Superconductivity

The phonon-mediated attractive electron-electroninteraction leads to the formation of Cooper-pairs

which undergo a k-space condensation

Superconductivity

kF k

F

k

k

Metal

F

Vk ' k

h

D

V

H

kc

k† c

kk , V

k ' kc

k '† c k '

† ck c

k k

n

1 3 =

The superconducting transition at

produces a gap

in the electronic excitation spectrum, thus removing all low energy excitations.

Tc1.13 h

De 1 N0V

1.764 Tc

This condensate is a charged quantum liquid described by a

macroscopic wave function

,, .r tir t e

Josephson junctions

Josephson junctions,are weak links connecting two superconducting leads/islands. They appear in various forms, e.g., as

tunnel junctions

lr

constrictions

l rg

g

VV

l l

r r

l

r

i

i

e

e

( ) 1 cos 1 cos

/ 2

J JE Const E Er l l r r l

d dt eV

energy

dynamic phase

Josephson relations describing the junction energy and the phase evolution

Two energy scales

phase 0

Q Qcharge

charging (capacitive) energy EC

e2

2C

current (inductive) energy EJ

Ic0

2c

The particle number N = Q/2e and the phase are conjugate variables, i.e., we have a particle phase duality [N,] = i (Anderson)

J CE Eclassical limit

current I

J CE Equantum limit

Fock space

N

N 1

N 2

N 1

N 2

exp i exp i

fixed phase fixed charge Q=2eN

Classical & quantum limits

phase 0

Q Qcharge

charging (capacitive) energy EC

e2

2C

current (inductive) energy EJ

Ic0

2c

[N,] = i

current I

2 22 24 1 cosJ

CeL E

2

2 1 cosC Jd

dH E E ~ E

J

2

2

N i d

d

action

Hamiltonian

2capacitive

222

2

24

C

Ce

E V

Einductive EJ 1 cos

Einductive : Ic sin

Classical limit: gauge invariance and fluxoid quantization in a loop

Free energy of a loop with inductance L:

F E1 1 cos E2 1 cos2 .... 12L ext 1 2 .... 2 .

0

0

2

1 22

0 1 2

2 ...

.....

leads junctions

junctions

A ds ds A ds

n A ds

n

02

gauge invariantphases

02,j A A

0s 2 in the leads,

hence

and

H

ext

self

2

F

0

1 junction: 2 junctions:

kinetic energy of currents ()

In a small inductance loop,

ii

ext

example :

ext 0 2

unique

In a large inductance loop,

n0

Quantum limit: Coulomb blockade and charge quantization on an island

V

Vn

V2

Visl ViCi

i Ne

Cii

C

C1

C2

Cn

electrostatic energy U 1

2 Cii

Visl Vi 2

U 12C

Cij i C j Vi Vj 2

(Ne)2

2C

Visl

U 0 & add N electrons

Quantum limit: Coulomb blockade and charge quantization on an island

= 0

Vg

V2 = 0

Visl ViCi

i Ne

Cii

C

C1

C2

Cg

electrostatic energy U 1

2 Cii

Visl Vi 2

U 12C

Cij i C j Vi Vj 2

(Ne)2

2C

Visl

U 0 & add N electrons

Account for the work done by the batterieswhen changing the island charge N

E 12C

CgVg Ne 2 const.

Vg

N N 1

E

EC e2

2C

0Vg

N

0

… in general

Vg

N N 1

E

EC

0

Vg

E

EC

chargemixing by EJ

0

0J CE E 0 J CE E

F

0

N N 1

F

0

fluxmixing by EC

ext 0 2

Qext 2e 2

H E

Cd2

d2 EJ

1 cos H EC Next Q 2e 2 EJ2 Q Q .

1 4

8

J C

J C

E Ep J C

E E

E E e

RCSJ model, adding dissipation

2

2 2sin ,

4.

Q

Q

cQ e e

R IR

Re

IRC

with the quantum resistance

I

R

C

Icadditional shunt resistor, e.g., accountingfor quasi-particle tunneling.

