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Basics of RF Superconductivity and QC-Related Microwave Issues Short Course Tutorial

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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

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)

Outline

Essentials of

Superconductivity

- Hallmarks of Superconductivity
- Superconducting Transmission Lines
- Network Analysis
- Superconducting Microwave Resonators for QC
- Microwave Losses
- Microwave Modeling and Simulation
- QC-Related Microwave Technology

Fundamentals of

Microwave

Measurements

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

The Three Hallmarks of Superconductivity

Macroscopic Quantum Effects

Complete Diamagnetism

Flux F

T>Tc

T<Tc

Magnetic Induction

0

Flux quantization F = nF0

Josephson Effects

Tc

Temperature

Zero Resistance

I

V

DC Resistance

B

0

Tc

Temperature

[BCS Theory]

[Mesoscopic Superconductivity]

I. Hallmarks of

Superconductivity

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

Cooper Pairs

I. Hallmarks of

Superconductivity

superconductor

l

l(T)

l

magnetic

penetration

depth

B=0

surface

screening

currents

l(0)

T

Tc

The Yamanashi MLX01 MagLevtest vehicle achieved

a speed of 361 mph (581 kph) in 2003

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

Perfect Diamagnetism

Magnetic Fields and Superconductors are not generally compatible

The Meissner Effect

Super-

conductor

T>Tc

T<Tc

Spontaneous exclusion of magnetic flux

I. Hallmarks of

Superconductivity

have quantized flux

B

A vortex

B(x)

vortex

lattice

Line cut

|Y(x)|

Type II

screening

currents

x

vortex

core

0

x

l

Sachdev and Zhang, Science

Macroscopic Quantum Effects

Superconductor is described by a single

Macroscopic Quantum Wavefunction

Consequences:

Flux F

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

[DETAILS]

I

superconductor

I. Hallmarks of

Superconductivity

Macroscopic Quantum Effects Continued

Josephson Effects (Tunneling of Cooper Pairs)

DC Josephson

Effect (VDC=0)

2

1

I

AC Josephson

Effect (VDC0)

(Tunnel barrier)

VDC

Quantum VCO:

Gauge-invariant

phase difference:

[DC SQUID Detail]

[RF SQUID Detail]

I. Hallmarks of

Superconductivity

Superconductors in a Magnetic Field

Lorentz

Force

vortex

Vortices also experience

a viscous drag force:

Moving vortices

create a longitudinal voltage

I

V>0

The Vortex State

H

Normal

State

Hc2(0)

Abrikosov

Vortex Lattice

B 0, R 0

B = 0, R = 0

Hc1(0)

Meissner State

T

Tc

Type II SC

[Phase diagram Details]

I. Hallmarks of

Superconductivity

II. Superconducting Transmission Lines

- Property I: Low-Loss
- Two-Fluid Model
- Surface Impedance andComplex Conductivity[Details]
- BCS Electrodynamics [Details]
- Property II: Low-Dispersion
- Kinetic Inductance
- Josephson Inductance

II. Superconducting

Transmission Lines

Inductance

Lgeo

Kinetic

Inductance

Lkin

Superconducting Transmission Lines

microstrip

(thickness t)

C

B

J

dielectric

E

ground

plane

propagating TEM wave

attenuation a ~ 0

L = Lkin + Lgeois frequency independent

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:

- [RSFQ Details]

II. Superconducting

Transmission Lines

Electrodynamics of Superconductors

in the MeissnerState (Two-Fluid Model)

AC Current-carrying superconductor

Js

J = Js + Jn

J

Jn

n= ns(T)+ nn(T)

nn(T)

ns(T)

J = s E

n

nn = number of QPs

ns = number of SC electrons

t = QP momentum relaxation time

m = carrier mass

w = frequency

sn = nne2t/m

s2 = nse2/mw

s = sn – i s2

0

T

Tc

Normal Fluid channel

E

sn

Quasiparticles

(Normal Fluid)

2D

Energy

Gap

Ls

0

Cooper Pairs

(Super Fluid)

