The bright side of Coulomb blockade: Radiation from a Josephson - - PowerPoint PPT Presentation

The bright side of Coulomb blockade:

Radiation from a Josephson junction in the single Cooper pair regime

Max Hofheinz, Fabien Portier, Carles Altimiras, Patrice Roche, Philippe Joyez, Patrice Bertet, Denis Vion, Daniel Estève Quantronics + Nanoelectronics groups, SPEC, CEA Saclay, France Nanoelectronics beyond the roadmap, Lake Balaton, 2011, June 16th

Normal tunnel junction

V V NIN Tunnel Junction e Forbidden by Pauli principle Pauli principle  number of allowed transitions  V  I  V

= /

t

I V R =

t

R

tunnel resistance +

+ + + + + + + + + + + + + + + + + +

/ I dI dV

Coulomb blockade of a tunnel junction

V V

Large R

NIN Tunnel Junction e

⇒ =

2

2

C

e E C

Coulomb blockade of a tunnel junction

Forbidden by Pauli principle +

+ +

I

V V

Large R

NIN Tunnel Junction e

C

E

+ + + + + + + + + + + + + + + +

2 e C

→ ∞ = both and

t

R R T

+

Coulomb blockade of a tunnel junction

Forbidden by Pauli principle Pauli principle + Charging energy  conductance suppression for

< 2 e V C

+ / dI dV V V

Large R

NIN Tunnel Junction e

C

E

+

2 e C

→ ∞ = both and

t

R R T

Dynamical Coulomb blockade

= + +

hole electron photon

eV E E E

Energy balance probability to emit Ephoton into

ν ν ν ν ⇒ = ≈ 

2 2 photon

Re[ ( )] / 2 [ ]( ) Re[ ( )] Z h e e P Z E h Z h h

V V

Delsing et al., PRL 63, 1180 (1989)

ν ( ) Z

photon

( ): P E

ν ( ) Z

NIN Tunnel Junction

Ingold & Nazarov,arxiv:0508728 (1992)

e

Geerligs et al., EuroPhys. Lett. 10, 79 (1989) Celand et al., PRL 64, 1565 (1990)

Coulomb blockade of a tunnel junction

+ / dI dV V V

Large R

NIN Tunnel Junction e

+

2 e C

= = −

∫0

1 ( ) ( )

eV t

T I eV E P E dE eR δ − = − ⇒ = Θ − 

2 2

, / ( ) ( /2 /2 2 ) ( )

t t

e e C e C C R R h e V P E E I V R

A simpler system

• Environment: single mode
• No quasiparticles: use Josephson junction polarized

below the gap voltage

NIN Tunnel Junction SIS Josephson Junction

Dynamical Coulomb Blockade of a Josephson Junction

V E

ω =  2eV

2e

Cooper pair

Cooper pair

eV

ω ϕ ϕ π = + − = + + = = 

† † 2

( 1/ 2) cos 2 / ( ) / 4

J

H h a a E eVt r a a Z L r Z h e C

+ Fermi golden rule calculation

• H. Pothier,
• Ph. D. dissertation (1991)

Dynamical Coulomb Blockade of a Josephson Junction

V E

ω =  2eV

2e

Cooper pair

Cooper pair

V

ϕ ν

π δ ω π δ ω ω

→ → →

Γ = − − = − Γ = = Γ

∑ ∑

    

2 2 2 2 2

( ) (2 ) 2 exp( ) (2 ) 2 ! ( /2 )

e i J n n J n h e

E V n e eV n r r E eV n n V n e n

Dynamical Coulomb Blockade of a Josephson Junction

V E

ω =  2eV

2e

Cooper pair

Cooper pair

V

1 2 3 4 1E-4 1E-3 0,01 0,1 1

Γn n Z=160Ω

π δ ω

→

− Γ = − = Ω ⇒

∑

  

2 2

exp( ) ( ) (2 ) 2 ! 160 0.08

n e J n

r r E V eV n n Z r

⇒

Josephson junction and resonator

500

Z (Ω) Holst et al, PRL 73, 3455 (1994)

