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 16 th

Normal tunnel junction I NIN + + + + + + + + + + + + + + + + + + e V Tunnel V dI dV / Junction + Forbidden by Pauli principle Pauli principle  number of allowed transitions  V  I  V = = R I V R / tunnel resistance t t

Coulomb blockade of a tunnel junction Large R NIN e V Tunnel V Junction 2 e ⇒ = E C 2 C

Coulomb blockade of a tunnel junction Large R I NIN + e V Tunnel V + + + + Junction + + + + + + + + + + + + + + + e 2 C Forbidden by Pauli principle E C → ∞ both and R R t = T 0

Coulomb blockade of a tunnel junction Large R NIN + e V Tunnel V dI dV / Junction + Forbidden by Pauli principle E e C 2 C → ∞ both and R R t = T 0 Pauli principle + Charging energy  conductance suppression for e < 2 V C

Dynamical Coulomb blockade Delsing et al., PRL 63 , 1180 (1989) ν Z ( ) NIN e V Tunnel V Junction Geerligs et al., EuroPhys. Lett. 10 , 79 (1989) Celand et al., PRL 64 , 1565 (1990) ν Z ( ) P E ( ): probability to emit E photon into photon ν ⇒  2 Re[ ( )] Z h e / Energy balance 2 2 e = ν ≈ ν P Z E [ ]( h ) Re[ ( )] Z = + + photon eV E E E ν hole electron photon h h Ingold & Nazarov,arxiv:0508728 (1992)

Coulomb blockade of a tunnel junction Large R NIN + e V Tunnel V dI dV / Junction + = T 0 1 e eV ∫ 0 = − I ( eV E P E dE ) ( ) eR 2 C t  2 R R , h e / t − 2 e V e /2 C = δ − ⇒ = Θ − P E ( ) ( E 2 ) I ( V e /2 C ) C R t

A simpler system • Environment: single mode • No quasiparticles: use Josephson junction polarized below the gap voltage NIN SIS Tunnel Josephson Junction Junction

Dynamical Coulomb Blockade of a Josephson Junction E Cooper V pair 2e eV ω =  2 eV Cooper pair = ω + − ϕ † H h ( a a 1/ 2) E cos J H. Pothier, ϕ = + +  † 2 eVt / r a ( a ) Ph. D. dissertation (1991) π Z L = = r Z 2 h / 4 e C + Fermi golden rule calculation

Dynamical Coulomb Blockade of a Josephson Junction E Cooper V pair 2e V ω =  2 eV Cooper pair π 2 E ∑ 2 Γ = ϕ δ − ω 2 e i  J ( ) V n e 0 (2 eV n ) →  2 n − π 2 n E exp( r r ) ∑ = δ − ω  J (2 eV n )  2 n ! n ν Γ = ω = Γ  h 2 e ( V n /2 ) e n → →

Dynamical Coulomb Blockade of a Josephson Junction E Cooper V pair 2e V ω =  2 eV Cooper pair − π 1 n 2 E exp( r r ) ∑ Γ = δ − ω 2 e  J ( ) V (2 eV n ) ⇒ → 0,1  2 n ! n = Ω ⇒ 0,01  Z 160 r 0.08 Γ n Z=160 Ω 1E-3 1E-4 0 1 2 3 4 n

Josephson junction and resonator ν = ≈ ≈ Ω  2 25 GHz, Q 5, Z 120 h /4 e 1 1 ν 3 ν 5 Z 1 =100 Ω Z 2 =28 Ω ν 1 500 Z ( Ω ) 50 Ω on res: 640 Ω 16 Ω 50 Ω 0 Holst et al, PRL 73 , 3455 (1994)

Josephson junction and resonator ν = 25 GHz 1 ν 3 ν 5 Z 1 =100 Ω Z 2 =28 Ω ν 1 500 Z ( Ω ) 50 Ω on res: 640 Ω 16 Ω 50 Ω 0 Holst et al, PRL 73 , 3455 (1994)

Josephson junction and resonator ν 1 = 25 GHz ν 3 ν 5 Z 1 =100 Ω Z 2 =28 Ω ν 1 500 Z ( Ω ) 50 Ω on res: 640 Ω 16 Ω 50 Ω 0 Holst et al, PRL 73 , 3455 (1994) Two photon processes weak  2 Z h /4 e because 1

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 Φ = π Φ Φ 0 E ( ) E cos( / ) V P I J J 0 300 K 10M 4 K Φ 50 15 mK 1000 6 GHz 100

Quarter-wave resonator Φ 50 Ω  25 Ω 135 Ω Designed Lorentzian 1,5 Fit ν 1 ν 3 ν 5 Re(Z) [k Ω ] 1,0 L = = Ω Z 160 0 C = Q 10 0,5 0,0 0 6 12 18 24 30 36 f(GHz)

Calibration of the detection impedance ∆ ν  ⇒ quasi-particle shot noise eV , h , k T Apply B ν Re[ ( )] Z R ∫ = ⇒ = ν ⇒ ν t S 2 eI P 2 eV d Re[ ( )] Z II + ν 2 R Z ( ) t = Ω R 18 k t 2,0 Measured Designed 1,5 Re(Z env ) [k Ω ] 1,0 0,5 0,0 5,0 5,5 6,0 6,5 7,0 frequency [GHz]

Cooper pair and photon rate match Cooper pairs Photons (5-7 GHz)