Effective action describing the Resistively and Capacitively Shunted Josephson junction:

2

22

2 '

4 '1 cos '16 C

JQR

REd E dS

ohmic dissipation (Caldeira-Leggett)

Schmid transition: T=0 quantum phase transition, driven by the environment

H

0 2 2

small inductance loop: two well potential

large inductance loop:particle in periodic potential

0

F

F

0

F

0 0

F

2 RQ R = 1, weak dissipation limit with delocalized phase

2 RQ R 1, strong dissipation limit with localized phase -- superconducting junction

(Leggett-Chakravarti)

classical computing

Charles Babbage1791-1871

Ada Byron, Lady Lovelace

1815-1852

First `Programmer’ and

Enigma,cracked by

Alan Turing with help of COLOSSUS

Inventor of theDifference Engine

1834

Mechanics

The `full’ version of this machine

was built in1991 by the Science

Museum,London

Electronics

Built at University of Pennsylvania,it included 18’000 tubes, weighed 30 tons, required 6 operators,and 160 m2 of space.

The ENIAC (Electronic Numerical

Integrator and Computer) computer

was built in 1946

HardwareClassical computer

Si-wafer

Capacitors:

Transistors:

Bits

Gates

1-bit gate: NOT

2-bit gate: AND

Rinput

Vg

output

Vsd

0 : V 0,

1 : V 0.

Vg

0, closed,

Vg

0, open.

R

Vg1

input

Vg2 output

Vsd

Transistors

Packed DeviceFirst Integrated Circuit, 1958 Jack Kilby, Texas Instruments

First Transistor, 1947Bell Laboratories

Bardeen, Brattain, & Shockley

Pentium Processor, 1997, Intel

in 50 Years

from 1 to 107

transistors

Nanoscale Technology

Ultra-short channel Si-MOSFET, IBM

0.5 m wide, 0.1 m channel

source

drain

gate insulator

Si-wafer

V

channel

Switch a MOSFET with 1000 electrons, while a SET requires only one!

2 m

Single Electron Transistor (SET), AlTechnology

PTB gate

source drain

box

Applications

Classical computers solve any computational task …..

Your washing machine

Your agenda

…. but some are really hard !

Your science

Your bank account

Your track controlin the car

Computational Complexity An input x is quantified via its information content L = log2 x.

A calculation is characterized by the number s of steps (logical gates) involved.

A problem is class P (efficiently solvable) if s is polynomial in L,

A `classic’ hard problem is that of prime factorization: given a non-prime number N, find its factors;

the best known algorithm scales as s ~ exp (2 L1/3 (lnL)2/3).

~s L

~ exps LA problem is deemed `hard’ (not in P) if s scales exponentially in L,

A modern computer can factor a 130-decimal-digits number (L = 300) in a few weeks days;

1827365426354265930284950398726453672819048374987653426354857645283905612849667483920396069782635471628694637109586756325221365901

doubling L would take millions of years to carry out this calculation.

A quantum computer would do the job within minutes

Public Key Encryption

,,

, ,

mod

mod

( 1) 1o )1 m d (

pt s q

s

M

E M

M E

p q

p qt

s N

s

N

N

message

( public keyno )n-Encoding

Decoding

A quantum computer would crack this encryption scheme

(Rivest, Shamir & Adleman, 1978)

Quantum computing

``…nature isn’t classical, dammit, and if you want to make asimulation of nature, you’d better make it quantum mechanical…”

R. Feynman

2LIf one has N quantum two level systems (e.g. L spins) they can have differentstates. To describe such a system in classical computer one needs to have complex numbers, that requires exponentially large computational resources. Thus modeling even small quantum system on a classical computer is practically impossibletask. But since Nature does it very efficiently one can try to use its ability to deal withquantum systems and to apply it also for computational problem.

2L

Bits and Qubits

A quantum bit (qubit) is the quantum mechanical generalization of a classical bit, a two-level system such as a spin, thepolarization of a photon, or ring currents in a superconductor.

0, 1

Classical bit

Physical realization via a charged/uncharged capacitor

0 V

1 V

Q Q

Quantum bit

Physical realization via a quantum two-level system

spins ring-currentspolarizations

21, 0 1ia aea

spin language

Classical & quantum gates IThe possibilities to manipulate a classical bit are quite

limited: The NOT-gate simply interchanges the two values 0 and 1 of the classical bit.

i f

0

01

1

On the other hand, manipulation of a quantum bit is much richer! For a spin / two-level system we can perform rotations around the x -, y -, and z - axis; placing the `spin’ S (with magnetic moment ) into a magnetic field H, the Hamiltonian

produces the desired rotation. …. S H

phaseshifter

/ 2

/ 2

0,

0

z

z

i H t

i H t

eU

e

/ 2 / 2 20 1 1 .z zi H t i H te ea at

H = Hz we obtain the time evolution

E.g., with

H = Hx we obtain the time evolution

cos sin

sin cos,x x

x x

H t i H t

i H t H tU

0 1tancos .xxH t i H tt

amplitudeshifter (a = 0)

Classical & quantum gates II

NOT AND OR The combination of the classicalgates allows us to construct all manipulations on classical bits.

i f0

011

i f00

i0 0

010011

1

1

i f00

i0 0

011111

1

1Is there a set of universal quantum gates ?How does such a set look like ?

irreversible

phaseshifter

cos 2 sin 2,

sin 2 cos 2

i

i

ieU

ie

0 a ei 1 1 a2 .

