Introduction to Quantum Mechanics of Superconducting

Electrical Circuits

• What is superconductivity?

• What is a Josephson junction?

• What is a Cooper Pair Box Qubit?

• Quantum Modes of Superconducting Transmission Lines•See R.-S. Huang PhD thesis on Boulder 2004 web page

What is superconductivity?

See: Boulder Lectures 2000

Cooper pairs are like bosons in a BEC -- except --size of Cooper pair spacing between electrons

F 1k ξ >>

k

k−Complex order parameter like BEC

( ) k kr c cψ↑ − ↓

∼( )( ) i rr e ϕψ ψ∼

Excitation Gap for Fermions

broken pair

k

In a semiconductor the gap is tied to the spatial lattice(only occurs at one special density commensurate with lattice)

In a superconductor, gap is tied to fermi level. Compressible.

WHY SUPERCONDUCTIVITY?

E few electrons N ~ 109

total numberof electrons

N even“forest” of states

~ 1 eV 2∆ ~ 1 meVsuperconducting gap

SUPERCONDUCTINGNANOELECTRODE

ATOM

Energy vs. Particle Number on Grain

…. N-1 N N+1 N+2 N+3 N+4 ....

∆

Josephson tunneling between two grains

2∆

1 2,N N 1 21, 1N N− + 1 22, 2N N− +

Josephson Tunneling ICoherent tunneling of Cooper pairs

Restrict Hilbert space to quantum ground states of the form

1 22 , 2 , n ... 3, 2, 1,0, 1, 2, 3...N n N n− + = − − − + + +

…. –4 -3 -2 -1 0 +1 +2 +3 +4 ….. n

Josephson Tunneling IIRestrict Hilbert space to quantum ground states of the form

1 22 , 2 , n ... 3, 2, 1,0, 1, 2, 3...nN n N n− + ⇒ = − − − + + +

…. –4 -3 -2 -1 0 +1 +2 +3 +4 …..

J 1 12

n

n

ET n n n n=+∞

=−∞

= − ⎡ + + + ⎤⎣ ⎦∑

Tight binding model: single ‘particle’ hopping on a 1D lattice

28 /t

JGEe h

∆=

normal state conductance

sc gap

N.B. not 1/JE ∝ ∆ ∆

Josephson Tunneling III

…. –4 -3 -2 -1 0 +1 +2 +3 +4 …..

J 1 12

n

n

ET n n n n=+∞

=−∞

= − ⎡ + + + ⎤⎣ ⎦∑Tight binding model

‘position’ n‘wave vector’ (compact!)

‘plane wave eigenstate’

ϕ

" "n

i n ikx

ne n eϕϕ

=+∞

=−∞

= =∑'

J'

' 1 ' ' ' 1n n

i n

n n

T E n n n n e nϕϕ=+∞ =+∞

=−∞ =−∞

= − ⎡ + + + ⎤⎣ ⎦∑ ∑

Josephson Tunneling IV

…. –4 -3 -2 -1 0 +1 +2 +3 +4 …..'

J'

' 1 ' ' ' 1n n

i n

n n

T E n n n n e nϕϕ=+∞ =+∞

=−∞ =−∞

= − ⎡ + + + ⎤⎣ ⎦∑ ∑

J 1 12

ni n

n

ET e n nϕϕ=+∞

=−∞

= − ⎡ + + − ⎤⎣ ⎦∑

J cos( )T Eϕ ϕ ϕ= −

ϕ

Supercurrent through a JJ

…. –4 -3 -2 -1 0 +1 +2 +3 +4 …..

‘position’ n‘momentum’ ϕ J cos( )T Eϕ ϕ ϕ= −

c c

1 sin

current:2(2 ) sin

2sin

J

J

J

Edn dTdt d

dn eI e Edt

eI I I E

ϕϕ

ϕ

ϕ

= =

= =

= ≡

Wave packet group velocity

ϕ

Charging Energy

1 22 , 2 , ... 3, 2, 1,0, 1, 2, 3...nN n N n n− + ⇒ = − − − + + +

2

2

(2 ) 2 2c2

c 2

4eC

eC

U n E n

E

= =

≡

n

U

‘Second’ Quantization

" "n

i n ikx

ne n eϕϕ

=+∞

=−∞

= =∑

Number Operator: n̂ iϕ∂

≡ −∂

2c J

2

c J2

ˆ4 cos( )

