basics of rf superconductivity and qc-related microwave issues short course tutorial

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StevenAnlage

Basics of RF Superconductivity and QC-Related Microwave Issues

Short Course Tutorial Superconducting Quantum Computing

2014 Applied Superconductivity ConferenceCharlotte, North Carolina USA

Steven M. Anlage

Center for Nanophysics and Advanced MaterialsPhysics Department

University of MarylandCollege Park, MD 20742-4111 USA

[email protected]

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StevenAnlage

Objective

To give a basic introduction to superconductivity, superconducting electrodynamics, and microwave measurements as background for the Short CourseTutorial “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:

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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 ofSuperconductivity

Fundamentals ofMicrowave

Measurements

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Please Ask Questions!

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I. Hallmarks of Superconductivity

• The Three Hallmarks of Superconductivity• Zero Resistance• Complete* Diamagnetism• Macroscopic Quantum Effects

• Superconductors in a Magnetic Field• Vortices and Dissipation


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The Three Hallmarks of Superconductivity

Zero Resistance

I

V

DC

Res

ista

nce

TemperatureTc

0

Complete Diamagnetism

Mag

neti

c In

duct

ion

TemperatureTc

0

T>Tc T<Tc

Macroscopic Quantum Effects

Flux F

Flux quantization F = nF0

Josephson Effects

B


[BCS Theory]

[Mesoscopic Superconductivity]

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Zero Resistance

R = 0 only at w = 0 (DC)

R > 0 for w > 0 and T > 0

EQuasiparticles

Cooper Pairs

2D

0

The Kamerlingh Onnes resistance measurement of mercury. At 4.15K the resistance suddenly dropped to zero

EnergyGap


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Perfect DiamagnetismMagnetic 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 2003

Super-conductor

T>Tc T<Tc

H

H

Spontaneous exclusion of magnetic flux

H

H

00 MHB

T

l(T)

l(0)

Tc

LzeHH /0

zH

z

vacuum superconductor

l

l is independent of frequency (w < 2 /D ħ)

magneticpenetration

depth =0B

surfacescreeningcurrents

l


https://www.youtube.com/watch?v=IPduAYKk_6I

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Macroscopic Quantum EffectsSuperconductor is described by a singleMacroscopic Quantum Wavefunction

ie

Consequences:Magnetic flux is quantized in units of F0 = h/2e (= 2.07 x 10-15 Tm2)

R = 0 allows persistent currentsCurrent I flows to maintain F = n F0 in loop n = integer, h = Planck’s const., 2e = Cooper pair charge[DETAILS]

Flux F

Isuperconductor

Sachdev and Zhang, Science

Magnetic vortices have quantized flux

vortexlattice

B

screeningcurrentsx

B(x)|Y(x)|

0x l

Type II

vortexcore

A vortexLine cut

0n

2


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Φ=∯𝐵 ∙𝑑𝐴=∯𝐵⊥𝑑𝐴

𝐁⊥

Definition of “Flux”

surface surface


Ad

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Macroscopic Quantum Effects Continued

Josephson Effects (Tunneling of Cooper Pairs)

111

ie 222

ie

I

DC Josephson Effect (VDC=0)

sincII

DCVe

dt

d

AC Josephson Effect (VDC0)

Quantum VCO:VDC

1 2

(Tunnel barrier)

0sin tVe

II DCc

μV

MHz593420.483

1*

0

h

e

2

1

21

2ldA

e

Gauge-invariantphase difference:

A

AB


[DC SQUID Detail] [RF SQUID Detail]

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AbrikosovVortex Lattice

Superconductors in a Magnetic FieldThe Vortex State

T

H

Hc1(0)

Tc

Meissner State

Hc2(0)

B = 0, R = 0

B 0, R 0

NormalState

Type II SC

vortex

LorentzForce

Moving vorticescreate a longitudinal voltage

V>0

I

0̂JFL

J

vortexDrag vF

Vortices also experiencea viscous drag force:

2


[Phase diagram Details]

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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


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Why are Superconductors so Useful at High Frequencies?