Singlequbit gates:Rotations

amplitudeshifter

Hadamard (basis change):

H 1

2

1 1

1 1

:

0

1

0 1 2,

0 1 2.

H

U

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

Twoqubit gate:XOR (CNOT)

i f00

i0 0

011101

1

1

control

target

The target flips if the control is on 1

Entangling two qubits

H

0 0 00 11 2 HXOR

H 0 0 1 2

put control qubit C into superposition state, then future gates act on two states

simultaneously

i.e., target qubit T gets flipped AND non-flipped

maximally entangledBell state

C

T

And subsequently: flipping a qubit in an entangled state modifies all its components

0

0

00 11

Quantum Algorithms

Quantum algorithms

Shor’s Factorization algorithm (1994)

finds prime factors in polynomial, rather than exponential time.

N

Grover’s Search algorithm (1997)

searches unstructured database (e.g. telephone book) of N entries by steps.

Although Grover’s algorithm doesn’t change complexity class it is not less fundamental, than Shor’s algorithm.

It is not hard to prove, that classical algorithms can do no better, than just straight search through the list, requiring on average N/2 steps

(i)

NThe `quantum speed-up’ ~ is greater than that achieved by Shor’s factorization algorithm

(ii) 1/3~ exp 2(ln )( )N

One can show, that no quantum algorithm can do better, than , thus it is optimal!

(iii) O N

0xAssume that the database contains N elements, N is some power of 2. Let there be only one solution, that we are looking for .

We start with a homogeneous superposition of all basis states

Given an unstructured set of elements, find the one, that corresponds to the answer of some question.

Task:

Grover’s Search algorithm

1

0

1 N

xNs x

1

20 0 0 1

n

nn s sH H H

H 1

2

1 1

1 1

:

0

The Algorithm

This can be achieved, e.g., by starting with n qubits in the state and applying the Hadamard transform

The goal is to to increase the amplitude of the component 0x

s

The oracle call is just quantum implementation of searching. There must exist unitary operator

s

The algorithm will iterate the Grover rotation G a certain number of times to obtain state very close to . The Grover rotation consists of two parts: an oracle call and a reflection about .

0x

0x

0 0 0, for O x x O x x x x

O s

We want

s

O s

2

G s

0x

0xEach Grover rotation rotates our state by an angle towards . 2

0 , sin( 2 ) 1tG s x t

Thus we should stop and measure after steps 1 1

( )4 2

Nt

N

Quantum Hardware

Physical implementation

All hardware implementations of quantum computers have to deal with the

conflicting requirements of

controllability

while minimizing the coupling to the environment in order to

avoid decoherence.

Solid state implementations

enjoy good scalability & variability

but require careful designs in order to avoid decoherencewhen trying to build Schrödinger cats

Network model of quantum computing

(David Deutsch, 1985)

00000K 0000 .

initial state

final state

Perturbations from the environment

destroy the parallel evolution of the computation

i

f

Parallel evolutionproviding the

quantum speedup.

01100K 1010 .

• each qubit can be prepared in some known state,

• each qubit can be measured in a basis,

• the qubits can be manipulated through quantum gates

• the qubits are protected from decoherence

Physical implementations

• trapped atoms (Cirac & Zoller)• photons in QED cavities (Monroe ea, Turchette ea)• molecular NMR (Gershenfeld & Chuang)• 31P in silicon (Kane)

• spins on quantum dots (Loss & DiVincenzo)• 31P in silicon (Kane)• Josephson junctions, charge (Schön ea, Averin) phase (Bocko ea, Mooij ea)

All hardware implementations of quantum computers have to deal with the conflicting requirements of controllability

while minimizing the coupling to the environment in order

to avoid decoherence.

Quantum optics, NMR-schemes Good decoupling & precision:

Solid state implementations Good scalability & variability:

Have to deal with individualatoms, photons, spins,……Problems with control, interconnections,measurements.

Have to deal with many degrees of freedom.Problems with decoherence.

