meet the transmon and his friends
DESCRIPTION
Departments of Physics and Applied Physics, Yale University. Chalmers University of Technology, Feb. 2009. Meet the transmon and his friends. Jens Koch. Outline. Bullwinkle. Transmon qubit ► from the CPB to the transmon ► advantages of the transmon - PowerPoint PPT PresentationTRANSCRIPT
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