Circuit Quantum Electrodynamics - CoQuS

CircuitQuantum ElectrodynamicsJohannes Majer

Motivation

Overview• Cavity: Transmission Line Resonator• “Atom”: Superconducting Qubits,Josephson junctionssolid-state physicsmicrowave engineering• Experimentscavity quantumelectrodynamics

circuit QEDThe ChipNb on a sapphire7 mm 2 mm

Transmission Line ResonatorCoplanar WaveguideWaveguideTE/TM ModesE-B+ + -TEM Modes

Numberslength ~ 20mmwidth ~ 10µmfrequency 6 GHzsuperconductivityQmax~5 10 5single photonenergy 10 -24 Jelectrical field 0.1 V/mmagnetic field 40 µGaussmode volume reduced by dλ 2Q/V mode =2· 10 13 (λ −3 )

Coupling Capacitor‘mirror’The ChipQubit‘Atom’

Josephson junctionCooper PairSISϕ 1 ϕ 2Josephson lawI = I 0 sin(∆ϕ)Capacitanceq = CVcharge and phase are conjugate variables[q/e, ϕ] = 1

Josephson Effectfirst Josephson lawI = I 0 sin(∆ϕ)tunnel current is proportional to thesin of the phase differencesecond Josephson lawV = φ 02π∂γ∂tJosephson energyE =IV dt = I 0φ 02πE = E J (1 − cos(γ))∆ϕ = ϕ 2 − ϕ 1changing phase leads to a voltageφ 0 = h 2e =2· 10−15 Wbsuperconducting flux quantumsin(γ) dγdt dt E J = I 0φ 02π

Josephson JunctionCooper pairJosephson effectI = I 0 sin(∆ϕ)SISϕ 1 ϕ 2Josephson energyE J = I 0Φ 02πcharge and phaseare conjugate variables[Φ,Q]=icapacitanceE c = e22CJosephson energy and charging energy can be engineered• size of the junction• oxide thickness

Josephson junctionL J = Φ 02πI 01cos(ϕ)L effC JR

Other Superconducting QubitsCooper-PairBoxNakamura, NECDevoret, Estève, SaclayTransmonSchoelkopf, YaleFlux QubitMooij, DelftPhase QubitMartinis, Santa Barbara

Cavity QEDgκγvacuum Rabifrequencycavity decayrateatom decayrateH = ω r a † a + 1 2quantizedfieldstrong couplingg γ, κ+ ω q2 σz + g(a † σ − + aσ + ) +Hγ + H κqubit‘atom’electric dipolinteraction

circuit QEDDriveAmplifier~6 GHz n 1number of photons in theresonatorphase and amplitudeT noise ≈ 5K

Measurement Setup

Dilution Refrigeratorbase temperature15mKhν = k B T5GHz ≡ 240mK

Resonator Transmission

Vacuum Rabi Splitting|↓ |1 − |↑ |0 |↓ |1 + |↑ |0qutonphobit

Vacuum Rabi SplittingDrive Frequency (GHz)magnetic field2g = 2π · 350MHzsplitting =300 x linewidthTransmission6.8 6.9Frequency (GHz)7.0

Dispersive Measurement

Coupled Qubitshomodynedetectiontransmon qubitcoupling capacitor

Spectroscopy

Avoided Qubit-Qubit Crossing|1, −1|↓↑ − |↑↓|0, 0|↓↑ + |↑↓ν = 8E J E C|1, 0ν = ν max| cos(πφ/φ 0 )|2 g |↑↑1g 2∆|1, +126MHz2 g 1g 2∆ = 26 MHz (σ1 x − σ2 x ) (|↓↑ + |↑↓) = 0

Dispersive Coupling2J = 2 g 1g 2∆coupling strength= 26 MHz < κ = 33 MHzvirtual photonscavity decay rate

State Transferno Stark pulseTextJwith Stark pulseJ. Majer, et. al. Nature 2007MA. Sillanpaa, et. al. Nature (2007)

Two-Qubit QuantumAlgorithmsDeutsch-JoszaGroverL. DiCarlo, ..., J. M, ..., and R. J. Schoelkopf, Demonstration of twoqubitalgorithms with a superconducting quantum processor, Nature460, 240-244 (2009)

Generating Fock States√N2gMax Hofheinz, E. M. Weig, M. Ansmann, Radoslaw C. Bialczak, ErikLucero, M. Neeley, A. D. O/'Connell, H. Wang, John M. Martinis, andA. N. Cleland,Generation of Fock states in a superconducting quantum circuit,Nature 454, 310--314 (2008)

Outlookn spine spinatoms moleculesRydbergatomssuperconductingqubitsmallSizelarge10 -16 m1Å100nm1µmslowCoupling Ratefast0.1Hz100Hz10kHz 100kHz10MHz 100MHzlongCoherence Timeshort1h1sec1µs

Hybrid Quantum Systemspolar moleculesmechanical resonatorA. Andre, et. al., Nature Physics 2, 636--642 (2006)electrons, ions, C60,...A. D. Connell, ... John M. Martinis, and A. N. Cleland,Nature 464, 697--7ultracold Rb atomsdiamond color centersJ. Verdu, ...PRL. 103, 043603 (2009)arXiv:1006.0251, Y. Kubo, et. alarXiv:1006.0242 Schuster et. al

LiteratureR. J. Schoelkopf and S. M. Girvin, Wiring up quantum systems, Nature, 2008M. H. Devoret and J. M. Martinis, Implementing Qubits with SuperconductingIntegrated Circuits, Quantum Information Processing, SpringerNetherlands,Volume 3, Numbers 1-5 / October, 2004J. Q. You and F. Nori, Superconducting Circuits and Quantum Information,Physics Today, 2005J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature 453,1031-1042TU Wien VO 141.246