The rules of the game achievements

Find a system which emulates a spin / quantum two-level system and

• which remains coherent

• which can be manipulated (rotations)

• which can be interconnected and entangled with other qubits

• which can be projected (measured)

• which carries out an algorithm (e.g., Shor’s prime factorization)

Condensed Matter2 charge qubits, interacting

Quantum optics4 9Be ions, deterministic

NMR15 = 3 * 5

Quantum opticsnot a problem

Condensed Matterup to Q ~ 104

Quantum opticsBell inequailty checks

Superconducting quantum bits

Superconducting qubits

Currents in a superconducting ring

VCharges on a superconducting island

Josephson junctions

Superconducting quantum bits

Loop vs Island In a superconducting ring,

the wave function

satisfies periodic boundary conditions. 0

Re

Im

zero flux state

flux one state

The macroscopic wave function winds oncearound the ring; the ring

carries a current

x x exp i x

j x ~ h

2mi * x cc

These two states are degenerate

at half-flux frustration

A finite bias draws a Cooper-pair

onto the island

CP

These two states are degenerate

at half-Cooper-pair frustration

V

superconductorsuperconductor

phase 0 E

e

CC 2

2 EI

cJc0

2

C JE EJ CE E

Produce a weak spot toflip between flux states:

Josephson junction.

221 cos16 J

CL EE 2

2 1 cosC Jd

dH E E

Frustrate a ring withhalf-flux and obtain

two degenerate flux states

Frustrate an island withhalf-Cooper pair and obtain two degenerate

charge statesOR

CP/2

Connect the box to allow charge hopping:Josephson junction.

Superconducting quantum bits

The Josephson junctions are key ingredients in any superconducting qubit design. Or, in other words,

the Josephson junctions introduce the quantum dynamics into the superconducting structure.

Three typesCharge

Schön et al.Averin 1997

Vgn

EC

C

C

EJ

n

Flux/Phase

Ig

EJEC

Bocko et al. 1997

Josephson

Ioffe et al.Orlando et al. 1999

Vg

n n 1

EJ

EC

chargemixing by EJ

0

2EJ

2

20

02

1/ 4~ exp

C Jp J CE E E E

Vphase states

CE

superconductor

phase 0

8p J CE E

h

p

EJ

EJ

Manipulation

Manipulation

V 0

V Vg

Vg

N N

I CJ J,

Charge Phase

EC

Ig

nEJEC

Vg

N N 1

EJ

E

EC

chargemixing by EJ

C

C

EJ

2EJ

E

2

20

Ig

flux mixing by EC

tunneling gap

1/ 48

~ C

J

JCE

p E

EEe

+ ac-microwave voltage / current

induces transitionsacross the gap

OR+ fast (non-adiabatic)

switching induces(incomplete)

Zener tunneling.0

phase shifter

amplitude shifter

Manipulation (general)

q

E

one-parameter (q)qubit

two-parameter (q,Q)qubit

phase shift

fast non-adiabaticswitching, amplitude shift

ac-microwaveinduced

transitions(NMR scheme)

q

E

q

trivialidle state

decoupledstates

coupledstatesQ

Q

mixing

phase shift

potential or dynamical

About the NMR scheme:With Nqu qubits the distance between resonances

is ~ / Nqu. The transition time of the k-th qubit

is related to the ac-signal V via

t Vkop ~ .

Other qubits nearby are excited withprobability

and a precise addressing requireslong times

V Vk k

2 2 2

t Nop qu .

Coherent devices

fast gate voltage pulse

0

2 time

Q

Q

Charge (Nakamura et al., 1999)

1mpulse gate

box

detector

SQUID-loop

detector

tt

Coherent devices

fast gate voltage pulse

0

2 time

Q

Q

Charge (Nakamura et al., 1999)

1mpulse gate

box

detector

SQUID-loop

detector

t

This time domain experiment shows coherent charge oscillations of 50 100 ps duration during a totalcoherence time of 2 ns.

Second generation

~100 coherent oscillations observed via Ramsey

interference Chiorescu et al., 2003

PhaseCharge

104 coherent charge oscillations

observed via

Ramsey fringes, Vion et al.,2002

Balanced energy scalesEJ ~ EC

mI

V

Box

2 pulse

2 pulse

free evolution

Ramseyinterference

Qubit duet, entanglement

Spectral analysis of interacting

J-qubitsBerkley et al., 2003

Spectral analysis ofinteracting q-qubitsPashkin et al., 2003

Josephson-qubitsCharge-qubits

q0q 1q

Two-qubit

gate

CNOT (XOR)Device

Puls sequence

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

U

Result

Yamamoto et al., 2003

The End