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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
[email protected]
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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)
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Anlage
Outline
•
•
•
•
•
•
•
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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!
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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
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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]
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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
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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  H 0 e  z / lL
l
T<Tc
Spontaneous exclusion of magnetic flux
B0
z
surface
screening
currents
l(0)
l is independent of frequency (w < 2D/ħ)
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
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)|
Line cut
Type II
x  2l
0x
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l
x
Steven
Anlage
vortex
core
vortex
lattice
screening
currents
Sachdev and Zhang, Science
I. Hallmarks of
Superconductivity
Definition of “Flux”

dA
𝐁⊥
Φ=
surface
10
𝐵 ∙ 𝑑𝐴 =
𝐵⊥ 𝑑𝐴
surface
Steven
Anlage
I. Hallmarks of
Superconductivity
Macroscopic Quantum Effects
Continued
Josephson Effects (Tunneling of Cooper Pairs)
1  1 e
2  2 e
1
2

A
I
(Tunnel barrier)

B   A
VDC
Gauge-invariant
phase difference:
2
2e  
  1  2   A  dl
 1
11
i2
i1
I  I c sin  
DC Josephson
Effect (VDC=0)
d eVDC

dt

 eVDC

I  I c sin 
t  0 
 

AC Josephson
Effect (VDC0)
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)


FL  J  F̂ 0
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
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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
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Js
J = Js + Jn
Jn
e2t/m
sn = nn
s2 = nse2/mw
n
0
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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
   1
2
1 
1 2
PDissipated  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 = wl
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[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
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Anlage
II. Superconducting
Transmission Lines
Kinetic Inductance
𝐿 = 𝐿𝑔𝑒𝑜 + 𝐿𝑘𝑖𝑛𝑒𝑡𝑖𝑐
Flux integral surface
A measure of energy stored in magnetic fields
both outside and inside the conductor
𝐿𝑔𝑒𝑜 = Φ/𝐼
𝑈𝑚𝑎𝑔
Φ
Lkinetic 
0
I
2
 l
2
1
= 𝐿𝑔𝑒𝑜 𝐼 2
2
𝐽𝑠
( x, y, z ) J s2 ( x, y, z ) dV
𝑈𝑘𝑖𝑛𝑒𝑡𝑖𝑐 =
𝑡
1
𝐿𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝐼 2
2
𝑤
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 /  
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l
(valid when l << w, see Orlando+Delin)
0 l2
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)
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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
𝜱
F  F applied  F induced  nF 0
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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
𝜱
RCSJ model
M. Trepanier, PRX 2013
Maximum Tuning: 80 THz/Gauss @ 12 GHz, 6.5 K
Total Tunability: 56%
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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]
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Anlage
III. Network Analysis
Network vs. Spectrum Analysis
Agilent – Back to Basics Seminar
2424
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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
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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?
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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
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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
V N
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  Z 0 ) 1 ( Z  Z 0 )
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
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Anlage
III. Network Analysis
31
Scattering Parameter of Resonators
32
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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
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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
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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
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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
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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
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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
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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)
𝑚𝑥 + 𝜂 𝑥 + 𝐹𝑝𝑖𝑛 = 𝐽 × Φ
Effective mass of vortex
Vortex viscosity ~ sn
Pinning
potential
vortex
Dissipated Power P / P0
𝑚
𝜂
Equation of motion for vortex
in a rigid lattice (vortex-vortex force is constant)
𝐹𝑝𝑖𝑛 = −𝑘𝑥
𝜂𝑥
Ignore vortex inertia
Pinning frequency 𝑓 0
2𝜋 𝑓0 ≡ 𝑘/𝜂
Frequency f / f0
42
𝐽×Φ
Steven
Anlage
Gittleman, PRL (1966)
V. Microwave Losses
What Limits the Q of Resonators?
𝑄~
Stored Energy
Energy Dissipated per Cycle
Assumption: Loss mechanisms add linearly
1
𝑄𝑇𝑜𝑡𝑎𝑙
1
𝑅𝑠
~
𝑄𝑂ℎ𝑚𝑖𝑐 2
=
1
𝑄𝑂ℎ𝑚𝑖𝑐
+
1
𝑄𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
+
1
𝑄𝐶𝑜𝑢𝑝𝑙𝑖𝑛𝑔
+
𝑄𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛
1
2
𝑄𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛
𝐻𝑡𝑎𝑛𝑔 𝑑𝑆
1
𝜔𝜖′′
~
𝑄𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
2
Steven
Anlage
+⋯
~𝑃𝑅𝑎𝑑𝑖𝑎𝑡𝑒𝑑
2
𝐸 𝑑𝑉
1
𝑄𝐶𝑜𝑢𝑝𝑙𝑖𝑛𝑔
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
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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
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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
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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
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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
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Anlage
VII. QC-Related
Microwave Technology
Cryogenic Microwave Calibration
J.-H. Yeh, RSI 84, 034706 (2013); L. Ranzani, RSI 84, 034704 (2013)
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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
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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
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Anlage
60
Steven
Anlage
Details and
Backup Slides
61
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[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 e1/ 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/mw
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) Z s  Rs  iX s  iw 0 / s
Superconducting State (w < 2D)
Normal State
Rs  X s 
w 0
1

2s 1 s 1
Rs ~ s 1  0
X s  0wl
Penetration depth
l(0) ~ 20 – 200 nm
Finite-temperature: Xs(T) = wL = w0l(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
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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
Ds
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
10
Rs  Rs ,residual  b T
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
90 40
Surface resistance R (W)
Rs  e
 D 0  / k BT
M. Hein, Wuppertal
[Return]
The London Equations

 mv
dv
m
 eE 
dt
t
Newton’s 2nd Law for
a charge carrier

dJ s ns e 2 
1 

E
E
2
dt
m
 0 lL
Superconductor:
1/t  0
1st London
Eq. and


B (Faraday) yield:
 E  
t
t = momentum relaxation time
Js = n s e vs
 ns e 2
d 
  J s 
dt 
m

B  0

1st London Equation
London
surmise
 ns e 2 
 Js 
B0
m
2nd London Equation
These equations yield the Meissner screening
vacuum

H z 
superconductor
  z/l
H  H 0e L
 1 
 H 2 H
2
lL
lL
lL is frequency independent (w < 2D/ħ)
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Anlage
lL 
m
0 ns e 2
z
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  0lL   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
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[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
 
 
0l J s  d    B  dS  nF 0

2

C
S
Flux F
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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
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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
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[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
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Anlage
III. Network Analysis
Waveguides
[Return]
H
Rectangular metallic waveguide
74
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III. Network Analysis
Transmission Line Resonators
[Return]
Transmission Line Model
Transmission Line
Unit Cell
Transmission Line Resonator Model
Ccoupling
Ln
fn  n
75
Steven
Anlage
l
Ccoupling
2
c
2L
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
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
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
  e

 vk ck  ck  0
i
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  4 EC  / g  E J cosg 
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      E J cosg 
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
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Steven
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EJ 
I c
2e