4 10 YBCO T=0.85T -5 10 epitaxial -6 Rs ~ w2 10 Nb Sn -7 HTS: Rs crossover at f ~ 100 GHz at 77 K 10 3 T=0.5T -8 Rs ~ s n 10 R n ~ 1/ s n c -1 10 0 c 1 10 10 10 frequency f (GHz) 2 M. Hein, Wuppertal 17 Steven Anlage II. Superconducting Transmission Lines Kinetic Inductance = + Flux integral surface A measure of energy stored in magnetic fields both outside and inside the conductor = Φ/ Φ Lkinetic 0 I2 l 2 1 = 2 2 ( x, y, z ) J s2 ( x, y, z ) dV = 1 2 2 A measure of energy stored in dissipation-less currents inside the superconductor For a current-carrying strip conductor: Lkinetic / 0l w t coth( ) In the limit of t << l or w << l : Lkinetic / 18 Steven Anlage l (valid when l << w, see Orlando+Delin) 0l2 t … and this can get very large for low-carrier density metals (e.g. TiN or Mo1-xGex), or near Tc II. Superconducting Transmission Lines Kinetic Inductance (Continued) Microstrip HTS transmission line resonator (B. W. Langley, RSI, 1991) (C. Kurter, PRB, 2013) Microwave Kinetic Inductance Detectors (MKIDs) J. Baselmans, JLTP (2012) 19 Steven Anlage II. Superconducting Transmission Lines Josephson Inductance Josephson Inductance is large, tunable and nonlinear Here is a non-rigorous derivation of LJJ Start with the dc Josephson relation: LJJJJ C R Resistively and Capacitively Shunted Junction (RCSJ) Model is the gauge-invariant phase difference across the junction I I c sin( ) 2e I I c cos( ) I c V cos( ) V LJJ I Take the time derivative and use the ac Josephson relation: Solving for voltage as: Yields: LJJ F0 2 I c cos( ) The Josephson inductance can be tuned e.g. when the JJ is incorporated into a loop and flux F is applied rf SQUID F Fapplied Finduced nF0 20 Steven Anlage II. Superconducting Transmission Lines Single-rf-SQUID Resonance Tuning with DC Magnetic Flux Comparison to Model rf SQUID RF power = -80 dBm, @6.5K Red represents resonance dip RCSJ model M. Trepanier, PRX 2013 Maximum Tuning: 80 THz/Gauss @ 12 GHz, 6.5 K Total Tunability: 56% 21 Steven Anlage II. Superconducting Transmission Lines Superconducting Quantum Computation Half Day Short Course Schedule 22 1:00 PM - 1:05 PM Welcome and Overview Fred Wellstood, University of Maryland 1:05 PM - 1:40 PM Essentials of Superconductivity Steve Anlage, University of Maryland 1:40 PM - 1:45 PM Short session break 1:45 PM - 2:45 PM Introduction to Qubits and Quantum Computation Fred Wellstood, University of Maryland 2:45 PM - 3:00 PM Short session break 3:00 PM - 3:50 PM Fundamentals of Microwave Measurements Steve Anlage, University of Maryland 3:50 PM - 4:00 PM Short session break 4:00 PM - 5:00 PM Quantum Superconducting Devices Ben Palmer, LPS Steven Anlage III. Network Analysis • Network vs. Spectrum Analysis • Scattering (S) Parameters • Quality Factor Q • Cavity Perturbation Theory [Details] 23 Steven Anlage III. Network Analysis Network vs. Spectrum Analysis Agilent – Back to Basics Seminar 2424 Steven Anlage III. Network Analysis Network Analysis Assumes linearity! 