4 cos( )

( ) ( ) ( )j j j

H E n E

H E E

H E

ϕ

ϕϕ

ϕ ϕ ϕ

= −

∂= − −

∂Ψ = Ψ

ϕ Now viewed as coordinate

( )j ϕΨ Describes quantum amplitude for (quantum) phase

Weak Charging Limit: Josephson Plasma Oscillations

c JE E 2c J

2

c J2

2 2

c J2

ˆ4 cos( )

4 cos( )

4 12

H E n E

H E E

H E E

ϕ

ϕϕ

ϕϕ

= −

∂= − −

∂

⎡ ⎤∂≈ − + − +⎢ ⎥∂ ⎣ ⎦

Simple harmonic oscillator

spring constant JK E=

J c J8E Eω =2c

18

ME

=mass

Weak Charging Limit: Josephson Plasma Oscillations

c JE E (heavy ‘mass’, small quantum fluctuations)

J c J8E Eω =

ϕ

spring constant JK E=

2c

18

ME

=mass

Weak anharmonicity can be usedto form qubit and readout levels

Strong Charging Limit: Cooper Pair Box

ϕ Fluctuates too wildly;better to use number (position) representationc J4E E≥

n

U

-1 0 +1 +2

C

500aF2K

CE∼∼

Random offset charge due to built in asymmetry

THE BARE JOSEPHSON JUNCTION: SIMPLEST SOLID STATE ATOM

Josephson junction with no wires to rest of world:superconductor

tunnel oxide layersuperconductor

( )2ˆ4 / 2 cos .....C r JH E n Q e E ϕ= − − +Hamiltonian:

2

2

8 /

2

r

tJ

CJ

QGEe heEC

∆=

=

negligibleterms atsmall T

residual offset charge3 parameters:

Josephson energy

Coulomb energy

IMPERFECTIONS OF JUNCTION PARAMETERS

TUNNEL CHANNEL ELECTRIC DIPOLESCHARGED IMPURITY

_+

-1nm

CAN IN PRINCIPLE MAKE JUNCTION PERFECT, BUT MEANWHILE.....

JUNCTION PARAMETER FLUCTUATIONS

( )( )( )

statr r r

statJ J J

statC C C

Q Q Q t

E E E t

E E E t

= + ∆

= + ∆

= + ∆

param. dispers.

Qrstat random!

EJstat 10%

ECstat 10%

noise sd@1Hz

∆Qr/2e ~10-3Hz-1/2

∆EJ /EJ 10-5-10-6Hz-1/2

∆EC /EC <10-6Hz-1/2?

Oth order problem: get rid of randomness of static offset charge!

Cooper pair box (charge qubit)2 solutions: - control offset charge with a gateor

- make EJ /EC very large RF-SQUID (flux qubit)Cur. Biased J. (phase qubit)

SENSITIVITY TO NOISE OF VARIOUS QUBITS

junction noise port noise

qubits ∆Qr ∆EJ ∆EC technical quantum back-action

charge

flux

phase

chargequbit

fluxqubit

phasequbit

COOPER BOX STRATEGY

gate charge can be controlledNg=CgU/2e

U

compensatestatic

offset chargewith gate

( )2

gˆ4 cos .....C JH E n N E ϕ= − − +

n

U

n

U

Jcos 1 12

n

Jn

ET E n n n nϕ=+∞

=−∞

= − = − ⎡ + + + ⎤⎣ ⎦∑

charge deg. point

Mapping to two level system

n

U

↑ ↓

0 1

1ˆ2

z

n σ−=

( )2

gˆ4 cos .....C JH E n N E ϕ= − − +

JJ

2cos 1 1

2

n

Jx

n

ET E n n En nϕ σ=+∞

=−∞

= − = − ⎡ + + + −⎤ ⇒⎣ ⎦∑

Mapping to two level system

n

U

↑ ↓

0 1

1ˆ2

z

n σ−=

( )g2 1 2 constant.....2

z xJC

EH E N σ σ= − − − +

2ˆ ˆ[ ]n n=

Rotate:

x zz x

→ −→ + x

z( )g2 1 2

2constant.....x zJ

CEH E N σ σ += − − +

Mapping to two level system

( )g2 1 22

constant.....x zJC

EH E N σ σ += − − +

2 2x z

H B

B B

σ

ε±

=

= ± +

i

EJ

( )g2 1N −

B2

JE

0

n

U

n

U

SPLIT COOPER PAIR BOX

φIcoil

Ng

Q

Ugate

I

maxJ J

0

cos 2E E φπφ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

JJ is actually a SQUID

Split Cooper Pair Box

( )g2 1 22

constant.....x zJC

EH E N σ σ += − − +

Vg

JJ is actually a SQUID

maxJ J

0

cos 2E E φπφ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

(Buttiker ’87; Bouchiat et al., 98)

