Quantum dynamics in Josephson junction circuits Wiebke Guichard Université Joseph Fourier/ Néel Institute Nano Department Equipe Cohérence quantique Josephson junction team Olivier Buisson, Bernard Pannetier, Laurent Lévy Current PhD students: Ioan Pop (Josephson junction arrays, Topologically protected qubit) Florent Lecocq (Qubit de phase) Zhihui Peng (Epitaxial Josephson junction) Former PhD students and postdocs: Aurélien Fay (Coupled qubit circuit) Emile Hoskinson Julien Claudon Franck Balestro Collaborations : Frank Hekking, Lev Ioffe, Bénoit Doucot, Léonid Glazman, Ivan Protopopov, Michael Gershenson Journée Information quantique, Monday 15 June 2009
Outline Quantum dynamics in a coupled qubit circuit Josephson junction arrays: Towards the realization of a topologically protected qubit?
Ultrasmall Josephson junction E J = R R Q T 2 - - + + j - + 2 j 1 - R T C + 1 S = Surface R T ~ S C ~ S 2 e E C = 2C Josephson effect: ˆ 2 Q 2C Charging effects 2e I( ϕ) = EJ sin( ϕ) Coulomb blockade: h Hˆ = E J cos ϕˆ I=0 for V<e/C ϕ N 1/ 2
E J >>E C Bloch Bands 2 d ψ E E J + cos = 2 + ϕ ψ dϕ Ec EC 0 Tight Binding Model H = E 2 d 2 dϕ c E J cosϕ Movement of a particle in a periodic potential E Pot E J cos(ϕ) E n (q) 2> 1> 0> ϕ q/2e 2π 2π Phase-slip! ( ) ( ) 3 1/ 4 Tunnelling amplitude for phase slip: v E E exp 8E / E J C J C
E C >>E J Weak Binding Model E(q) E Pot E J cos(ϕ) E J ϕ E C 0> 1> q
The coupled circuit Asymmetric Cooper pair transistor 10µm 500nm V G δ(ι b,φ b ) Φ b I b SQUID H ˆ T = (2e)2 ( n ) n g ) 2 Σe 2C j cos(δ /2)cos( Θ ) d ) Σ 0.5 µm e j sin(δ /2)sin( Θ ) d ) H ˆ S = 2 1 hω ( ˆ p P 2 + X ˆ 2 ) σhω ˆ P X 3 Phase qubit E J >> E C Charge qubit E J ~ E C hω S (I b,φ b ) U (I b,φ b )
Asymmetric Cooper pair transistor: charge qubit δ Charging energy Josephson energy symmetric asymmetric V G Microwave A charge qubit: with [ n ˆ, Θ ˆ d ]= i n g =1/2 Josephson energy Charge energy
Manipulation of the qubit at the optimal point δ=π ν MW =ν ->-> +> Measurement on the SQUID T 2 =15ns T MW : tunable T delay Rabi oscillations : Energy relaxation T decay,rabi =110ns
Spectroscopy measurement of the two quantum systems n g =1/2 I b =1910nA ACPT SQUID
Spectroscopy versus V g Resonant coupling Dispersive coupling 0,+> 1,-> g~110mhz δν T ~40MHz (charge noise limitation) δν S ~20MHz (fluctuator and flux noise limitation) 0,+> -> +> 200ΜΗz χ g 2 /4 10MHz
Spectroscopy versus flux Spectroscopy at I bias = 1890 2140 107 na na I 50% (µa) Frequency (GHz) Frequency (GHz) ν r ~ 16.6 GHz g ~ 250 MHz ν r ~ 10.4 GHz g ~ 900 MHz ν r ~ 18.1 GHz g ~ 160 MHz Two qubits can be in resonance from 9 GHz to 20 GHz. Strong variation of the coupling strength.