2-port system input (1) output (2) Resonant Cavity B 25 2 |S21(f)|2 |S21| (dB) Co-Planar Waveguide (CPW) Resonator 1 resonator transmission f0 frequency (f) P. K. Day, Nature, 2003 Steven Anlage III. Network Analysis 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 26 Steven Anlage III. Network Analysis Transmission Lines Take the ratio of the voltage and current waves at any given point in the transmission line: Wave Speed = Z0 The characteristic impedance Z0 of the T-line Reflections from a terminated transmission line Z0 ZL Reflection coefficient Vleft Vright b Z L Z0 a Z L Z0 Open Circuit ZL = ∞, = 1 ei0 Some interesting special cases: Short Circuit ZL = 0, = 1 ei Perfect Load ZL = Z0, = 0 ei? 27 Steven Anlage III. Network Analysis Transmission Lines and Their Characteristic Impedances [Transmission Line Detail] Normalized Values Attenuation is lowest at 77 W 50 W Standard Power handling capacity peaks at 30 W 28 Steven Anlage Characteristic Impedance for coaxial cable (W 28 Agilent – Back to Basics Seminar N-Port Description of an Arbitrary System V1 V1 N-Port System V1 , I1 Z0,1 Described equally well by: N – Port ► Voltages and Currents, or System ► Incoming and Outgoing Waves VN VN S matrix V 1 V 1 V 2 V 2 [S ] V N V N 29 VN , IN Z0,N Z matrix V1 I1 V I 2 2 [ ] VN I N Steven Anlage S (Z Z0 )1 (Z Z0 ) Z (w ), S (w ) S = Scattering Matrix Z = Impedance Matrix Z0 = (diagonal) Characteristic Impedance Matrix III. Network Analysis Linear vs. Nonlinear Behavior Device Under Test Agilent – Back to Basics Seminar 30 Steven Anlage III. Network Analysis Quality Factor • Two important quantities characterise a resonator: The resonance frequency f0 and the quality factor Q Stored Energy U f 0 w0U Q Df Pc U U 0 exp( t t L ) Df Df Df 0 Frequency Offset from Resonance f – f0 • [Q of a shunt-coupled resonator Detail] Where U is the energy stored in the cavity volume and Pc/w0 is the energy lost per RF period by the induced surface currents Some typical Q-values: SRF accelerator cavity Q ~ 1011 3D qubit cavity Q ~ 108 31 Steven Anlage III. Network Analysis 31 Scattering Parameter of Resonators 32 Steven Anlage III. Network Analysis IV. Superconducting Microwave Resonators for QC • Thin Film Resonators • Co-planar Waveguide • Lumped-Element • SQUID-based • Bulk Resonators • Coupling to Resonators 33 Steven Anlage IV. Superconducting Microwave Resonators for QC Resonators … the building block of superconducting applications … CPW Field Structure Microwave surface impedance measurements Cavity Quantum Electrodynamics of Qubits Superconducting RF Accelerators Metamaterials (eff < 0 ‘atoms’) etc. co-planar waveguide (CPW) resonator Pout Pin Port 2 Port 1 |S21(f)|2 resonator transmission Transmitted Power f0 34 frequency Steven Anlage IV. Superconducting Microwave Resonators for QC The Inductor-Capacitor Circuit Resonator Animation link 10 8 http://www.phys.unsw.edu.au Impedance of a Resonator Re[Z] 6 4 Im[Z] Im[Z] = 0 on resonance 2 2 f0 frequency (f) 4 35 Steven Anlage IV. Superconducting Microwave Resonators for QC Lumped-Element LC-Resonator Pout Z. Kim 36 Steven Anlage IV. Superconducting Microwave Resonators for QC Resonators (continued) YBCO/LaAlO3 CPW Resonator Excited in Fundamental Mode Imaged by Laser Scanning Microscopy* YBCO Ground Plane Scanned Area STO Substrate 1 x 8 mm scan RF output RF input YBCO Ground Plane T = 79 K P = - 10 dBm f = 5.285 GHz Wstrip = 500 m [Trans. Line Resonator Detail] *A. P. Zhuravel, et al., J. Appl. Phys. 108, 033920 (2010) G. Ciovati, et al., Rev. Sci. Instrum. 