Z2=28 Ω Z1=100 Ω

• n res: 640 Ω

16 Ω 50 Ω 50 Ω

ν = ≈ ≈ Ω 

2 1 1

25 GHz, 5, 120 /4 Q Z h e

ν1 ν 3 ν 5

Josephson junction and resonator

500

Z (Ω)

Z2=28 Ω Z1=100 Ω

• n res: 640 Ω

16 Ω 50 Ω 50 Ω

Holst et al, PRL 73, 3455 (1994)

ν =

1

25 GHz

ν1 ν 3 ν 5

Josephson junction and resonator

500

Z (Ω)

Z2=28 Ω Z1=100 Ω

• n res: 640 Ω

16 Ω 50 Ω 50 Ω

ν1 = 25 GHz

Holst et al, PRL 73, 3455 (1994) Two photon processes weak because

ν1 ν 3 ν 5



2 1

/4 Z h e

Goal

Dynamical Coulomb blockade:

• Effect due to photons
• DC side well established
• But no one has seen photons

Look on the bright side of…. Coulomb blockade

Setup

I V

15 mK 4 K 300 K 10M 100 1000 50

P

6 GHz Φ

π Φ = Φ Φ ( ) cos( / )

J J

E E

Quarter-wave resonator

160 10 L Z C Q = = Ω =

25Ω 135Ω

 Φ

50Ω

ν1 ν 3 ν 5

6 12 18 24 30 36 0,0 0,5 1,0 1,5

Designed Lorentzian Fit

Re(Z) [kΩ] f(GHz)

Calibration of the detection impedance

5,0 5,5 6,0 6,5 7,0 0,0 0,5 1,0 1,5 2,0

Measured Designed Re(Zenv) [kΩ] frequency [GHz]

Apply ⇒ quasi-particle shot noise

2

Re[ ( )] 2 2 Re[ ( )] ( ) 18 k

t II t t

Z R S eI P eV d Z R Z R ν ν ν ν = ⇒ = ⇒ + = Ω

∫

, ,

B

eV h k T ν ∆ 

Cooper pair and photon rate match

Cooper pairs Photons (5-7 GHz)

Second order processes

Photons in mode ν0 (5 – 7 GHz) Cooper pairs

ν1 ν 3 ν 5 ν 7 ν1 ν 3 ν 5 ν 7 ν1 ν ν +

1 3

ν1

2

ν ν +

1 5

ν ν +

1 7

ν1

Spectral properties of emitted radiation

2eV h ν =

1

2 ( ) eV h ν ν = +

V

Coulomb blockade with an arbitrary environment

ν ϕ = + − −

∑

†

ˆ ( 1/ 2) cos 2

i i i J t i

H h a a E eV N ϕ π

+

= + +

∑



2

4

2 ( )

e i i i h i

eVt Z a a

ϕ

π δ ν π π   Γ = −     = ≈

∑ ∑



   

1 2

2 2 J 1 2 , , 2 J 2 2 J

, , 2 2 2 Re (2 / ) 4 ) 2 (2

i i i n n i

E n n e eV n h E Z P eV h e V E h e eV

Alternate calculation of P(E): Ingold & Nazarov, arxiv:0508728 (1992)

ϕ ϕ ϕ

δ ν δν ν ν ν π δ π ν

− +

− − + ≠

  = − −   −   = Γ Γ ×

∑ ∑

 

   

1 1

2 2 2 ( ) J 1 1 , , , 2 2 J

, , 2 R (2 ) 4 e 2 2 2 ( )

m m m m m

i i m m m m i i m m n n i n m m

E n e n n e eV n h n h e P eV E h Z h Cooper pair rate: Photon rate at ν=νm: exclude mode m from sum δν → 0:

m

probability of photon emission at ν tunneling rate while absorbing rest of energy