Superfluid channel

II. Superconducting

Transmission Lines

Surface Resistance Rs: Measure of Ohmic power dissipation

Rs

Surface Reactance Xs: Measure of stored energy per period

Xs

Lgeo

Lkinetic

Xs = wLs = wml

Surface Impedance

H

y

E

x

J

conductor

-z

Local Limit

- [London Eqs. Details1, Details2]

II. Superconducting

Transmission Lines

Normal Fluid channel

sn

Rn ~ w1/2

Ls

Superfluid channel

Because Rs ~ w2:

The advantage of SC over Cu

diminishes with increasing frequency

HTS: Rscrossover at f ~ 100 GHz at 77 K

Rs ~ w2

M. Hein, Wuppertal

II. Superconducting

Transmission Lines

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

For a current-carrying strip conductor:

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

… and this can get very large for

low-carrier density metals (e.g. TiN

or Mo1-xGex), or near Tc

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

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)

II. Superconducting

Transmission Lines

Josephson Inductance is large, tunable and nonlinear

Here is a non-rigorous derivation of LJJ

Start with the dc Josephson relation:

Take the time derivative and

use the ac Josephson relation:

Resistively and Capacitively Shunted Junction

(RCSJ) Model

Solving for voltage as:

Yields:

is the gauge-invariant

phase difference across

the junction

The Josephson inductance can be

tuned e.g. when the JJ is incorporated

into a loop and flux F is applied

rf SQUID

II. Superconducting

Transmission Lines

Single-rf-SQUID Resonance Tuning with DC Magnetic Flux

Comparison to Model

RF power = -80 dBm, @6.5K

rf SQUID

Red represents resonance dip

RCSJ model

M. Trepanier, PRX 2013

Maximum Tuning: 80 THz/Gauss @ 12 GHz, 6.5 K

Total Tunability: 56%

II. Superconducting

Transmission Lines

- Network vs. Spectrum Analysis
- Scattering (S) Parameters
- Quality Factor Q
- Cavity Perturbation Theory [Details]

III. Network Analysis

Assumes linearity!

2-port system

output (2)

input (1)

B

Resonant

Cavity

1

2

|S21(f)|2

resonator

transmission

|S21| (dB)

Co-Planar Waveguide (CPW)

Resonator

f0

frequency (f)

P. K. Day, Nature, 2003

III. Network Analysis

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

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

Reflection

coefficient

Z0

ZL

Open Circuit ZL = ∞, G = 1 ei0

Some interesting special cases:

Short Circuit ZL = 0, G = 1 eip

Perfect Load ZL = Z0, G = 0 ei?

III. Network Analysis

Transmission Lines and Their Characteristic Impedances

[Transmission Line Detail]

Attenuation is lowest

at 77 W

50 W Standard

Normalized Values

Power handling capacity

peaks at 30 W

Characteristic Impedance

for coaxial cable (W)

Agilent – Back to Basics Seminar

N-Port Description of an Arbitrary System

Z matrix

S matrix

S = Scattering Matrix

Z = Impedance Matrix

Z0 = (diagonal)

Characteristic

Impedance Matrix

V1 , I1

N-Port System

Described equally well by:

► Voltages and Currents, or

► Incoming and Outgoing Waves

Z0,1

N – Port

System

VN , IN

Z0,N

III. Network Analysis

- Two important quantities characterise a resonator:

The resonance frequency f0 and the quality factor Q

Stored Energy U

= Df

-Df

Df

0

[Q of a shunt-coupled resonator Detail]

Frequency Offset from Resonance f – f0

- 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

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

III. Network Analysis

Scattering Parameter of Resonators

III. Network Analysis

Resonators for QC

- Thin Film Resonators
- Co-planar Waveguide
- Lumped-Element
- SQUID-based
- Bulk Resonators
- Coupling to Resonators

IV. Superconducting

Microwave Resonators

for QC

… the building block of superconducting applications …

Microwave surface impedance measurements

Cavity Quantum Electrodynamics of Qubits

Superconducting RF Accelerators

Metamaterials (meff < 0 ‘atoms’)

etc.