83, 034704 (2012) 37 Steven Anlage IV. Superconducting Microwave Resonators for QC Three-Dimensional Resonator Rahul Gogna, UMD Inductively-coupled cylindrical cavity with sapphire “hot-finger” TE011 mode Electric fields Al cylindrical cavity TE011 mode Q = 6 x 108 T = 20 mK M. Reagor, APL 102, 192604 (2013) 38 Steven Anlage IV. Superconducting Microwave Resonators for QC Coupling to Resonators Inductive Capacitive Loop antennas: Coax cables Nb spiral on quartz Lumped Inductor Lumped Capacitor P. Bertet SPEC, CEA Saclay 39 Steven Anlage IV. Superconducting Microwave Resonators for QC V. Microwave Losses • Microscopic Sources of Loss • 2-Level Systems (TLS) in Dielectrics • Flux Motion • What Limits the Q of Resonators? 40 Steven Anlage V. Microwave Losses Microwave Losses / 2-Level Systems (TLS) in Dielectrics Classic reference: Effective Loss @ 2.3 GHz (W) TLS in MgO substrates T = 5K Nb/MgO -60 dBm YBCO/MgO -20 dBm YBCO film on MgO M. Hein, APL 80, 1007 (2002) Energy Low Power Two-Level System (TLS) Energy 41 High Power Steven Anlage V. Microwave Losses Microwave Losses / Flux Motion Single-vortex response (Gittleman-Rosenblum model) + + = × Φ Effective mass of vortex Vortex viscosity ~ sn Pinning potential Dissipated Power P / P0 Equation of motion for vortex in a rigid lattice (vortex-vortex force is constant) = − ×Φ Ignore vortex inertia Pinning frequency 0 2 0 ≡ / Frequency f / f0 42 vortex Steven Anlage Gittleman, PRL (1966) V. Microwave Losses What Limits the Q of Resonators? ~ Stored Energy Energy Dissipated per Cycle Assumption: Loss mechanisms add linearly 1 1 ~ ℎ 2 = 1 ℎ + 1 + 1 + 1 2 1 ′′ ~ 2 Steven Anlage +⋯ ~ 2 1 43 1 ~ Power dissipated in load impedance(s) V. Microwave Losses VI. Microwave Modeling and Simulation • Computational Electromagnetics (CEM) • Finite Element Approach (FEM) • Finite Difference Time Domain (FDTD) • Solvers • Eigenmode • Driven • Transient Time-Domain • Examples 44 of Use Steven Anlage VI. Microwave Modeling and Simulation Computational EM: FEM and FDTD The Maxwell curl equations 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. 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. 45 Method Advantages Disadvantages Examples 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 Steven parallelize and solve with Anlage GPUs. Not good for high-Q devices or dispersive materials. Staircased grid does not model curved surfaces well. CST Microwave Studio, XFDTD Dave Morris, Agilent 5990-9759 Example CEM Mesh and Grid Example FDTD grid with ‘Yee’ cells Grid approximation for a sphere Examples FEM triangular / tetrahedral meshes 46 Steven Anlage VI. Microwave Modeling and Simulation CEM Solvers Eigenmode: Closed system, finds the eigen-frequencies and Q values Bo Xiao, UMD Master 2 Harita Tenneti, UMD Slave 2 Frequency (GHz) Slave 1 Dirac point Master 1 Wavenumber ( ) 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 Gaussian wave-packet excitation with antenna array Rahul Gogna, UMD 47 Propagation simulates time-evolution Steven Anlage VI. Microwave Modeling and Simulation CEM Solvers (Continued) 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. Domain wall Loop antenna (3 turns) Bo Xiao, UMD 48 Steven Anlage VI. Microwave Modeling and Simulation Computational Electromagnetics • 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 TE311 49 Steven