Photons emitted in

Spectral properties of emitted radiation

hν ≈ Lorentzian with

δ ν δν ν ν π − = × Γ 

2 2 J

4 2 2 (2 Re ( ) ) P eV E h h e Z

∆ =

2

4 2 (0) e E kT Z h π

Spectral properties of emitted radiation

hν

≈

2

4 Re (2 / ) 2 2 e Z eV h h eV

δ ν δν ν ν π − = × Γ 

2 2 J

4 2 2 (2 Re ( ) ) P eV E h h e Z

Emitting photon pairs

V Za Zb + E E Z Z 2eV 2eV Emission of photon pairs Bell-like state

Setup

I V

15 mK 4 K 300 K 10M 100 1000 50

P

6 GHz Φ

π Φ = Φ Φ ( ) cos( / )

J J

E E

Emitting photon pairs

5,0 5,5 6,0 6,5 7,0 0,0 0,5 1,0 1,5 2,0 2,5 Re(Zenv) [kΩ] frequency [GHz]

a b

quasi-particle shot noise

Emitting photon pairs

1 2

Re(Z) [kΩ]

Filter a Filter b

Setup

V

15 mK 4 K 300 K 100k 10 50

Pa

6 GHz Φ

Pb

ADC

δ δ δ δ ⇒ < > < > < >

2 2

, ,

b a b a

P P P P

π Φ = Φ Φ ( ) cos( / )

J J

E E

Tuning the photon emission rate

π Γ ∝ Φ = Φ Φ

2

( ) cos( / )

J J J

E E E

0,0 0,5 1,0 0,00 0,05 0,10 0,15

Γ

a

Γ

b

Γ

a,Γ b [GHz]

Φ/Φ |Γ

a-Γ b| / (Γ a+Γ b) < 5%

Power fluctuations cross correlation

ν ν ν ν ν ν = = Γ = = Γ = = Γ = Γ

2 2 2

2 2 2 2( ) 2 2( ) 2

P PaP b a a a a Pa a II Pb b b b Pb b

S eI P e S h h S h h h P S h

• Poissonian source of electrons  electronic shot noise due the

charge granularity

• Poissonian source of photons
• Poissonian source of photon pairs

ν ν ( , )

b a

ν ν ⇔ = = Γ /( ) 2

a Pa a b Pb b

S S h h

Correlated photon pairs

Evidence of Poissonian emission of photon pairs



J

E

0,00 0,05 0,0 0,1

2photon regime (2eVDS~hν

a +hν b)

Shot Noise regime (eVDS>2∆) Theory: fully correlated Poissonian emission

• f photon pairs (ν

a, ν b)

Sab [GHz] (Γ

aΓ b) 0.5 [GHz]

2Γ

0,00 0,05 0,10 0,15 0,0 0,1 0,2 0,3 0,4 0,5

2photon regime (2eVDS~hν

a +hν b)

Shot Noise regime (eVDS>2∆) Theory: fully correlated Poissonian emission

• f photon pairs (ν

a, ν b)

Sab [GHz] (Γ

aΓ b) 0.5 [GHz]

2Γ

Correlated photon pairs

Deviations due to stimulated emission?



J

E

Limits of Coulomb blockade theory

• So far good agreement with P(E) theory
• But need very low EJ to fulfill assumptions

– environment at equilibrium – single Cooper pair regime: EJ P(2eV) << 1

• What happens if assumptions are violated?

Γ = −

∑ ∑



 

1

2 2 J 1 , ,

, , (2 ) 2

i i i n n i

E n n e eV n h

ϕ

π δ ν

Out of equilibrium environment

increase EJ Lasing-like transition ? Incoherent pair tunneling classical AC Josephson effect Transition ?

Conclusions

• Photon side of Coulomb blockade:

– Cooper pair vs. photon rate – multi photon processes – spectral properties

• M. Hofheinz et al. Phys. Rev. Lett. 106, 217005 (2011)
• Perspectives:

– interesting for quantum optics with microwave photons, need for a deeper characterization of the emitted radiation – out of equilibrium environment (no theory yet)