from Josephson effects to quantum computing

by Pascal Febvre and Paul Seidel

1

2

Outline

• Basic principles for superconducting electronics

• The Josephson junction

• The SQUID (Superconducting QUantum Interference Device)

• Digital electronics based on Josephson junctions1. The superconducting latching logic of the 80’s2. The Rapid-Single-Flux-Quantum (RSFQ) logic

• Introduction to comparators

• Analog-to-digital conversion with Josephson junctions1. hysteretic junctions2. non-hysteretic junctions

• The DC-to-SFQ interface as a comparator

• The Quasi-One-Junction SQUID comparator

• The balanced comparator

• Applications of RSFQ logic

3

Vector potential & electrical potential

with time-dependent Schrödinger equation, lead to :

where is the quantum of magnetic flux

B & E uniques though the number of couples (A,U) is infinite(depends on the chosen gauge)

Lagrange function

Hamilton function ( ) [ ]21

, , ( , ) ( , )2

H r p t p q A r t q U r tm

= - +

Maxwell equations + &

F0=

h

2e

4

Relation between magnetic flux & phase (1/2)

Case of normal metals:

metal iinduit

=1

R

¶f(t)¶t B

iinduced

Faraday's law:

5

Relation between magnetic flux & phase (2/2)

: London penetration depth

B

superconductor

Phase – flux relationAharonov-Bohm effect

0

2( ) ( ) length dll dl l

pj j+ - = F

FFlux quantization F

loop= nF

0

Meissner currents

(diamagnetism) l

L

j(l + dl)

j(l + dl)

j(l)

j(l)

G

6

Phases, flux and inductances

j(l + dl) - j(l)=2p

F0

Fdl

j (l + dl) j (l)

dl

F

dl= L

dlI

(1) j (l + dl) j (l)

dl

(2)

I1

I2

+F

1+ F

2

-F

1- F

2

-F

1+ F

2I1 + I2

Dj (dl)

-F1

+ F2

= 2np

® L1

I1

= L2

I2

if no flux in the loop ; samecurrent distribution as with

R

1I

1= R

2I

2

7

The Josephson junction

IJ = Ic sinj(t)Josephson equations:

DC Josephson effect at V = 0 :

IJ 0 = Ic sinj0

2000 Å

2000 Å

~10 Å

superconductor 1

superconductor 2

insulating barrier

: phase difference between the two wavefunctionsy i =y i0eiji (t )

JJCJ

I(t)

V(t)

¶j(t)¶t

=2pF

0

V(t)

ac Josephson effect at V0 = cste :

IJ

= Icsin

2p

F0

V0t

æ

èç

ö

ø÷ = I

csin 2p f

Jt( )

with fJ = 484 GHz/mV (Josephson relation)

j =j1 -j2

8

Josephson junction: equivalent to the transistor for semiconductor electronics

3 µm

Materials commonly used: Nb/Al-AlOx/Nb @4.2 K

THIRD WORKSHOP ON SPECTRUM MONITORING OF SPACE SIGNALS - Institut AéroSpatial - Toulouse - 25-26 September 2014

The Josephson junction

I-V characteristic of a Josephson junction

Ic

Vg

RN

for niobium:Vg ~ 2.85 mV and fg ~ 680 GHz

quasi-particle branchJosephson branch

SS I

V=0

« normal » branch

VJUNCTION

IJUNCTION

S

S I

V=Vg

eVg = 2D

10

The superconducting latching logic of the 80's

Ijunction

Vjunction

Ic

Vg

RN

Absence of resistance of superconductors combined with characteristic frequencies of a few hundreds GHz

• development of a static logic in the eighties (IBM ; MITI)• based on hysteretic Josephson junctions

Outcome:

• clock frequencies of a few GHz limited by the punch-through effect

• with the drawback of cryogenic cooling

• period of intense research & development still used today

End of the eighties: Konstantin K. Likharev & his team (Moscow State University)proposed a superconducting logic based on quanta of magnetic flux :

a dynamic logic which relies on the use of shunted Josephson junctions

state “ 0 ” state “ 1 ”

11

Josephson junction electrodynamics

IJ (t) = Ic sinj(t)

IC (t) = CJ

¶V (t)

¶t

IR (t)=V (t)