Low Losses: Filters have low insertion loss Better S/N, filters can be made smallNMR/MRI SC RF pickup coils x10 improvement in speed of spectrometerHigh Q Filters with steep skirts, good out-of-band rejection

Low Dispersion: SC transmission lines can carry short pulses with little distortionRSFQ logic pulses – 2 ps long, ~1 mV in amplitude: psmV07.20 dttV

Superconducting Transmission Lines

E

BJ

microstrip(thickness t)

groundplane

C

KineticInductanceLkin

GeometricalInductanceLgeo

LCvphase

1

dielectric

propagating TEM wave tLkin

2

~

L = Lkin + Lgeo is frequency independentattenuation a ~ 0


[RSFQ Details]

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Electrodynamics of Superconductorsin the Meissner State (Two-Fluid Model)

EQuasiparticles(Normal Fluid)

Cooper Pairs(Super Fluid)

2D

0Ls

sn

Superfluid channel

Normal Fluid channel

EnergyGap

s = sn – i s2

J = Js + JnJs

Jn

AC Current-carrying superconductor

J = s E sn = nne2t/ms2 = nse2/mw

nn = number of QPsns = number of SC electronst = QP momentum relaxation timem = carrier massw = frequencyT0

nnn(T)

ns(T)

Tc

Jn = ns(T) + nn(T)


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Surface Impedance

x

-z

y

J

EH

i

dzzJ

EiXRZ sss

Surface Resistance Rs: Measure of Ohmic power dissipation

s

Surface

s

Volume

Dissipated RIdAHRdVEJP 22

2

1~

2

1Re

2

1

Rs

conductorLocal Limit

Surface Reactance Xs: Measure of stored energy per period

222

2

1~

2

1Im

2

1LIdAHXdVEJHW

Surface

s

Volume

Stored

Xs

Xs = wLs = wmlLgeo Lkinetic


[London Eqs. Details1, Details2]

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10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

10-1 100 101 102

Rs-epiRs-polyRs-Nb3SnRscalc-epiRscalc-poyRscalc-Nb3SnS

urfa

ce r

esis

tanc

e R

s (

)

frequency f (GHz)

Cu(77K)

poly

epitaxial

Nb3Sn

T=0.85Tc

T=0.5Tc

YBCO

Two-Fluid Surface Impedance

Because Rs ~ w2:The advantage of SC over Cu diminishes with increasing frequency

HTS: Rs crossover at f ~ 100 GHz at 77 K

Ls

sn

Superfluid channel

Normal Fluid channel

nsR 30

2

2

1

0sX

sss iXRZ

M. Hein, Wuppertal

Rs ~ w2

Rn ~ w1/2

nn /1~R~R ns


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Kinetic Inductance𝐿=𝐿𝑔𝑒𝑜+𝐿𝑘𝑖𝑛𝑒𝑡𝑖𝑐

𝐿𝑔𝑒𝑜=Φ / 𝐼Flux integral surface A measure of energy stored in magnetic fields

both outside and inside the conductor

𝑈𝑚𝑎𝑔=12𝐿

𝑔𝑒𝑜

𝐼 2

Φ

dVzyxJzyxI

L skinetic ),,(),,( 22

20

A measure of energy stored in dissipation-less currents inside the superconductor

)coth(/ 0

t

wLkinetic

𝑤

𝑡

𝐽 𝑠

(valid when l << w, see Orlando+Delin)

𝑈𝑘𝑖𝑛𝑒𝑡𝑖𝑐=12𝐿

𝑘𝑖𝑛𝑒𝑡𝑖𝑐

𝐼 2

For a current-carrying strip conductor:

In the limit of t << l or w << l : tLkinetic

20/

… and this can get very large forlow-carrier density metals (e.g. TiNor Mo1-xGex), or near Tc


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J. Baselmans, JLTP (2012)

