Transcript Life after Charge Noise: An Introduction to the Transmon Qubit

Departments of Physics
and Applied Physics,
Yale University
Chalmers University of Technology,
Feb. 2009
Meet the transmon
and his friends
Jens Koch
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
3 parameters:
offset charge
(tunable by gate)
Josephson energy
(tunable by flux in split CPB)
charging energy
(fixed by geometry)
charge basis:
numerical
diagonalization
phase basis:
exact solution with
Mathieu functions
CPB as a charge qubit
Charge limit:
CPB as a charge qubit
Charge limit:
big
small perturbation
Noise from the environment
Superconducting qubits are affected by
charge noise
flux noise
critical current noise
•
Noise can lead to energy relaxation (
dephasing (
•
Persistent problem with superconducting qubits: short
Reduce noise itself
► materials science approach
► eliminate two-level fluctuators
J. Martinis et al.,
PRL 95, 210503 (2005)
)
)
bad for qubit!
Reduce sensitivity to noise
► design improved quantum circuits
► find smart ways to beat the noise!
Paradigmatic example:
sweet spot for the
Cooper Pair Box
Outsmarting noise: CPB sweet spot
◄ charge fluctuations
energy
energy
sweet spot
only sensitive
to 2nd order
fluctuations in
gate charge!
ng (gate charge)
ng
Vion et al.,
Science 296, 886 (2002)
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)
disadvantages:
► need feedback
► still no good long-term stability
► does not help with “violent” charge fluctuations
How to make a sweeter spot?
Towards the transmon: increasing EJ/EC
► charge dispersion
becomes flat
(peak to peak)
sweet spot
everywhere!
► anharmonicity
decreases
Harmonic oscillator approximation
quantum rotor
(charged, in constant
magnetic field
)
• Consequences of
► strong “gravitational pull”
► small angles
dominate
expand
ignore periodic
boundary conditions
eliminate vector potential
by “gauge” transformation
Harmonic oscillator approximation
• resulting Schrödinger equation:
► harmonic spectrum
► no charge dispersion
Anharmonic oscillator
• Anharmonic oscillator approximation
expand
like before
perturbation
Perturbation theory in quartic term
• anharmonic spectrum
• still no charge dispersion
Charge dispersion
► full 2p rotation,
Aharonov-Bohm type phase
► quantum tunneling with
periodic boundary conditions
- WKB with periodic b.c.
- instantons
- asymptotics of
Mathieu characteristic values
Coherence and operation times
charge
regime
T2 from 1/f charge noise at sweet
spot
Top due to anharmonicity
transmon
regime
the “anharmonicity barrier”
at EJ/EC = 9
Increase EJ/EC
Increase the
by decreasing
ratio
Island volume ~1000 times bigger
than conventional CPB island
Experimental characterization of the transmon
Reduction of charge dispersion:
Improved coherence times
theory
Strong coupling
2g ~ 350 MHz
vacuum Rabi splitting
THEORY: J. Koch et al., PRA 76, 042319 (2007),
EXPERIMENT: J. A. Schreier et al., Phys. Rev. B 77, 180502(R) (2008)
Cavity & circuit quantum electrodynamics
►coupling an atom to discrete mode of EM field
cavity QED
Haroche (ENS), Kimble (Caltech)
J.M. Raimond, M. Brun, S. Haroche,
Rev. Mod. Phys. 73, 565 (2001)
circuit QED
A. 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)
k = cavity decay rate
2g = vacuum Rabi freq.
g = “transverse” decay rate
Jaynes-Cummings Hamiltonian
atom/qubit
resonator
mode
coupling
Circuit QED
atom
artificial atom: SC qubit
cavity
2D transmission line resonator
paradigm for study of
open quantum systems
► coherent control
► quantum information processing
► conditional quantum evolution
► quantum feedback
► decoherence
integrated on
microchip
Coupling transmon - resonator
Cooper pair box / transmon:
qubit
resonator mode
coupling to resonator:
Control and QND readout: the dispersive limit
•
Control and readout of the qubit:
dispersive limit
(detune qubit from resonator)
: detuning
canonical transformation
dynamical Stark shift Hamiltonian
dispersive shift:
Circuit QED with transmons
Probing photon states via the
numbersplitting effect
2006/7
►transmon as a detector for photon states
J. Gambetta et al., PRA 74, 042318 (2006);
D. Schuster et al., Nature 445, 515 (2007)
2007
Realization of a two-qubit gate
► two transmons coupled via exchange of
virtual photons
J. Majer et al., Nature 449, 443 (2007)
Observing the √n nonlinearity
of the JC ladder
A. Wallraff et al. (ETH Zurich)
L. S. Bishop et al. (Yale)
2008
Rob Schoelkopf
Steve Girvin
Michel Devoret