R/ /

I(t) = IC (t)+ IJ (t)+ IR (t)

IJUNCTION

VJUNCTION

IC

RN

V (t)=F0

2p

¶j(t)

¶t2nd Josephson equation (Faraday's law):

V (t) =F0

2p

1

Ic cosj(t)

¶IJ (t)

¶t

é

ëê

ù

ûú= LJ

¶IJ (t)

¶t

LJ =LJ0

cosj(t)with LJ0

=F0

2p Ic

Josephson inductance

JJ = non-linear parallel RLC circuit

Vg

12

Characteristic time constants & frequencies

Physical approach(BCS theory)

t c =2F0

p 2 Vg

t p =2p F0CS

jC

t RC =p Vg CS

4 jC

Nb: 0.15 psNbN: 0.07 ps

R-C circuit time constant : tRC = R/ / CJ

L-R circuit time constant : t c =LJ 0

R/ /

Plasma periodof the L-C circuit :

t p =2p LJ 0CJ

Electricalapproach

I(t)

Ic

= LJ0CJ

¶2j(t)

¶t 2+

LJ0

R/ /

¶j(t)

¶t+sinj(t)

Anharmonic oscillator

13

I(t)

Ic

= LJ0CJ

¶2j(t)

¶t 2+

LJ0

R/ /

¶j(t)

¶t+j(t)

For damped oscillatory regimes, transients evolve as: e

-t

2 R// CJ =e-

t

2t RC

Linearized approach

Damping coefficient: x =1

2R/ /

LJ0

CJ

Quality factor: Q=1

2x= R/ /

CJ

LJ0

If j t( ) <<1

Example:SIS junction used in radio-astronomy:RN=12 ohms ; CJ = 190 fF ; Ic = 200 µA

t LR = 0.13pst p = 3.5 pst RC = 2.3ps

Josephson junction dynamics

-0,5

0

0,5

1

1,5

2

2,5

3

3,5

4

0 5 10 15 20

Time (ps)

JJ

vo

lta

ge (

mV

)

Ifinal = 300 µA

Ifinal = 50 µA0

50

100

150

200

250

300

350

400

450

500

0 1 2 3 4 5 6voltage (mV)

cu

rren

t (µ

A)

6 t RC

Criteria : fmax =1

12 t RC

14

0

10

20

30

40

50

60

70

0 10 20 30 40 50

current density (kA/cm2)

freq

uen

cy

(G

Hz)

Limitations of the static logic

Reduce the switching timeReduce the R-C time constant

Reduce R

15

R-C circuit time constant

tRC = R/ / CJ

L-R circuit time constant

tLR

=L

J 0

R/ /

L-C circuit plasma period

t p =2p LJ 0CJ

McCumber parameter defined by:

bc

=t

RC

tLR

=R

/ /

2 CJ

LJ

0

=2p R

/ /

2 CJ

Ic

F0

R/ / » Rshunt

Minimum switching time obtained for :

tRC

= tLR

=t

p

2p

æ

èç

ö

ø÷ : b

c= 1

New time constant

t 0 =F0 CS

2p jc

=F0

2p Rshunt Ic

t 0 ps( )»1

p Vc mV( )with Vc = Rshunt Ic

Minimising the switching time

fmax(GHz) = 500 x Vc(mV)

Criteria: fmax = 1/(2πτ0)

16

0

100

200

300

400

500

600

0 10 20 30 40 50

current density (kA/cm2)

Ma

xim

um

freq

uen

cy

of

op

erati

on

(G

Hz)

Maximum frequency of operation

Valid for externally-shunted SIS junctions

Nb

NbN

17

Ijunction

Vjunction

c >> 1

Vg

Rshunt

c > 1

VcJunctions NbN/MgO/NbN - CEA-INAC Grenoble

c = 1

Vc

c < 1

Vc

c 1: fmax

but the hysteresis disappears:no more switching between 2 states

Influence of the McCumber parameter

18

I-V characteristics of Josephson junctions

0

0,5

1

1,5

2

2,5

3

3,5

4

0 5 10 15 20time (ps)

Jose

phso

n ju

nctio

n vo

ltage

(m

V)Junction parameters: RN=12 ; CJ = 190 fF ; Ic = 200 µA

shunted junction such that c =10 = 0.56 ps

unshunted Josephson junction

c= 0.13 ps - p=3.5 ps - RC=2.3 ps - c =17

I = CJ

¶V (t)