Microwave Kinetic Inductance Detectors (MKIDs)

Kinetic Inductance (Continued)

Microstrip HTS transmission lineresonator (B. W. Langley, RSI, 1991)

(C. Kurter, PRB, 2013)


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Josephson InductanceJosephson Inductance is large, tunable and nonlinear

)cos(20

cJJ I

L

LJJ R C

Resistively and Capacitively Shunted Junction (RCSJ) Model

The Josephson inductance can betuned e.g. when the JJ is incorporated

into a loop and flux F is applied

JJL

rf SQUID

𝜱

)sin(cII Here is a non-rigorous derivation of LJJ

Start with the dc Josephson relation:

Take the time derivative anduse the ac Josephson relation:

)cos(2

)cos( Ve

III cc

ILV JJSolving for voltage as:

Yields:

is the gauge-invariantphase difference acrossthe junction

0 ninducedapplied


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Single-rf-SQUID Resonance Tuning with DC Magnetic FluxComparison to Model

RF power = -80 dBm, @6.5K

Maximum Tuning: 80 THz/Gauss @ 12 GHz, 6.5 KTotal Tunability: 56%

rf SQUID

𝜱


M. Trepanier, PRX 2013

Red represents resonance dip

RCSJ model

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Superconducting Quantum Computation Half Day Short Course Schedule

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

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III. Network Analysis

• Network vs. Spectrum Analysis

• Scattering (S) Parameters• Quality Factor Q

• Cavity Perturbation Theory [Details]


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Network vs. Spectrum Analysis

Agilent – Back to Basics Seminar


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Network Analysis

Assumes linearity!2-port system

frequency (f)f0

|S21(f)|2 resonator transmission

P. K. Day, Nature, 2003

|S21

| (dB

)input (1) output (2)

B

1 2


ResonantCavity

Co-

Pla

nar

Wav

egui

de (

CP

W)

Res

onat

or

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Transmission LinesTransmission lines carry microwave signals from one point to another

They are important because the wavelength is much smaller than the length of typical T-linesused in the lab

You have to look at them as distributed circuits, rather than lumped circuits

The wave equations

V


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Transmission Lines

Wave Speed Take the ratio of the voltage and current wavesat any given point in the transmission line:

= Z0

The characteristic impedance Z0 of the T-line

Reflections from a terminated transmission line

ZLZ0 0

0

ZZ

ZZ

a

b

V

V

L

L

right

left

Reflectioncoefficient

Some interesting special cases:

Open Circuit ZL = ∞, G = 1 ei0

Short Circuit ZL = 0, G = 1 eip

Perfect Load ZL = Z0, G = 0 ei?


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Transmission Lines and Their Characteristic Impedances


Attenuation is lowestat 77 W

50 W Standard

Power handling capacitypeaks at 30 W

Nor

mal

ized

Val

ues

Characteristic Impedancefor coaxial cable ( )W

[Transmission Line Detail]

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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 matrix

NN I

I

I

V

V

V

2

1

2

1

][

S matrix

NN V

V

V

S

V

V

V

2

1

2

1

][

1V

1VV1 , I1

VN , IN

NV

NV

S = Scattering MatrixZ = Impedance MatrixZ0 = (diagonal) Characteristic Impedance Matrix

)()( 01

0 ZZZZS

)(),( SZ


Z0,1

Z0,N

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Linear vs. Nonlinear Behavior


DeviceUnderTest


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Quality Factor• Two important quantities characterise a resonator:

The resonance frequency f0 and the quality factor Q

cP

U

f

fQ 00

• Where U is the energy stored in the cavity volume and Pc/0 is the energy lost per RF period by the induced surface currents

)exp(0 LtUU

31

Frequency Offset from Resonance f – f0

0 Df-Df

= Df

Sto

red

Ene

rgy

U

Some typical Q-values: SRF accelerator cavity Q ~ 1011 3D qubit cavity Q ~ 108


[Q of a shunt-coupled resonator Detail]

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Scattering Parameter of Resonators


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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

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Resonators… the building block of superconducting applications …

Microwave surface impedance measurementsCavity Quantum Electrodynamics of QubitsSuperconducting RF AcceleratorsMetamaterials (meff < 0 ‘atoms’)etc.