Resonant coupling Coupling varies from 60 MHz to 1100 MHz g (GHz) factor of 18 ν r (GHz)
Electrical schematic of the circuit
Coupling in resonance Ĥ COUPL = Capacitive coupling Josephson coupling Capacitance asymmetry Josephson energy asymmetry
Resonant coupling A. Fay et al., PRL 100, 187003 (2008) We consider λ = µ = 41.6% g (GHz) ν r (GHz) If transistor was symmetric (λ=µ=0) coupling would be zero
Resonant coupling A. Fay et al., PRL 100, 187003 (2008) We consider λ = 37.7% and µ = 41.6% g (GHz) ν r (GHz)
Two qubits read-out Aurelien Fay Thesis U (I b,φ b ) Γ 1 Flux pulse SQUID ω p (I b,φ b ) A nano-second flux pulse reduces the barrier Escape Quantronium read-out : classical Josephson junction ω S <<ω T In our case: ω S ω T!!!
Presence of an anti-level crossing ν (GHz) 18 16 14 12 10 With anti-level crossing SQUID alone 8 Very weak Landau-Zener transition Adiabatic quantum transfer!
Absence of an anti-level crossing ν (GHz) 18 16 14 12 10 8 Without anti-level crossing Squid alone Very weak Landau-Zener transition
Quantum dynamics in Josephson junction arrays Candidate for the realisation of a topologically protected qubit Dual of Shapiro steps in a Josephson junction array δ Φ c Phase bias and frustration: Φ R f = 2πΦ Φ δ = Φ c 0 0, f=0 f=0.5 eff J E, E eff c Chain of Josephson junctions with effective E J and E C Chain of N spins 2 N possible states
Phase biased Josephson junction array ϕ 1 ϕ 2 δ ϕ N E Pot =-E J cos(ϕ) Tunneling-amplitude = Bloch band width Quantum Tunneling ( ) 1/ 4 E E exp( 8E E ) 3 v / J C J C ψ(ϕ) ψ(ϕ+2π) ϕ
Chain of single Josephson junctions: classical regime E J >>E C δ = ϕ 1 ϕ 2 i= 1 ϕ N N ϕ i ϕ = i δ N NE J E(δ) E pot = = i NE E J J EJ 2 = δ 2N [1 cos( ϕ )] [1 cos( δ / N)] i 2πN δ
Phase biased chain: Effect of phase slip in phase biased chain δ = N i= 1 δ i ϕ i where ϕ = i δ N ϕ i Phase slip: energy unchanged but contraint violated! Phase slip combined with small adjustments: constraint satisfied! δ i ϕ ϕ + 2π i i j i δ δ + 2π n δ i δ 2π ϕ j = + 2π N δ 2π ϕi = N j ϕ ϕ + 2π i i i δ n
Chain of single Josephson junctions: classical regime E J >>E C ϕ 1 ϕ 2 N ϕ N δ = i= 1 ϕ i ϕ = i δ N δ 2π ϕ j = + 2π N δ 2π ϕi = N NE J E(δ) E pot = i E J [1 cos( ϕ )] = NEJ [1 cos(( δ -2π )/ N)] Phase-slip i 2πN δ
Chain of single Josephson junctions: classical regime E J >>E C ϕ 1 ϕ 2 N ϕ N δ = i= 1 ϕ i ϕ = i δ N NE J E(δ) E m EJ 2N ( δ + 2πm ) Critical current of chain: I c πi N c 2 Phase-slip 2πN δ
Phase biased Josephson junction array δ = ϕi ϕ 1 ϕ 2 ϕ N ~δ/n N i= 1 Hψ m ψ Nv( ψ ψ ) 1 = E + m m m 1 m+ E(δ) E m Matveev et al (2002) EJ = m 2N ( 2π ˆ δ ) 2 m-1 m m+1 δ/2π E L L = 2 π = 2L N E J 2 h 2e Quantum variable
d Mathieu equation for Josephson array ψ ( q) 2 dq 2 E 2Nν + + cos 2q ψ ( q) = 0 EL EL E Pot =-ev c cos(2q) E(δ) 1> q/2e m-1 m m+1 0> 2e L q&& + Rq& + V sin( 2πq /(2e)) = c V bias δ/2π