¶t+

1

Rshunt

V (t)+ Ic sinj(t)

A B

V (t)dtA

B

ò =F0

Dj =2p

Josephson junction dynamics

area=0

area =2 mV.ps

20

h/2e

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

time (ps)

Voltage n

orm

alized t

o V

c

Dynamics of shunted Josephson junctions

This dynamic logic is called Rapid Single Flux Quantum (RSFQ)

Magnetic flux quantum: h/2e = 2.07 mV.ps = 2.07 mA.pH

I=1.2 Ic

I=3 Ic

I=5 Ic

21

On-chip RSFQ clock

Technological status

Source: ITRS 2004

How to manage single pulses?

Rshunt

Vc

IbiasIsignal

voltage

current

time

critical current Ic

Kirchhoff's laws:

Iapplied = IL + IJNodal rule:

Loop rule: Vi

loop

å =0

Integration over a loop:

Vi

loop

åæ

èç

ö

ø÷ dtò = 0dtò =constant , " time interval

Vi dtòloop

å =F0

2pDji

loop

å = constant = nF0

Modified Kirchhoff's law (law of phases): Dji

loop

å =2np

Processing magnetic flux quanta

25

2p LIapplied

F0

=2p LIL

F0

+2p LIc

F0

sinj

DC-SFQ converteror

ac SQUID (e.g. rf SQUID)or

Schmidt trigger

jL =2p

F0

FLjext =2p

F0

L Iapplied

jext

= 2 np +j + l sinj

l =2p L I

c

F0

2 * maximum number of flux quanta storablebefore JJ switching

Processing flux quanta - static analysis (1/2)

Iapplied = IL + IC sinjNodal rule :

jL -j =2npLoop rule :

jext

= jL

+ l sinj

26

0

1

2

3

4

5

0 1 2 3 4 5

j/2p

jex

t/2p

l =2p L IC

F0

£1 1£l =2p L IC

F0

£4.6

0

1

2

3

4

5

0 1 2 3 4 5

j/2p

jex

t /2p

1st fluxon

2nd fluxon

1st antifluxon

2nd antifluxon

Processing flux quanta - static analysis (2/2)

27

Generation of quantized pulses (1/2)

1£l =2p L IC

F0

£4.6

0

1

2

3

4

5

0 1 2 3 4 5

j/2p

jex

t /2p

jext

1st fluxon( ) = arccos -1l

æ

èçö

ø÷+ l2 -1The first fluxon enters the loop for:

1st fluxon

jext

2nd fluxon( ) =jext

1st fluxon( ) + 2pThe second fluxon enters the loop for:

2nd fluxon

The first antifluxon enters the loop for:

jext

1st antifluxon( ) = 2p - arccos -1l

æ

èçö

ø÷- l2 -1

1st antifluxon

jext

2nd antifluxon( ) =jext

1st antifluxon( ) - 2p

The second antifluxon enters the loop for:

2nd antifluxon

28

A

Iext

IC

pointA( ) =1

larccos -

1

l

æ

èçö

ø÷+ l2 -1

é

ëê

ù

ûú

1£l =2p L IC

F0

£4.6

0

1

2

3

4

5

0 1 2 3 4 5

j/2p

jex

t /2p

1st fluxon

2nd fluxon

1st antifluxon

2nd antifluxon

Generation of quantized pulses (2/2)

Iext

IC

point B( ) =Iext

IC

point A( ) +2p

l

B

29

Iapplie

d

Iapplied

stored flux

Quantized pulses of area h/2e =2.07 10-15 Wb = 2.07 mV.ps = 2.07 mA.pH

Picosecond pulse and magnetic flux quantum

30

Simulation of a pulse generator

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

50 100 150 200 250

time (ps)

curr

ent

(mA)

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

voltage (

mV)

applied current junction current junction voltage

supercurrent in the loopof the DC-SFQ converter

quantized area pulse = 2.07 mV.ps = 2.07 mA.pH = 2.07 10-15 Wb

31

Pulse transmission

L .Ic < Φ0

Pulse storage

L .Ic > Φ0

The digital signal is coded with 2 states, between two clock pulses : ‘0’ : no pulse ‘1’ : one quantized pulse (SFQ pulse)