Pout

co-planar waveguide (CPW) resonator

Pin

frequencyf0

|S21(f)|2 resonator transmission

Port 1

Port 2

TransmittedPower


for QC

CPW Field Structure

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http://www.phys.unsw.edu.au

Animation link

The Inductor-Capacitor Circuit Resonator

4

2

2

4

6

8

10

frequency (f)f0

Re[Z]

Im[Z]

Impedance ofa Resonator

Im[Z] = 0 on resonance


for QC

http://www.softwareforeducation.com/wikileki/index.php?title=L_C_Tuned_Circuit

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Lumped-Element LC-Resonator

Z. Kim

Pout


for QC

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Resonators (continued)

T = 79 KP = - 10 dBmf = 5.285 GHz

YBCO/LaAlO3

CPW ResonatorExcited in Fundamental ModeImaged by Laser Scanning Microscopy*

Wstrip = 500 mm

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)

Scanned Area

RF inputRF output

YBCO Ground Plane

YBCO Ground Plane

STO Substrate


for QC

[Trans. Line Resonator Detail]

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IV. Superconducting

Microwave Resonatorsfor QC

Three-Dimensional Resonator

Inductively-coupled cylindrical cavity with sapphire “hot-finger” TE011 mode Electric fields

Rahul Gogna, UMD

M. Reagor, APL 102, 192604 (2013)

Al cylindrical cavityTE011 modeQ = 6 x 108

T = 20 mK

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Coupling to Resonators

Inductive Capacitive

Loop antennas: Coax cables

Nb spiralon quartz

LumpedInductor

LumpedCapacitor

P. Bertet SPEC, CEA Saclay


for QC

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V. Microwave Losses

• Microscopic Sources of Loss• 2-Level Systems (TLS) in Dielectrics• Flux Motion

• What Limits the Q of Resonators?

V. Microwave Losses

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V. Microwave Losses

Microwave Losses / 2-Level Systems (TLS) in Dielectrics

M. Hein, APL 80, 1007 (2002)

Eff

ecti

ve L

oss

@ 2

.3 G

Hz

(W)

-60 dBm

-20 dBm

TLS in MgO substrates

YBCO film on MgO

YBCO/MgO

T = 5K

Nb/MgO

Energy

Energy

Low Power

High Power

Two-LevelSystem (TLS)

Classic reference:

photon

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Microwave Losses / Flux MotionSingle-vortex response (Gittleman-Rosenblum model)

𝑚 ̈𝑥+𝜂 ̇𝑥+𝐹𝑝𝑖𝑛=𝐽 ×Φ𝑚

Equation of motion for vortex in a rigid lattice (vortex-vortex force is constant)

Effective mass of vortex

𝜂 Vortex viscosity ~ sn

𝐽×Φ𝐹 𝑝𝑖𝑛=−𝑘𝑥𝜂 �̇�

Ignore vortex inertia

vortex

Pinningpotential

Frequency f / f0

Dis

sipa

ted

Pow

er P

/ P

0

2 𝜋 𝑓 0 ≡𝑘/𝜂Pinning frequency 0

Gittleman, PRL (1966)

V. Microwave Losses

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What Limits the Q of Resonators?