Anlage VI. Microwave Modeling and Simulation 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 50 Steven Anlage VII. QC-Related Microwave Technology Isolating the Qubit from External Radiation The experiments take place in a shielded room with filtered AC lines and electrical feedthroughs Grounding the leads prevents electro-static discharge in to the device Cu patch box helps to thermalize the leads Low-Pass LC-filter (fc = 10 MHz) A. J. Przybysz, Ph.D. Thesis, UMD (2010) Cu Powder Filter (strong suppression of microwaves) 51 Steven Anlage VII. QC-Related Microwave Technology Typical SC QC Experimental set-up Attenuators 300K Directional coupler (HP87301D, 1-40 GHz) T = 25 mK Bias Tee Pi Po 4K 300K Mixer 3 dB Attenuator dc block LNA Isolators fpump fprobe Pulse IF amp Isolator 4-8 GHz fLO Resonator CPB Vg (slide courtesy Z. Kim, LPS) 52 LO LNA for qubit for resonator IF RF Amplitude and phase Steven Anlage Digital oscilloscope VII. QC-Related Microwave Technology 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 Bias Tee RF input Coupled Signal (-x dB) x = 6, 10, 20, etc. 53 RF+DC output DC input Steven Anlage Isolator RF input Output “Left-Going” reflected signals are absorbed by the matched load VII. QC-Related Microwave Technology Active Cryogenic Microwave Hardware Low-Noise Amplifier (e.g. Low-Noise Factory LNF-LNC6_20B) LNA A measure of noise added to signal 13 mW dissipated power Mixer (e.g. Marki Microwave T3-0838) Mixer RF The ‘RF’ signal is mixed with a Local Oscillator signal to produce an Intermediate Frequency signal at the difference frequency IF LO 54 Steven Anlage VII. QC-Related Microwave Technology Cryogenic Microwave Calibration J.-H. Yeh, RSI 84, 034706 (2013); L. Ranzani, RSI 84, 034704 (2013) 55 Steven Anlage VII. QC-Related Microwave Technology 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. Steven Anlage 56 Superconductivity Links Wikipedia article on Superconductivity http://en.wikipedia.org/wiki/Superconductivity Gallery of Abrikosov Vortex Lattices http://www.fys.uio.no/super/vortex/ Graduate course on Superconductivity (Anlage) http://www.physics.umd.edu/courses/Phys798I/anlage/AnlageFall12/index.html YouTube videos of Superconductivity (a classic film by Prof. Alfred Leitner) http://www.youtube.com/watch?v=nLWUtUZvOP8 57 Steven Anlage Superconducting Quantum Computation Half Day Short Course Schedule 58 1:00 PM - 1:05 PM Welcome and Overview Fred Wellstood, University of Maryland 1:05 PM - 1:40 PM Essentials of Superconductivity Steve Anlage, University of Maryland 1:40 PM - 1:45 PM Short session break 1:45 PM - 2:45 PM Introduction to Qubits and Quantum Computation Fred Wellstood, University of Maryland 2:45 PM - 3:00 PM Short session break 3:00 PM - 3:50 PM Fundamentals of Microwave Measurements Steve Anlage, University of Maryland 3:50 PM - 4:00 PM Short session break 4:00 PM - 5:00 PM Quantum Superconducting Devices Ben Palmer, LPS Steven Anlage Please Ask Questions! 