Example of SFQ gate: the Delay-Flip-Flop (DFF)

34

Drawing for the NbN/MgO/NbN technology- CEA INAC - Grenoble

Schematics of the Delay-Flip-Flop cellFLUXONICS Cell Library

www.FLUXONICS.eu

35

Layout for the Nb/Al-AlOx/Nb technology – FLUXONICS Foundry – IPHT Jena - Germany

Layout of the Delay-Flip-Flop cellFLUXONICS Cell Library

33 µm

J1 J2

J3

J4

J5

J6

J7

www.FLUXONICS.eu

36

Simulation of the Delay-Flip-Flop cellFLUXONICS Cell Library

www.FLUXONICS.eu

37

Margins of the Delay-Flip-Flop cellFLUXONICS Cell Library

www.FLUXONICS.eu

38

Yield of the Delay-Flip-Flop cellFLUXONICS Cell Library

www.FLUXONICS.eu

39

RSFQ process cross-sectionFLUXONICS Foundry

40

RSFQ technology parameters and tolerancesFLUXONICS Foundry

FLUXONICS octagonal Josephson junctions

d Josephson junctionspecific capacitance :50±2 fF/µm2

41

RSFQ technology parameters and tolerancesFLUXONICS Foundry

✦ Current density of 1 kA/cm² based on Nb/Al-AlOx/Nb tri-layers

✦ 3 metal layers M0, M1 & M2 : 1st for ground plane and 2nd/3rd

for wiring

✦ Characteristic voltage Vc=256 µV

✦ McCumber parameter βc = 1

✦ 13 shunted octagonal Josephson junctions for digital operation

Junction Lb100 Lb125 Lb150 Lb175 Lb200 Lb225 Lb250 Lb275 Lb300 Lb325 Lb350 Lb375 Lb400

Ic (µA) 100 125 150 175 200 225 250 275 300 325 350 375 400

area (µm2)

9.94 12.44 15.11 17.59 20.09 22.54 25.13 27.59 29.99 32.47 35.03 37.42 40.18

d (µm) 3.4 3.8 4.3 4.7 4.9 5.2 5.5 5.9 6.1 6.3 6.5 6.8 7.0

Rshunt (Ω) 2.58 2.06 1.69 1.46 1.27 1.14 1.02 0.93 0.85 0.79 0.73 0.68 0.64

42

RSFQ technology parameters and tolerancesFLUXONICS Foundry

www.FLUXONICS.eu

43

The Toggle Flip-Flop cell

The Toggle FlipFlop (TFF) celldelivers an SFQoutput pulseonce every twoSFQ pulses sentto its input.

For an SFQ clocksignal at a givenfrequency, theoutput pulsetrain is an SFQclock at half-frequency.

www.FLUXONICS.eu

The Toggle Flip-Flop cell

www.FLUXONICS.eu

The FLUXONICS Toggle Flip-Flop cellmargins and yield

www.FLUXONICS.eu

Picture of Toggle Flip-FLop cell

www.FLUXONICS.euJ1

25 µm Cell size : 200 x 300 µm

Toggle Flip-FLop experimental results

www.FLUXONICS.eu

FLUXONICS cell library

DCSFQ JTL SPLITTER MERGER-1 MERGER-2

TFF SFQDCBUFFERJTL DFFC

TFFC-A TFFC-B NDRO

www.FLUXONICS.eu

Experimentally-verifiedfrequency limit

The physics of the Josephson junction gives

easy access to the frequency of oscillations

Performances & applications of RSFQ logic

Applications:Telecommunication domain:

• analog-to-digital converters• routers• base-station of mobile phones

Security & intelligence:• analog-to-digital converters• super-computers • software-defined radio - SDR

Science:• super-computers• astronomy: signal processing of imagers• geophysics: study of Earthquakes, geophysical prospection• (bio)-medical: encephalo- & cardio- magnetography• archeology

Performances:• clock frequencies in the 30-110 GHz range (state-of-the-art)• ultimate clock frequencies up to 770 GHz : demonstrated on simple circuit• clock frequency objective for the next decade: 160-300 GHz• very low consumption with new eSFQ and eRSFQ concepts

51

Analog-to-Digital Converters

1 cm2 chip, fabricated with HYPRES’ standard Nb process with Jc = 4.5 kA/cm2.