1𝑄𝑇𝑜𝑡𝑎𝑙

=1

𝑄 h𝑂 𝑚𝑖𝑐

+1

𝑄𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐

+1

𝑄𝐶𝑜𝑢𝑝𝑙𝑖𝑛𝑔

+1

𝑄𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛

+…

Assumption: Loss mechanisms add linearly

𝑄Stored EnergyEnergy Dissipated per Cycle

1𝑄 h𝑂 𝑚𝑖𝑐

𝑅𝑠

2∯|𝐻𝑡𝑎𝑛𝑔|

2𝑑𝑆

1𝑄𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟 𝑖𝑐

𝜔𝜖 ′ ′2 ∰|𝐸|2𝑑𝑉

1𝑄𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛

𝑃𝑅𝑎𝑑𝑖𝑎𝑡𝑒𝑑

1𝑄𝐶𝑜𝑢𝑝𝑙𝑖𝑛𝑔

Power dissipated in load impedance(s)

V. Microwave Losses

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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 Modelingand Simulation

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Finite-Element Method (FEM): Create a finite-element mesh (triangles and tetrahedra), expand the fieldsin 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 gridstaggered 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

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 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

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Example FDTD grid with ‘Yee’ cells

Examples FEM triangular / tetrahedral meshes

Dav

e M

orri

s, A

gile

nt 5

990-

9759

Example CEM Mesh and Grid


Grid approximation for a sphere

http://cp.literature.agilent.com/litweb/pdf/5990-9759EN.pdf

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CEM SolversEigenmode: Closed system, finds the eigen-frequencies and Q values

Driven: The system has one or more ‘ports’ connected to infinity by a transmission line orfree-space propagating mode. Calculate the Scattering (S) Parameters.

Driven anharmonic billiard

Slave 1

Slave 2

Master 2

Master 1

Dirac point

Fre

quen

cy (G

Hz)

Wavenumber ( )


Gaussian wave-packet excitation with antenna array

Rahul Gogna, UMD

Bo Xiao, UMD

Harita Tenneti, UMD

Propagation simulates time-evolution

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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)


Bo Xiao, UMD

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• 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

TE311


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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


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A. J. Przybysz, Ph.D. Thesis, UMD (2010)

Grounding the leads preventselectro-static discharge in tothe device

Isolating the Qubit from External Radiation

Cu patch box helps to thermalize the leads

Low-Pass LC-filter (fc = 10 MHz)

Cu Powder Filter(strong suppression of microwaves)

The experiments take place in a shielded roomwith filtered AC lines and electrical feedthroughs

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Typical SC QC Experimental set-up

LNA

T = 25 mK

fLO

IFdc block

Mixer

IF amp

Bias Tee

Resonator CPB

Isolators

RF

4 K 300K

fpump for

qubit

Vg

Directional coupler (HP87301D, 1-40 GHz)

300K

LNA

3 dB AttenuatorPi Po

Pulse

Isolator 4-8 GHz

LO

Attenuators

fprobe for

resonator

Digital oscilloscopeAmplitude and phase(slide courtesy Z. Kim, LPS)


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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

CoupledSignal (-x dB)

x = 6, 10, 20, etc.

Bias Tee

DC input

RF input RF+DC output

Isolator

RF input Output

“Left-Going” reflectedsignals are absorbed

by the matched load


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Active Cryogenic Microwave Hardware

Low-Noise Amplifier (e.g. Low-Noise Factory LNF-LNC6_20B)

LNA

13 mW dissipatedpower

A measure ofnoise added to signal

MixerRF

LO

IF

Mixer (e.g. Marki Microwave T3-0838)

The ‘RF’ signal is mixed with a Local Oscillatorsignal to produce an Intermediate Frequencysignal at the difference frequency


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Cryogenic Microwave CalibrationJ.-H. Yeh, RSI 84, 034706 (2013); L. Ranzani, RSI 84, 034704 (2013)


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References and Further ReadingZ. 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.

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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 Superconductivityhttp://en.wikipedia.org/wiki/Superconductivity

YouTube videos of Superconductivity (a classic film by Prof. Alfred Leitner)http://www.youtube.com/watch?v=nLWUtUZvOP8

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Superconducting Quantum Computation Half Day Short Course Schedule

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

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Please Ask Questions!

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Details and Backup Slides

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What are the Limits of Superconductivity?