CPW Field Structure

co-planar waveguide (CPW)

resonator

Pout

Pin

Port 2

Port 1

|S21(f)|2

resonator

transmission

Transmitted

Power

f0

frequency

IV. Superconducting

Microwave Resonators

for QC

The Inductor-Capacitor Circuit Resonator

Animation link

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

Impedance of

a Resonator

Re[Z]

Im[Z]

Im[Z] = 0 on resonance

frequency (f)

f0

IV. Superconducting

Microwave Resonators

for QC

YBCO/LaAlO3

CPW Resonator

Excited in Fundamental Mode

Imaged by Laser Scanning Microscopy*

1 x 8 mm scan

T = 79 K

P = - 10 dBm

f = 5.285 GHz

Wstrip = 500 mm

[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)

IV. Superconducting

Microwave Resonators

for QC

Rahul Gogna, UMD

TE011 mode Electric fields

Inductively-coupled cylindrical cavity with sapphire “hot-finger”

Al cylindrical cavity

TE011 mode

Q = 6 x 108

T = 20 mK

M. Reagor, APL 102, 192604 (2013)

IV. Superconducting

Microwave Resonators

for QC

Inductive

Capacitive

Lumped

Inductor

Lumped

Capacitor

P. Bertet SPEC, CEA Saclay

IV. Superconducting

Microwave Resonators

for QC

- Microscopic Sources of Loss
- 2-Level Systems (TLS) in Dielectrics
- Flux Motion
- What Limits the Q of Resonators?

V. Microwave Losses

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

TLS in MgO substrates

T = 5K

-60 dBm

Nb/MgO

Classic reference:

YBCO/MgO

Effective Loss @ 2.3 GHz (W)

-20 dBm

YBCO film on MgO

M. Hein, APL 80, 1007 (2002)

photon

Energy

Low Power

Two-Level

System (TLS)

High Power

Energy

V. Microwave Losses

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 ~ sn

Pinning

potential

vortex

Ignore vortex inertia

Dissipated Power P / P0

Pinning frequency 0

Frequency f / f0

Gittleman, PRL (1966)

V. Microwave Losses

What Limits the Q of Resonators?

Assumption: Loss mechanisms add linearly

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 of Use

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.

Dave Morris, Agilent 5990-9759

Example FDTD grid with ‘Yee’ cells

Grid approximation for a sphere

Examples FEM triangular / tetrahedral meshes

VI. Microwave Modeling

and Simulation

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

Bo Xiao, UMD

HaritaTenneti, UMD

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

Propagation simulates time-evolution

VI. Microwave Modeling

and Simulation

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

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

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

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)

VII. QC-Related

Microwave Technology

Typical SC QC Experimental set-up

Attenuators

300K

T = 25mK

4 K

300K

Mixer

Directional coupler (HP87301D, 1-40 GHz)

3 dB Attenuator

Pi

Po

Bias Tee

dc block

IF

RF

LNA

IF amp

LO

Isolator 4-8 GHz

Isolators

fpump for qubit

fLO

fprobe for resonator

LNA

Resonator

CPB

Pulse

Vg

Digital oscilloscope

Amplitude and phase

(slide courtesy Z. Kim, LPS)

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

Bias Tee

Isolator

Input

Output

Output

RF input

RF input

RF+DC output

Coupled

Signal (-x dB)

x = 6, 10, 20, etc.

DC input

“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

The ‘RF’ signal is mixed with a Local Oscillator

signal to produce an Intermediate Frequency

signal at the difference frequency

IF

RF

LO

VII. QC-Related

Microwave Technology

Cryogenic Microwave Calibration

J.-H. Yeh, RSI 84, 034706 (2013); L. Ranzani, RSI 84, 034704 (2013)

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.

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

Backup Slides

What are the Limits of Superconductivity?