Second order processes ν 1 ν 3 ν 5 ν 7 Cooper pairs ν 1 ν 3 ν 5 ν 7 ν 1 Photons in mode ν 0 (5 – 7 GHz) ν 1 ν 1 2 ν + ν ν + ν 1 3 ν + ν 1 5 1 7

Spectral properties of emitted radiation = h ν + ν 2 eV ( ) 1 = ν 2 eV h

Coulomb blockade with an arbitrary environment ∑ = ν + − ϕ − ˆ † H h ( a a 1/ 2) E cos 2 eV N i i i J t i V 2 eVt ∑ ϕ = + π + + 2 4 e Z a ( a ) h i i i  i Cooper pair rate: π   ∑ ∑ 2 Γ = ϕ δ − ν  2 i E n n , , e 0 2 eV n h   J 1 2 i i    2  n n , , i 1 2 π = Alternate calculation of P(E): 2 E P (2 e V ) J  2 Ingold & Nazarov, arxiv:0508728 (1992) π 2 4 e Re (2 Z eV h / ) ≈ 2 E J  h 2 eV Photon rate at ν = ν m : exclude mode m from sum π   ∑ ∑ 2 2 ϕ ϕ ϕ − Γ = δ − ν − ν 2 i   i ( ) E n e 0 n , n , e 0 2 eV n h n h   m m − + m , n J m m m 1 m 1 i i m m    2 m ≠   n , n , i m − + m 1 m 1 δν → 0: Photons emitted in m δ Γ ν π 2 4 e R e Z ( ) = × − ν 2 E P eV (2 h ) 2 δν ν J  h 2 probability of photon tunneling rate while emission at ν absorbing rest of energy

Spectral properties of emitted radiation h ν ≈ Lorentzian with 2 4 e ∆ = π E 2 kT Z (0) h δ Γ ν π 2 4 e Re ( Z ) = × − ν 2 2 E P eV (2 h ) δν ν J  h 2

Spectral properties of emitted radiation h ν 2 4 e Re (2 Z eV h / ) ≈ 2 h 2 eV δ Γ ν π 2 4 e Re ( Z ) = × − ν 2 2 E P eV (2 h ) δν ν J  h 2

Emitting photon pairs Z a Z b V Emission of photon pairs Z E 2 eV + Bell-like state Z E 2 eV

Setup Φ = π Φ Φ 0 E ( ) E cos( / ) V P I J J 0 300 K 10M 4 K Φ 50 15 mK 1000 6 GHz 100

Emitting photon pairs quasi-particle shot noise a 2,5 2,0 b Re(Z env ) [k Ω ] 1,5 1,0 0,5 0,0 5,0 5,5 6,0 6,5 7,0 frequency [GHz]

Emitting photon pairs Re(Z) [k Ω ] 0 1 2 Filter b Filter a

Setup ⇒ < δ > < δ > < δ δ > 2 2 P , P , P P ADC a b a b P a P b V 300 K 100k 4 K Φ 50 15 mK Φ = π Φ Φ 0 E ( ) E cos( / ) J J 0 6 GHz 10

Tuning the photon emission rate Γ ∝ 2 E J Φ = π Φ Φ 0 E ( ) E cos( / ) J J 0 | Γ a - Γ b | / (Γ a + Γ b ) < 5% 0,15 b [GHz] 0,10 a , Γ Γ 0,05 Γ a Γ b 0,00 0,0 0,5 1,0 Φ/Φ 0

Power fluctuations cross correlation • Poissonian source of electrons  electronic shot noise due the charge granularity = = Γ 2 S 2 eI 2 e II • Poissonian source of photons = ν = ν Γ 2 S 2 h P 2( h ) PaP a a a a = ν = ν Γ 2 S 2 h P 2( h ) P b Pb b b b ν ν ( , ) • Poissonian source of photon pairs b a = ν ν Γ S 2 h h Pa Pb a b ⇔ = ν ν = Γ S S /( h h ) 2 a b Pa Pb a b

Correlated photon pairs  E J 2photon regime (2eV DS ~h ν a +h ν b ) Shot Noise regime (eV DS >2 ∆ ) Theory: fully correlated Poissonian emission of photon pairs ( ν a , ν b ) 0,1 S ab [GHz] 2 Γ 0,0 0,00 0,05 ( Γ a Γ 0.5 [ GHz ] b ) Evidence of Poissonian emission of photon pairs

Correlated photon pairs  E J 0,5 2photon regime (2eV DS ~h ν a +h ν b ) Shot Noise regime (eV DS >2 ∆ ) 0,4 Theory: fully correlated Poissonian emission of photon pairs ( ν a , ν b ) 0,3 S ab [GHz] 2 Γ 0,2 0,1 0,0 0,00 0,05 0,10 0,15 ( Γ a Γ 0.5 [ GHz ] b ) Deviations due to stimulated emission?

Limits of Coulomb blockade theory • So far good agreement with P ( E ) theory • But need very low E J to fulfill assumptions – environment at equilibrium π ∑ ∑ 2 Γ = ϕ δ − ν 2  i E n n , , e 0 (2 eV n h ) J 0 1 i i  2  n n , , i 0 1 – single Cooper pair regime: E J P (2 eV ) << 1 • What happens if assumptions are violated?

Out of equilibrium environment increase E J Lasing-like transition ? Transition ? Incoherent classical AC pair tunneling Josephson effect

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)