It contains :- a bandpass delta-sigma ADC,- a digital channelizer,- output drivers.

5000 - 10 000 Josephson junctions.

Courtesy of Deep Gupta - HYPRES

RSFQ half-precision floating-point adder (FPA) successfully demonstrated at 20 GHz.

Circuit size is 5.9 mm × 5.7 mm.

10224 Josephson junctions.

Performance is 1.67 GFLOPS.

Total power consumption is 3.5 mW.

Courtesy of Noboyuki Yoshikawa - University of Yokohama - Japan

Floating-point units (Japan)

Superconducting digital electronicsintegrated systems

Courtesy of Deep Gupta - HYPRES

Complete cryocooled digital-RF receiver system prototype, assembled in a standard 1.8-meter tall 0.5-meter wide equipment rack.

Using the modular packaging approach, the system can currently host variety of chips.

The system includes a two-stage 4-K Gifford-McMahon cryocooler manufactured by Sumitomo, two sets of interface amplifiers for connecting chip outputs to an FPGA board (placed behind the vacuum enclosure, on the metal tray) for further digital processing and computer interface. The system also includes a current source and a temperature controller.

• Semiconductor technology: comparators based on operational ampshave specifications that depend weakly on environment

An introduction to comparators

+

-

Vin Vref

Vout

• Superconductor technology: comparators are based on Josephsonjunctions whose high non-linearity involves a design that is fullyenvironment-dependent

55

When Josephson junctions are shunted to be non-hysteretic and faster, digitalinformation is stored in the magnetic flux quantum and requires asuperconducting loop to be generated.

• Digital signal is based on two states, for a given time interval: ‘0’ : no pulse ‘1’ : one SFQ pulse

• Much faster than switching logic if shunt resistance is correctly chosen

Rshunt

c ≈ 1

Vc

Ibias

Isignal

voltage

current

time

Analog-to-digital conversion with Josephson junctions

56

The DC-to-SFQ interface used as a comparator

• The Josephson junction switches forspecific values of Iext;

• It can be used to know Iext at theswitching instants;

• If the temporal form of Iext iscomplex, the pulses are not equallyspaced in time and the post-processing of digital data iscomplex.

• A pulse added to the signal to sense, and chosen to force triggeringof the Josephson junction at a regular rate, transforms the DC-to-SFQ converter into a comparator.

• Then the generated pulse needs to be measured and processed.

57

The DC-to-SFQ interface used as a comparator

58

The DC-to-SFQ interface used as a QOJ comparator

• The second junction is chosento have a critical currentmuch higher that the one ofthe comparator junction

• The device is a quasi-one-junction SQUID (QOJS): Ko &Van Duzer, 1988

• It is biased so thatP(switching) = 50% inabsence of signal

• Ko & Van Duzer, 1988

• Networks of such QOJS madefor flash ADCs

• 6-bit ADC demonstrated (4 bits at 5 GHz and 3 bits at 10 GHz) : Bradley (1993)

59

The DC-to-SFQ interface used as a QOJ comparator

• 6-bit ADC demonstrated (4 bits at 5 GHz and 3 bits at 10 GHz) :Bradley (1993)

60

The SQUID comparator

Kratz & Jutzi, 1985

signal

pulsetrigger

61

The balanced comparator

Filippov & Kornev, 1991

DI =¶P

¶I I =0

-1

current

time

threshold

signal

62

Parameters influencing the comparator resolution

ΔI depends on:- noise in the system (<-- temperature)- trigerring pulse shape: time rise,…(<-- frequency of operation)- JJ parameters (capacitance, McCumber parameter)- loop inductances (and stray inductances)- bias current- environment impedance in a wide frequency range

63

The balanced comparator resolution

α is the switching rate

DI µ Ic

1/3IT

2/3 ln aRN IcT

a

æ

èçö

ø÷

-1/3

64

The balanced comparator resolution

Filippov et al, 1995Oelze et al, 1997

65

The balanced comparator resolution

Oelze et al, 1997 66

The current knowledge on balanced comparators

• The influence of noise, shape of triggering pulse and speed is fully understood, even versus the input signal inductance

