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

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

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: comment on the rewriting of E_J \cos \phi in the charge basis here

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 bad for qubit! Reduce noise itself Reduce sensitivity to noise ► materials science approach ► eliminate two-level fluctuators J. Martinis et al., PRL 95, 210503 (2005) ► 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 sweet spot only sensitive to 2nd order fluctuations in gate charge! energy ng (gate charge) Vion et al., Science 296, 886 (2002) ng

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) ► anharmonicity decreases sweet spot everywhere!

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 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.

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 ratio by decreasing 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) 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

Circuit QED integrated on microchip atom artificial atom: SC qubit cavity 2D 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

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

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:

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

Rob Schoelkopf Steve Girvin Michel Devoret