59 Steven Anlage 60 Steven Anlage Details and Backup Slides 61 Steven Anlage [Return] What are the Limits of Superconductivity? Phase Diagram Jc Normal State Superconducting State Tc Ginzburg-Landau free energy density 62 Temperature Dependence Steven Anlage Currents 0Hc2 Applied Magnetic Field [Return] BCS Theory of Superconductivity Bardeen-Cooper-Schrieffer (BCS) Cooper Pair s-wave ( = 0) pairing + + S + + Spin singlet pair v v + + + + S First electron polarizes the lattice Tc WDebye e1/ N ( EF )V Second electron is attracted to the concentration of positive charges left behind by the first electron WDebye is the characteristic phonon (lattice vibration) frequency N(EF) is the electronic density of states at the Fermi Energy V is the attractive electron-electron interaction A many-electron quantum wavefunction made up of Cooper pairs is constructed with these properties: An energy 2D(T) is required to break a Cooper pair into two quasiparticles (roughly speaking) Cooper pair ‘size’: x vF 63 D http://www.chemsoc.org/exemplarchem/entries/igrant/hightctheory_noflash.html Steven Anlage Superconductor Electrodynamics T=0 s1(w) s2(w) ~ 1/w ideal s-wave Normal State (T > Tc) (Drude Model) s2(w) ns(T) [Return] s s 1 i s 2 1.0 0.8 0.6 nse2/mw 0.4 s1(w) 0.2 0 Superfluid density l2 ~ m/ns ~ 1/wps2 0 0.0 0.5 1.0 0 1/t2D / 1.5 2.0 2.5 3.0 3.5 w 0 Tc “binding energy” of Cooper pair (100 GHz ~ few THz) Surface Impedance (w > 0) Zs Rs iX s iw 0 / s Superconducting State (w < 2D) Normal State Rs X s w 0 1 2s 1 s 1 Rs ~ s1 0 X s 0wl Penetration depth l(0) ~ 20 – 200 nm Finite-temperature: Xs(T) = wL = w0l(T) → ∞ as T →Tc (and wps(T) → 0) Narrow wire or thin film of thickness t : L(T) = 0l(T) coth(t/l(T)) → 0 l2(T)/t Kinetic Inductance 64 Steven Anlage T [Return] BCS Microwave Electrodynamics Low Microwave Dissipation Full energy gap → Rs can be made arbitrarily small for T < Tc/3 in a fully-gapped SC Rs,residual ~ 10-9 W at 1.5 GHz in Nb ky Filled Fermi Sea node HTS materials have nodes in Dd the energy gap. This leads to power-law behavior of l(T) and Rs(T) and residual losses 10 kx ky Filled Fermi Sea kx 15 T (K) 10 YBa Cu O -1 2 10 3 7-x -2 sputtered LaAlO 3 10 -3 coevaporated MgO 10 -4 Nb Sn on 3 l T l 0 a T Rs Rs,residual b T 10 log{R (T)} –D/kT ∙T /T s c Steven Anlage c sapphire -5 0 2 4 6 8 10 Inverse reduced temperature T /T c Rs,residual ~ 10-5 W at 10 GHz in YBa2Cu3O7- 65 20 @ f=87 GHz s Rs RBCS T Rs,residual Ds 90 40 Surface resistance R (W) Rs e D 0 / k BT M. Hein, Wuppertal [Return] The London Equations Newton’s 2nd Law for a charge carrier Superconductor: 1/t 0 1st London Eq. and B (Faraday) yield: E t mv dv m eE dt t t = momentum relaxation time Js = n s e vs dJ s ns e2 1 E E 2 dt m 0lL ns e 2 d J s dt m B 0 1st London Equation London surmise ns e 2 Js B0 m 2nd London Equation These equations yield the Meissner screening H z vacuum superconductor z/l 1 2 H H 0e L H 2 H lL z lL lL is frequency independent (w < 2D/ħ) 66 Steven Anlage lL m 0 ns e 2 lL ~ 20 – 200 nm [Return] The London Equations continued Normal metal E is the source of Jn Lenz’s Law Jn s nE d 1 J n dt 0 lL 2 Superconductor dJ s 1 E E=0: Js goes on forever 2 dt 0 lL B is the source of 2 B 0 0 lL J s B zero frequency Js, spontaneous flux exclusion 1st London Equation E is required to maintain an ac current in a SC Cooper pair has finite inertia QPs are accelerated and dissipation occurs 67 Steven Anlage [Return] Fluxoid Quantization Superconductor is described by a single Macroscopic Quantum Wavefunction ei Flux F Consequences: Magnetic flux is quantized in units of F0 = h/2e (= 2.07 x 10-15 Tm2) R = 0 