• The jitter of the device can be derived from this analysis

• Experiments and theory are in very good agreement regarding noise considerations (no extra noise needs to be added)

• The influence of the sensing impedance (in particular inductance) and of other parasitic elements in the sensing loop is still not totally sorted out (Gordeeva & Pankratov, JAP2008)

67

Sources of knowledge about Superconducting Electronics

Websites

The European Superconductivity News Forum (ESNF):http://www.ewh.ieee.org/tc/csc/europe/newsforum

http:// www.quantarctic.eu (under construction)

http:// www.fluxonics.eu (under reconstruction)

Books

A. Barone, G. Paterno ́, Physics and Applications of the Josephson Effect, John Wiley & Sons, 1982. ISBN: 0-471-01469-9.

T. Van Duzer, C.W. Turner, Principes of Superconductive Devices and Circuits, second ed., Prentice Hall PTR, Upper Saddle River, NJ, USA, 1998. ISBN: 0-13-262742-6.

K.K. Likharev, Dynamics of Josephson Junctions and Circuits, Gordon and Breach Publ., New York, 1986.

The SQUID Handbook, by John Clarke and Alex Braginski

Michael Tinkham, Introduction to Superconductivity, second ed., Dover Books on Physics, 2004. ISBN: 0-486-43503-2 Paperback. 68

Sources of knowledge about Superconducting Electronics

69

Sources of knowledge about SQUIDs

The SQUID Handbook, by John Clarke and Alex Braginski-------The European Superconductivity News Forum (ESNF):http://www.ewh.ieee.org/tc/csc/europe/newsforum

Volume 1 - Issue 1 - July 2007: High-Performance dc SQUID Sensors and Electronics by D. Drung et alhttp://www.ewh.ieee.org/tc/csc/europe/newsforum/pdf/Drung_ESNF_PTB_Magnicon_final_0626071.pdf

Volume 2 - Issue 6 - October 2008: section about SQUIDs, SQUIFs and related -http://www.ewh.ieee.org/tc/csc/europe/newsforum/Contents06.html

Volume 3 - Issue 8 - April 2009: Recent SQUID Activities in Europe, Part I: Devices by A. I. Braginski & G. B. Donaldson -http://www.ewh.ieee.org/tc/csc/europe/newsforum/pdf/CR-12.pdf

Volume 3 - Issue 9 - July 2009: Recent SQUID Activities in Europe, Part II: Applications by A. I. Braginski & G. B. Donaldson -http://www.ewh.ieee.org/tc/csc/europe/newsforum/pdf/CR12-II_Final_073009.pdf

Volume 4 - Issue 12 - April 2010: Simplified Analysis of Direct SQUID Readout Schemes by D. Drunghttp://www.ewh.ieee.org/tc/csc/europe/newsforum/pdf/MT-21_ST188.pdf

Volume 5 - Issue 15 - Jan. 2011: SQUIFs, Bi-SQUIDs & R-SQUIDs - http://www.ewh.ieee.org/tc/csc/europe/newsforum/Contents15.html

Volume 5 - Issue 18 - Oct. 2011: dc SQUID & SQIF Sensor with High Transfer Function Based on Sub-micrometer Cross-type Josephson Tunnel Junctions by T. Schönau et al - http://www.ewh.ieee.org/tc/csc/europe/newsforum/pdf/KRYO-Schonau.pdf

Volume 6 - Issue 19 - Jan. 2012: SQUID-based Systems for Co-registration of Ultra-Low Field Nuclear Magnetic Resonance Images and Magnetoencephalography by A. Matlashov - http://www.ewh.ieee.org/tc/csc/europe/newsforum/pdf/ST289.pdf

Volume 6 - Issue 20 - April 2012: SQUIDs: Then and Now by John Clarkehttp://www.ewh.ieee.org/tc/csc/europe/newsforum/pdf/issue20-Clarke.pdf-------Superconductor Science and Technology - Volume 22 - no. 6 - June 2009 - Special section: focus on nanosquids and their applications -http://iopscience.iop.org/0953-2048/22/6

70

Thank you for your attention !

phase Flux jump

x

time

Fontaine de Vaucluse - France - March 9, 2013

Cooper pair

71