Phase Diagram

Jc

Normal

State

Superconducting

State

m0Hc2

Tc

Ginzburg-Landau

free energy density

Temperature

Dependence

Currents

Applied Magnetic Field

+

+

+

+

v

v

+

+

+

+

S

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 Y 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’:

[Return]

BCS Theory of Superconductivity

Bardeen-Cooper-Schrieffer (BCS)

Cooper Pair

s-wave ( = 0)pairing

Spin singlet pair

Second electron is attracted to the

concentration of positive charges

left behind by the first electron

First electron polarizes the lattice

http://www.chemsoc.org/exemplarchem/entries/igrant/hightctheory_noflash.html

Superconductor Electrodynamics

[Return]

T = 0

ideal s-wave

s2(w) ~ 1/w

s1(w)

s = s1-is2

Normal State (T > Tc)

(Drude Model)

s2(w)

ns(T)

(p nse2/m)d(w)

s1(w)

T

0

Tc

0

Superfluid density

l2 ~ m/ns

~ 1/wps2

w

0

1/t

0

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

Surface Impedance (w > 0)

Superconducting State (w < 2D)

Normal State

Penetration depth

l(0) ~ 20 – 200 nm

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)) → m0l2(T)/t

Kinetic Inductance

ky

HTS materials have nodes in

the energy gap. This leads

to power-law behavior of

l(T) and Rs(T) and residual losses

Dd

Filled

Fermi

Sea

kx

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

BCS Microwave Electrodynamics

Low Microwave Dissipation

[Return]

Full energy gap → Rs can be made arbitrarily small

for T < Tc/3 in a

fully-gapped SC

ky

Ds

Filled

Fermi

Sea

kx

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

M. Hein, Wuppertal

These equations yield the Meissner screening

vacuum

superconductor

lL

lL is frequency independent (w < 2D/ħ)

lL ~ 20 – 200 nm

[Return]

The London Equations

Newton’s 2nd Law for

a charge carrier

t = momentum relaxation time

Js = ns e vs

Superconductor:

1/t 0

1st London Equation

1st London Eq. and

(Faraday) yield:

London

surmise

2nd London Equation

The London Equations continued

Normal metal

Superconductor

E is the source of Jn

E=0: Js goes on forever

B is the source of

zero frequency Js,

spontaneous flux

exclusion

Lenz’s Law

1st London Equation E is required to maintain an ac current in a SC

Cooper pair has finite inertia QPsare accelerated and dissipation occurs

Fluxoid Quantization

Superconductor is described by a single

Macroscopic Quantum Wavefunction

Consequences:

Flux F

Surface S

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

Circuit C

superconductor

B

Fluxoid

Quantization

n = 0, ±1, ±2, …

Flux F

Cavity Perturbation

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

of a resonant cavity containing the sample of interest

transmission

Input

Output

T1

T2

~ microwave

wavelength

l

B

df

df’

Microwave

Resonator

Sample at

Temperature T

frequency

f0’

f0

Quality Factor

Df = f0’ – f0 D(Stored Energy)

D(1/2Q) D(Dissipated Energy)

Cavity perturbation means Df << f0

G is the sample/cavity geometry factor

III. Network Analysis

Superconducting Quantum Interference Devices (SQUIDs)

The DC SQUID

A Sensitive Magnetic Flux-to-Voltage Transducer

[Return]

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

Superconducting Quantum Interference Devices (SQUIDs)

The RF SQUID

A High-Frequency Magnetic Flux-to-Voltage Transducer

[Return]

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

[Return]

The power absorbed in a termination is:

Model of a realistic transmission line including loss

Shunt

Conductance

Traveling

Wave

solutions

with

III. Network Analysis

[Return]

Transmission Line Model

Transmission Line

Unit Cell

Transmission Line Resonator Model

Ccoupling

Ccoupling

IV. Superconducting

Microwave Resonators

for QC

Rapid Single Flux Quantum Logic

superconducting “classical” digital computing

Ibias

~1mV

JJ

~2ps

Input

[Return]

- 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

- 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

[Return]

BCS Ground state wavefunction

Coherent state with no fixed number of Cooper pairs

Number-phase uncertainty (conjugate variables)

Typically:

Rbarrier > 2ħ/e2 ~ 6 kW

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

Bulk Superconductor

Small

superconducting

island

Classical energy

Josephson tunnel barrier

phase difference g

KE

Using Q = CV and Q/2e = N = i ∂/∂f = i∂/∂g

the Hamiltonian is:

PE

Two important limits:

EC << EJ well-define phase f utilize phase eigenstates, like

EC >> EJ all values of f equally probable utilize number eigenstates

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