Landau-Zener interferometry with superconducting qubits

Landau-Zener interferometrywith superconducting qubitsYuriy MakhlinLandau InstituteM. Sillanpää, T. Lehtinen, A. Paila, P. Hakonen (Helsinki)Thanks toM. Feigel‘man, R. Fazio

Outline• Landau, Zener and others (1932)(after Di Giacomo, Nikitin, Phys. Usp. 2005)• superconducting qubits• LZ interferometer• Stokes phase• continuous CSET quantum measurement• data and discussion

Landau, Zener and others (1932)atomic collisionsL. Landau, Phys. Z. Sowjetunion 1, 88 (1932)sudden limit: ζ¿1L. Landau, Phys. Z. Sowjetunion 2, 46 (1932)adiabatic limit: ζÀ1

C.Zener, Proc. R. Soc. London Ser. A 137, 696 (1932)one crossingdouble crossingh vs. ~

E.C.G. Stueckelberg, Helv. Phys. Acta 5, 369 (1932)semiclassical approximation, analytical continuationof the WKB solution across Stokes lines=> LZ expression

E. Majorana, Nuovo Cimento 9, 43 (1932)spin dynamics in a time-dependent magnetic field

Phase sensitivity and interferenceatomic collisionsL. Landau, Phys. Z. Sowjetunion 1, 88 (1932)Φ Lsudden limit: ζ¿1

Josephson charge qubitsNΦ xV gˆθ, ˆNθ, NΦ xV g2 degrees of freedomN, θ =−i[ ]2 energy scales E C , E Jcharge and phasecharging energy, Josephson couplingHCgVg2 Φx= EC( N − ) −EJcos(π ) cosθ2eΦtunableEJ02 control fields: V g and Φ xgate voltage, flux

Josephson charge qubitsNΦ xV gˆθ, ˆNθ, NΦ xV gHCgVg2 Φx= EC( N − ) −EJcos(π ) cosθ2eΦ0tunableEJ2 states only, e.g. for E C »E JH1=− E (V ) σ − E ( Φ ) σ2chgz12Jxx

Josephson charge qubitsNΦ xV gˆθ, ˆNθ, NΦ xV gHCgVg2 Φx= EC( N − ) −EJcos(π ) cosθ2eΦ0tunableEJ

Charge-phase qubitVion et al. (Saclay)Quantronium1 1=− E ch g σz−220.5E 10.250E-0.25 0H ( V ) E ( Φ ) σ--11 Ђ2 0Φ x /Φ 00.5121Ђ01Ђ22 V g1Jxxoperation at saddle pointswitching probability (%)55detuning=50MHz5045403530T 2= 300 ns250 200 400 600 800Delay between π/2 pulses (ns)

Charge-phase qubitVion et al. (Saclay)Quantronium1 1=− E ch g σz−220.5E 10.250E-0.25 0H ( V ) E ( Φ ) σ--11 ЂJxxoperation at saddle point2 0Φ x /Φ 00.5121Ђ01Ђ22 V g11 1 ∂E1 ∂ EH = − E V − δ2ch J2J( Φx0) σx−Vδgo gσz2 Φ Φxo xσx∂V∂Φgx2 2 4

LZ interferometryHenry, Lang `77Lopez-Castillo et al. `92Kayanuma 90‘sShytov, Ivanov, Feigelman `03probe decoherence in sc. qubits

One period:LZ:Φinterference:oscillating dependence on control parameters

Multipass interferometer (Mach-Zehnder)operation per period:phases picked away from the crossing:

in terms of spin-1/2 rotations:Euler anglesStokes phase modifies φ, φ‘ :

Scattering phasephase = „dynamic“ + ...

Transformation at the crossing U LZadiabaticity parameter:Stokes phase:

Many periods:period 2π in φ and φ‘φ, φ`=multiples of 2π –maxima- minimaNote: depends only on offset n g ,not rf amplitude !

Multiphoton resonancesin the rotating frame:Bloch eqs. give stationary solution with population response ∝NMR

Setup and readoutΓ - reflection coefficient

Population response

What is monitored?Read-out frequency f m >> relax. rate:too weakFast relaxation => account for(too strong response)In practice, they are comparable: f m ~ relax. rate

Simulations of the Bloch equationsXηApplicable? More or lessBloch eqs. in discrete time(cf. Shytov et al.)

Simulations of the Bloch equations

Stronger drive: quasiparticle states

Stokes phase?w/ Stokes w/o Stokes

Variation with frequency

Josephson flux qubits

LZ interference in flux qubitsOliver et al. (MIT)low ∆

Some applications• investigation of decoherence• sensitive readout of charge or flux• coherent manipulations