Superconducting Tunnel Junction as aCovalently Bonded Diatomic ‘Molecule’

(simplified view at charge deg. point)

1 pairsN +

pairsN1 pairsN +

pairsN810N ∼ 1 mµ∼tunnel barrier

aluminum island

aluminum island

Cooper Pair Josephson Tunneling Splits the Bonding and Anti-bonding ‘Molecular Orbitals’

bondinganti-bonding

Bonding Anti-bonding Splitting

( )12

ψ ± = ±810 1+

810 1+

810

810

anti-bonding bonding J 7 GHz 0.3 KE E E− = ∼ ∼Josephson coupling

J

2zEH σ= −

bonding

anti-bonding

↑ =

↓ =

Dipole Moment of the Cooper-Pair Box(determines polarizability)

2

1 2 3

1/(2 )1/ 1/ 1/

Cd e LC C C

=+ +

Vg

1 nm

L = 10 µm

+ ++ +

+ +

- -- -

- -

/gE LV=

~ 2 - md e µ

Vg

0

1C

2C

3C

J

2z x

gd VL

EH σσ −= − bonding

anti-bonding

↑ =

↓ =

Energy, Charge, and Capacitance of the CPB

↑

↓J

2z x

gE dH V

Lσ σ= − −JE

Ene

rgy

dEQdV

=

Cha

rge

Cap

acita

nce

no chargesignal

↑

↑

↓

↓

charge

dQCdV

= polarizability is state dependent

/g gC V e0 1 2deg. pt. = coherence sweet spot

sweet spot

E1

Ene

rgy

kK

)(

B

Ng=CgU/eφ/Φ0=δ/2π

E0

hν01

RESPONSES OF BOXIN EACH LEVEL

kk

EQU

∂=

∂k

kEIφ

∂=

∂island chargeloop current

12

2k

kELφ

−⎛ ⎞∂

= ⎜ ⎟∂⎝ ⎠

2

2k

kEC

U∂

=∂

box capacitancebox inductance

Dephasing, decoherence, decay

150.658 10 eV-s−≈ ×

151 nV acting on charge for 1 s 10 eV-se µ −=

Spins very coherent but hard to couple and very hard to measure.

Avoiding Dephasing1

0 1

0Ebox

2ggC V

e

→

←

1/2

2'sweet spot' for T

ν

( ) ( )( )( ) cos / 2 1 sin / 2 0i tt e ϕψ θ θ= +0

( ) ( )t

t i dt tϕ ν′ ′= − ∫ g

0Q QVν∂

= → → − ← ← =∂

Vion et al. Science 2002

16460 16462 16464 16466 16468

30

32

34

36

38

sw

itchi

ng p

roba

bilit

y (%

)

ν (MHz)

Lorentzian fit:ν01= 16463.64 MHzQ ~20 000

Ng = 0.5φ = 0τ = 4 µs

‘Atomic spectral line’

Courtesy M. Devoret

0 1 2 4 5 6 730

35

40

45

50

switc

hing

pro

babi

lity

(%)

time tw (µs)

RELAXATION TIME AT OPTIMAL POINT

Exponential fit:T1 = 1.84 µs

Q1 ~ 90 000

tw

PRINCIPLE OF RAMSEY EXPERIMENT

prepa-ration

freeevolution

measu-rement

90ºpulse

90ºpulse

time

prepa-ration

freeevolution

measu-rement

90ºpulse

90ºpulse

0.0 0.1 0.2 0.3 0.4 0.5 0.6

30

35

40

45

sw

itchi

ng p

roba

bilit

y (%

)

time between pulses ∆t (µs)

RAMSEY FRINGES

νRF = 16409.50 MHz

0.0 0.1 0.2 0.3 0.4 0.5 0.6

30

35

40

45

sw

itchi

ng p

roba

bilit

y (%

)

time between pulses ∆t (µs)

Qϕ ~ 25000 (8000 fringes)

∆t

Courtesy M. Devoret