1. Meet the transmon and his friends
Departments of Physics and Applied Physics, Yale University
Chalmers University of Technology, Feb. 2009
Meet the transmon and his friends
Jens Koch

2. Outline Transmon qubit Circuit QED with the transmon – examples
► from the CPB to the transmon
► advantages of the transmon
► theoretical predictions vs. experimental data
Circuit QED with the transmon
– examples
Bullwinkle

3. 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

4. CPB as a charge qubit Charge limit:
comment on the rewriting of E_J \cos \phi in the charge basis here

5. CPB as a charge qubit
Charge limit:
big
small perturbation

6. 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
bad for qubit!
Reduce noise itself
Reduce sensitivity to noise
► materials science approach ► eliminate two-level fluctuators
J. Martinis et al., PRL 95, (2005)
► design improved quantum circuits
► find smart ways to beat the noise!
Paradigmatic example: sweet spot for the Cooper Pair Box

7. Outsmarting noise: CPB sweet spot
◄ charge fluctuations
energy
sweet spot
only sensitive
to 2nd order fluctuations in gate charge!
energy
ng (gate charge)
Vion et al., Science 296, 886 (2002) ng

8. 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?

9. Towards the transmon: increasing EJ/EC
► charge dispersion becomes flat
(peak to peak)
► anharmonicity decreases
sweet spot
everywhere!

10. 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

11. Harmonic oscillator approximation
resulting Schrödinger equation:
► harmonic spectrum
► no charge dispersion

12. Anharmonic oscillator
Anharmonic oscillator approximation
expand
like before
perturbation
Perturbation theory in quartic term
anharmonic spectrum
still no charge dispersion

13. 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
WKB valid if relative change of the local classical momentum over the de Brogie wavelength is small. In other words, V(x) varies slowly on the scale of the de Broglie wavelength.

14. 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

15. Increase EJ/EC Increase the ratio by decreasing
Island volume ~1000 times bigger than conventional CPB island

16. 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, (2007), EXPERIMENT: J. A. Schreier et al., Phys. Rev. B 77, (R) (2008)

17. 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, (2004)
A. Wallraff et al., Nature 431,162 (2004)
R. J. Schoelkopf, S.M. Girvin, Nature 451, 664 (2008)
2g = vacuum Rabi freq.
k = cavity decay rate
Describe diagram
Describe each rate
Describe each term in Hamiltonian
Discuss strong coupling
g = “transverse” decay rate
Jaynes-Cummings Hamiltonian
resonator
mode
atom/qubit
coupling

18. Circuit QED integrated on microchip
atom artificial atom: SC qubit
cavity D transmission line resonator
integrated on
microchip
paradigm for study of open quantum systems
► coherent control
► quantum information processing
► conditional quantum evolution
► quantum feedback
► decoherence
Describe diagram
Describe each rate
Describe each term in Hamiltonian
Discuss strong coupling

19. Coupling transmon - resonator
qubit
resonator mode
Cooper pair box / transmon:
coupling to resonator:

20. Control and QND readout: the dispersive limit
Control and readout of the qubit: (detune qubit from resonator)
dispersive limit
: detuning
canonical transformation
dynamical Stark shift Hamiltonian
dispersive shift:

21. Circuit QED with transmons
2006/7
Probing photon states via the
numbersplitting effect
►transmon as a detector for photon states
J. Gambetta et al., PRA 74, (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)
2008
Observing the √n nonlinearity
of the JC ladder
A. Wallraff et al. (ETH Zurich)
L. S. Bishop et al. (Yale)

22. Rob Schoelkopf
Steve Girvin
Michel Devoret