Energy spectrum and current-phase relation of chain E J /E c =20 E J /E c =3 E J /E c =1.3
Sample set-up Sample caracteristics: Single junction in chain R J =6kΩ C J =1.2fF Read-out Josephson junction: R=1kΩ Area ratio between big loop and SQUID loop: 285
Measurement of the current-phase relation I I I bias bias = = I I chain chain + I J (γ) + I C sin( Γ) I << Chain c I J c I bias = I chain Φ 2π Φ 0 π + 2 I c π sin 2 Current-phase relation yields information on the ground state I S ( γ ) E γ = 0
Measurement of the current-phase relation II Large Josephson junction: I sw =710nA PhD: Ioan Pop, to be published
Measurement of the current-phase relation III Fitting parameters: R=16kΩ PhD: Ioan Pop, to be published
Josephson junction rhombi chain δ Phase bias and frustration: Φ c f = δ = Φ R 2πΦ Φ Φ c 0 0, f=0 eff J E, E eff c Chain of Josephson junctions with effective E J and E C f=0.5 Chain of N spins 2 N possible states
Measurement of the current-phase relation δ I Φ c S R Chain c I << I J c S L =10*S R Current-phase relation yields information on the ground state I S (I. Pop et al.,phys. Rev. B, 78, 104504 (2008)) E δ = 0
Current-phase relation at f=0: classical regime E J /E C ~20 E(δ) I S E δ = 0 δ (I. Pop et al.,phys. Rev. B, 78, 104504 (2008))
Current-phase relation at f=0: quantum regime E J /E C ~2 I S E δ = 0 Hψ υ S Rhombus 0 A m = 2 = Em m N ( m 1 + m+ 1 = 8E E ψ 4.50( E Aexp( S c J 3 J E C ) 1/ 4 υ ψ (I. Pop et al.,phys. Rev. B, 78, 104504 (2008)) 0 ) ψ )
Current phase relation at f=0.5 δ I(δ)
Measurement of the ground state energy in the classical regime (1) f=0 Phase slips across Rhombi m=-1 m=0 m=1 > (I. Pop et al.,phys. Rev. B, 78, 104504 (2008)) 0.5 f
Measurement of the ground state energy in the classical regime (2) f=0.48 > > (I. Pop et al.,phys. Rev. B, 78, 104504 (2008)) 0.5 f
Measurement of the ground state energy in the classical regime (3) (I. Pop et al.,phys. Rev. B, 78, 104504 (2008)) f=0.5 Odd number of spin up Even number of spin up 0.5 f
Measurement of the ground state energy in the classical regime (4) (I. Pop et al.,phys. Rev. B, 78, 104504 (2008)) > > > >
Towards a topologically protected qubit? Add quantum fluctuations at f=0.5 in order to lift the degeneracy of the states
Idea of topologically protected qubit [ M ] N Kitaev et al (2003) δe N 1
Energy spectrum of a rhombi chain at half flux frustration I + 0; γ = 0 1; γ = 0 + 0; γ = π 1; γ = π
Lev Ioffe and Benoit Doucot unpublished 1 0 ) ( 0 0 + Φ Φ = Ψ Ψ N N J t i qubit E e δ δϕ δϕ ϕ Energy spectrum of a rhombi chain at half flux frustration II
Proposed sample V(ϕ) E 2 (δ) 01 12 0 α 0> -β π> β 0> +α π> ϕ π Lev Ioffe and Benoit Doucot unpublished
Phase-charge qubit: Asymmetric Cooper pair transistor H ACPT 2 ( Qψ + CgVg ) = 2e j cos( Ψ δ / 2) cos( δ / 2) 2 e j 2C ψ sin( Ψ δ / 2) sin( δ / 2) Charging energy Josephson energy δ tunable -1> 0> 1> E J /E C =1