Meet the transmon and his friends

Jens Koch

Departments of Physics and Applied Physics, Yale University

Chalmers University of Technology,Feb. 2009

Outline

Transmon qubit► from the CPB to the transmon► advantages of the transmon► theoretical predictions vs. experimental data

Circuit QED with the transmon – examples

Bullwinkle

Review: Cooper pair box

charge basis:

phase basis:exact solution withMathieu functions

numerical diagonalization

3 parameters:

offset charge (tunable by gate)Josephson energy (tunable by flux in split CPB)charging energy (fixed by geometry)

CPB as a charge qubit

Charge limit:

CPB as a charge qubitCharge limit:

bigsmall perturbation

Noise from the environment

• Noise can lead to energy relaxation ( ) dephasing ( )

• Persistent problem with superconducting qubits: short

bad for qubit!

Reduce noise itself Reduce sensitivity to noise

► design improved quantum circuits► find smart ways to beat the noise!

Paradigmatic example: sweet spot for the Cooper Pair Box

► materials science approach► eliminate two-level fluctuators

J. Martinis et al., PRL 95, 210503 (2005)

Superconducting qubits are affected by

charge noise flux noise critical current noise

Outsmarting noise: CPB sweet spot

only sensitive to 2nd order fluctuations in gate charge!

ener

gy sweet spot

ng (gate charge)

ener

gy

ng

Vion et al., Science 296, 886 (2002)

◄ charge fluctuations

How to make a sweeter spot?

disadvantages:

► need feedback► still no good long-term stability► does not help with “violent” charge fluctuations

CPB sweet spot: the good and the bad

Linear noiseT2 ~ 1 nanosecond

(e.g. Nakamura)

Sweet spotT2 > 0.5 microsecond

(e.g. Saclay, Yale)

Towards the transmon: increasing EJ/EC

► charge dispersion becomes flat

(peak to peak)

► anharmonicity decreasessweet spot

everywhere!

Harmonic oscillator approximation

• Consequences of

► strong “gravitational pull”► small angles dominate

quantum rotor(charged, in constant magnetic field )

expand

ignore periodic boundary conditions

eliminate vector potential by “gauge” transformation

► harmonic spectrum

► no charge dispersion

Harmonic oscillator approximation

• resulting Schrödinger equation:

• Anharmonic oscillator approximation

expand

Perturbation theory in quartic termlike before perturbation

Anharmonic oscillator

• anharmonic spectrum

• still no charge dispersion

- WKB with periodic b.c.- instantons- asymptotics of Mathieu characteristic values

Charge dispersion

► full 2 rotation, Aharonov-Bohm type phase

► quantum tunneling with periodic boundary conditions

Coherence and operation times

T2 from 1/f charge noise at sweet spotTop due to anharmonicity

the “anharmonicity barrier” at EJ/EC = 9

chargeregime

transmonregime

Increase EJ/EC

Increase the ratioby decreasing

Island volume ~1000 times biggerthan conventional CPB island

Experimental characterization of the transmon

THEORY: J. Koch et al., PRA 76, 042319 (2007), EXPERIMENT: J. A. Schreier et al., Phys. Rev. B 77, 180502(R) (2008)

theory

Reduction of charge dispersion:

Strong coupling

vacuum Rabi splitting2g ~ 350 MHz

Improved coherence times

Cavity & circuit quantum electrodynamics►coupling an atom to discrete mode of EM field

2g = vacuum Rabi freq. = cavity decay rate = “transverse” decay rate

cavity QED Haroche (ENS), Kimble (Caltech)J.M. Raimond, M. Brun, S. Haroche, Rev. Mod. Phys. 73, 565 (2001)

circuit QEDA. Blais et al., Phys. Rev. A 69, 062320 (2004) A. Wallraff et al., Nature 431,162 (2004) R. J. Schoelkopf, S.M. Girvin, Nature 451, 664 (2008)

Jaynes-Cummings Hamiltonian

atom/qubitresonator

mode coupling

Circuit QED

atom artificial atom: SC qubit

cavity 2D transmission line resonator

integrated onmicrochip

► coherent control

► quantum information processing

► conditional quantum evolution

► quantum feedback

► decoherence

paradigm for study of open quantum systems

qubit

resonator mode

Coupling transmon - resonator

coupling to resonator:

Cooper pair box / transmon:

Control and QND readout: the dispersive limit

• Control and readout of the qubit: (detune qubit from resonator)

: detuning

canonical transformation

dynamical Stark shift Hamiltonian

dispersive shift:

dispersive limit

Circuit QED with transmons

Realization of a two-qubit gate ► two transmons coupled via exchange of

virtual photons

2007

2006/7Probing photon states via thenumbersplitting effect ►transmon as a detector for photon states

J. Gambetta et al., PRA 74, 042318 (2006); D. Schuster et al., Nature 445, 515 (2007)

J. Majer et al., Nature 449, 443 (2007)

2008Observing the √n nonlinearityof the JC ladder A. Wallraff et al. (ETH Zurich) L. S. Bishop et al. (Yale)

Rob Schoelkopf Steve Girvin

Michel Devoret