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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!
en
erg
y sweet spot
ng (gate charge)
en
erg
y
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 noise
T2 ~ 1 nanosecond (e.g. Nakamura)
Sweet spot
T2 > 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 term
like 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 ratio
by 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
modecoupling
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