allows persistent currents Current I flows to maintain F = n F0 in loop n = integer, h = Planck’s const., 2e = Cooper pair charge Surface S I Circuit C superconductor B Fluxoid Quantization 0 l J s d B dS nF 0 2 C S Flux F 68 Steven Anlage n = 0, ±1, ±2, … [Return] Cavity Perturbation Objective: determine Rs, Xs (or s1, s2) from f0 and Q measurements of a resonant cavity containing the sample of interest Input ~ microwave wavelength l Microwave Resonator Output transmission f f’ Sample at Temperature T f0 f0’ frequency Df = f0’ – f0 D(Stored Energy) D(1/2Q) D(Dissipated Energy) U Stored f 0 U Dissipated f Rs 69 T2 B Quality Factor Q T1 Cavity perturbation means Df << f0 Q DX s Steven Anlage 2 Df f is the sample/cavity geometry factor III. Network Analysis Superconducting Quantum Interference Devices (SQUIDs) [Return] The DC SQUID A Sensitive Magnetic Flux-to-Voltage Transducer Bias Current Flux Ic 2Ic Ic Wikipedia.org One can measure small fields (e.g. 5 aT) with low noise (e.g. 3 fT/Hz1/2) Used extensively for Magnetometry (m vs. H) Magnetoencephalography (MEG) Magnetic microscopy Low-field magnetic resonance imaging 70 Steven Anlage Superconducting Quantum Interference Devices (SQUIDs) [Return] The RF SQUID A High-Frequency Magnetic Flux-to-Voltage Transducer RF SQUID Tank Circuit 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 71 Steven Anlage [Return] Transmission Lines, continued The power absorbed in a termination is: Model of a realistic transmission line including loss Shunt Conductance Traveling Wave solutions 72 with Steven Anlage III. Network Analysis [Return] How Much Power Reaches the Load? Agilent – Back to Basics Seminar 73 Steven Anlage III. Network Analysis Waveguides [Return] H Rectangular metallic waveguide 74 Steven Anlage III. Network Analysis Transmission Line Resonators [Return] Transmission Line Model Transmission Line Unit Cell Transmission Line Resonator Model Ccoupling Ln l 2 c fn n 2L 75 Steven Anlage Ccoupling n 1, 2, 3, ... IV. Superconducting Microwave Resonators for QC [Return] Measuring the Q of a Shunt-Coupled Resonator Ch. Kaiser, Sup Sci Tech 23, 075008 (2010) 76 Steven Anlage [Return] Rapid Single Flux Quantum Logic superconducting “classical” digital computing Ibias ~1mV Input JJ ~2ps When (Input + Ibias) exceeds JJ critical current Ic, JJ “flips”, producing an SFQ pulse. Area of the pulse is F0=2.067 mV-ps Pulse width shrinks as JC increases SFQ logic is based on counting single flux quanta Courtesy Arnold Silver 77 Steven Anlage 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 Mesoscopic Superconductivity [Return] G , BCS k kM u k k k1 vk ck ck 0 i e DN D BCS Ground state wavefunction Coherent state with no fixed number of Cooper pairs 1 Number-phase uncertainty (conjugate variables) 2 Typically: Bulk Superconductor Rbarrier > 2ħ/e2 ~ 6 kW C Small superconducting island Josephson tunnel barrier phase difference g Using Q = CV and Q/2e = N = i ∂/∂ i ∂/∂g the Hamiltonian is: H 4EC / g EJ cosg 2 2 EC e2 2C e2/2C > kBT (requires C ~ fF at 1 K) Classical energy 1 E CV 2 E J cos g 2 KE PE 1 dg E C EJ cosg 2 2e dt 2 2 Two important limits: EC << EJ well-define phase utilize phase eigenstates, like G, BCS EC >> EJ all values of equally probable utilize number eigenstates N 78 Steven Anlage EJ I c 2e