Phase Diagram

SuperconductingState

NormalState

2

**2

1

2

20

242

normalsuper

hAe

im

TTff

Ginzburg-Landau free energy density Applied Magnetic FieldCurrentsTemperature

Dependence

Tcm0Hc2

Jc

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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 chargesleft behind by the first electron

Sv+ +

++

Sv + +

++

s-wave ( = 0) pairing

Spin singlet pair

Cooper Pair

VENDebyec

FeT )(/1WDebye is the characteristic phonon (lattice vibration) frequencyN(EF) is the electronic density of states at the Fermi EnergyV is the attractive electron-electron interaction

An energy 2D(T) is required to break a Cooper pair into two quasiparticles (roughly speaking)

FvCooper pair ‘size’:

A many-electron quantum wavefunction Y made up of Cooper pairs is constructedwith these properties:

Bardeen-Cooper-Schrieffer (BCS)

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0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5

0 .2

0 .4

0 .6

0 .8

1 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5

0 .2

0 .4

0 .6

0 .8

1 .0

00

s1(w)

s2(w)

s1(w)

s2(w) ~ 1/w

w

= s s1-is2

Superconductor Electrodynamics

“binding energy” of Cooper pair (100 GHz ~ few THz)

T = 0ideal s-wave

/0iiXRZ sss Surface Impedance ( w > 0)Normal State

11

0 1

2 ss XR

Superconducting State (w < 2D)

01 0~ ss XR

Penetration depthl(0) ~ 20 – 200 nm

( p nse2/m) ( )d w

Finite-temperature: Xs(T) = wL = wm0l(T) → ∞ as T →Tc (and wps(T) → 0)

Narrow wire or thin film of thickness t : L(T) = m0l(T) coth(t/l(T)) → m0 l2(T)/tKinetic Inductance

Superfluid densityl2 ~ m/ns

~ 1/wps2

T

ns(T)

Tc00

/2

Normal State (T > Tc)(Drude Model)

1/t

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BCS Microwave ElectrodynamicsLow Microwave Dissipation

Full energy gap → Rs can be made arbitrarily small

Tks

BeR/0 for T < Tc/3 in a

fully-gapped SC

10-5

10-4

10-3

10-2

10-1

0 2 4 6 8 10

Rs[Ohm]

Rs(59/162)

Rs(33b)

Sur

face

res

ista

nce

Rs (

)

Inverse reduced temperature Tc/T

YBa2Cu

3O

7-x

@ f=87 GHz

sputteredLaAlO

3

coevaporatedMgO

Nb3Sn on

sapphirelog{R

s(T)} –/kT

c∙T

c/T

90 1040 20 15 T (K)

residual,sBCSs RTRR

Rs,residual ~ 10-9 W at 1.5 GHz in Nb

M. Hein, Wuppertal

FilledFermiSea

Dskx

ky

HTS materials have nodes inthe energy gap. This leadsto power-law behavior ofl(T) and Rs(T) and residual losses

TaT 0TbRR ss residual,

Rs,residual ~ 10-5 W at 10 GHz in YBa2Cu3O7-d

FilledFermiSea

Dd

node

kx

ky

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The London Equations

vm

Eedt

vdm

Newton’s 2nd Law for

a charge carrier

Superconductor:1/ t 0

EEm

en

dt

Jd

L

ss

20

2 1

1st London Equation

t = momentum relaxation timeJs = ns e vs

1st London Eq. and (Faraday) yield:

t

BE

02

B

m

enJ

dt

d ss

London

surmise0

2

Bm

enJ s

s

2nd London Equation

These equations yield the Meissner screening

HHL

2

2 1

LzeHH /

0

zH

z

vacuum superconductor

lL

20 en

m

sL

lL ~ 20 – 200 nmlL is frequency independent (w < 2 /D ħ)

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BJBJdt

d

Edt

JdEJ

sL

L

n

L

snn

202

0

20

01

1

The London Equations continued

Normal metal Superconductor

B is the source of zero frequency Js,spontaneous fluxexclusion

E is the source of Jn E=0: Js goes on forever

Lenz’s Law

1st London Equation E is required to maintain an ac current in a SCCooper pair has finite inertia QPs are accelerated and dissipation occurs

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Fluxoid QuantizationSuperconductor is described by a singleMacroscopic Quantum Wavefunction

ie

Consequences:Magnetic flux is quantized in units of F0 = h/2e (= 2.07 x 10-15 Tm2)

R = 0 allows persistent currentsCurrent I flows to maintain F = n F0 in loopn = integer, h = Planck’s const., 2e = Cooper pair charge

Flux F

Isuperconductor

SC

s nSdBdJ 02

0

Flux F

Circuit C

Surface S

B

FluxoidQuantization

n = 0, ±1, ±2, …

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Cavity PerturbationObjective: determine Rs, Xs (or s1, s2) from f0 and Q measurements

of a resonant cavity containing the sample of interest

Sample atTemperature T

MicrowaveResonator

Input Output

frequency

transmission

f0

df

f0’

df ’

Df = f0’ – f0 D(Stored Energy)D(1/2Q) D(Dissipated Energy)

Quality Factor

~ microwavewavelength

l

T1 T2

B

f

f

U

UQ

0

Dissipated

Stored Cavity perturbation means Df << f0

QRs

f

fX s

2G is the sample/cavity geometry factor


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Superconducting Quantum Interference Devices (SQUIDs)The DC SQUID

A Sensitive Magnetic Flux-to-Voltage Transducer

FluxIc

Ic

BiasCurrent

2Ic

One can measure small fields (e.g. 5 aT) with low noise (e.g. 3 fT/Hz1/2)

Used extensively forMagnetometry (m vs. H)Magnetoencephalography (MEG)Magnetic microscopyLow-field magnetic resonance imaging

Wikipedia.org

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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 SQUIDsLess susceptible to noise at 77 K with HTS junctionsBasis for many qubit and meta-atom designs

RF SQUID

TankCircuit

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The power absorbed in a termination is:

Transmission Lines, continued

Model of a realistic transmission line including loss

TravelingWavesolutions

with

ShuntConductance


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How Much Power Reaches the Load?



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Waveguides

Rectangular metallic waveguide

H


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Transmission LineUnit Cell

Transmission Line ResonatorsTransmission Line Model

Transmission Line Resonator Model

Ccoupling Ccoupling

L

cnfn 2

2

nL

...,3,2,1n


for QC

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Measuring the Q of a Shunt-Coupled Resonator

Ch. Kaiser, Sup Sci Tech 23, 075008 (2010)

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http://stacks.iop.org/SUST/23/075008





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Rapid Single Flux Quantum Logicsuperconducting “classical” digital computing

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

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

Ibias

JJ

Courtesy Arnold Silver

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Mesoscopic Superconductivity

01

,

Mkk

kkkkkkBCSG ccvu BCS Ground state wavefunction

Coherent state with no fixed number of Cooper pairs

2

1 Nie Number-phase uncertainty (conjugate variables)

Bulk Superconductor

Small superconductingisland

Josephson tunnel barrierphase difference g cos

2

1 2JECVE

CTypically:

Rbarrier > 2ħ/e2 ~ 6 kW

e2/2C > kBT (requires C ~ fF at 1 K)

cos

22

122

JEdt

d

eCE

KE PEUsing Q = CV and Q/2e = N = i ∂/∂ = f i ∂/∂gthe Hamiltonian is:

e

IE c

J 2

cos/4 22JC EEH

Two important limits:EC << EJ well-define phase f utilize phase eigenstates, like EC >> EJ all values of f equally probable utilize number eigenstates

C

eEC 2

2

BCSG ,

Classical energy

N

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basics of rf superconductivity and qc-related microwave issues short course tutorial

Documents

stevenanlagejosephson

resonant cavity

jj flips

single flux quanta sfq

jj critical current

values of f

speed of light

q measurementsof