Measuring Quantum Coherence in the Cooper-Pair Box Konrad Lehnert Depts. of Applied Physics & Physics Yale University Yale Lafe Spietz Ryan Held Ben Turek Rob Schoelkopf Chalmers University Kevin Bladh David Gunnarsson Per Delsing And discussions w/: M. Devoret, S. Girvin, A. Clerk, K. Nguyen The David and Lucile Packard Foundation Funding:

Can Electrical Circuits be ‘Quantum?’ Macroscopic Quantum Coherence: Cooper-pair box Y. Nakamura et al, Nature 1999 New Challenges: Understand and minimize decoherence Develop efficient quantum readout New Opportunities: Create artificial atoms Quantum computation

Quantum Circuits for Quantum Computing Classical bit Quantum bit (or “qubit”) Information as state of a two-level quantum system values , or values 0 or 1 superposition: Prediction: a 2,000 bit quantum computer = a conventional computer the size of universe.

Quantum Computing Ion Traps Liquid State NMR Nuclear Spins in Semiconductors Coherent Scalable Controllable Measurable Cooper-pair box SQUID’s How coherent is a Cooper-pair box?

Single Electron Transistor Measuring Box Vds Box SET Electrometer Cg Cc Cge Box Vg Vge SET Superconducting tunnel junction Al/AlOx/Al junctions; 50 x 50 nm e-beam lithography; double-angle evaporation Tc ~ 1.5 K

Cooper-pair Box Vg Vg

Cooper-pair Box as Quasi-spin 1/2 Measure charge Ground state 1 b c a a b c Excited state 0.5 E a b c

NMR of a Single Spin Single Spin ½ Quantum Measurement Vds Cgb Cc Cge Box Vgb Vge SET

Single-electron Transistor: Electrometer SET drain Vds Cge Vge Ids 10 nA source Electrometer input gate Vds 1 mV

Radio-Frequency Single Electron Transistor (RF-SET) Response to step in Vge Transformer SET single time trace RF Reflected power Electrometer input gate Measure RF power reflected from LC transformer 10-5 e/Hz1/2 charge noise Sub-electron sensitivity for > 100 MHz bandwidth Schoelkopf et al., (Science 1998)

Dilution refrigerator Small, Cold and Fast Microwaves Dilution refrigerator T = 15 mK 1 mm Millikelvins Nanometers

Experiment Diagram

Continuous Measurement of a Single Spin Measured continuously by SET Theory: Cooper-pair box ground state 1 0.5 2e 1e 0.5 1 Measurement must cause additional dephasing uncertainty principle Measurement may also mix states, drive transitions from ground state

Cooper-Pair Resonance Spectroscopy Cg Vapp 38 GHz Vapp=Vg+Vacsinwt 1 w/2p=38 GHz 0.5 1

Determination of Box Hamiltonian “SQUID box” to vary EJ Peak location 32 GHz 0.29 Vapp B 35 GHz 38 GHz 0.25 -2 -1 1 2 E Fit parameters:

Saturation of the Cooper Pair Resonance Photon Peak Height 0.5 37 GHz 39 GHz 0.2 0.235 0.265 Peak width Peak height 50% saturated

Excited-state Lifetime 0.15 e t<0 t>0 time 10 ms t=20 ms 1e 0.3e t=1.6 ms t<0 t=0.4 ms Peak height (e) 0.5 1 time 10 ms

Spontaneous Emission Environment Box SET Vds Cc Cg Vg 2e E Relaxation

Spontaneous Emission into Environment Spontaneous Emission: Fermi’s golden rule Cg Box Vg 2e

Electrometer Input Impedance Cg Cc Cg Cc Vg 2e 2e SET 2e 0.6 185 W Peak Height (e) 0.3 370 740 Electrometer Operating Point (Vg)

Conclusions Cooper-pair Box: A quantum two-level system worst-case coherence Box Hamiltonian determined with spectroscopy Long excited-state lifetime while continuously measured. Box measures electrometer input impedance

Box State Depends on Electrometer Bias Vds (mV) 250 290 420 470 760 1200

Conclusions RF-SET measures charge states of box Spectroscopic determination of Hamiltonian of box Dephasing time ~ 1 ns : (w/ continuous measurement) Long Excited-state lifetime >1 ms : Electrometer affects T1

Outline Charge quantization on a normal-metal island Single-electron Box Superconducting island as quantum two-level system Cooper-pair Box Spectroscopy of the Cooper-pair box Single-electron Tranistor (SET) measures box Box Measures SET Quantum Spectrum Analyzer

The Single-Electron Box island Cg Vg ne e Cj Rj Normal tunnel junction E Ec ne to ne+1 electrons Ec/4 ne=-1 ne=0 ne=1

Single-electron Box: Coulomb Staircase First demonstrated by Lafarge et al, ’91 (Saclay) Ec Ec/4 ne=-1 ne=0 ne=1 200 mK 16 mK Coulomb Staircase Thermally broadened 1 e e kT/Ec -1 -1 -0.5 0.5 1

Cooper-pair Box Spectrum: Electrostatic and Josephson EJ Ec n=-1 n=0 n=1 Condition: Two level System EJ /4Ec 2e 2e -0.5 0.5

Cooper-pair Box as Spin 1/2 Time scales w01 Larmor frequency 10-40 GHz T1 Excited state lifetime 0.1-10 ms WR Rabi frequency T2* Ensemble decoherence time

The Quantum Spectrum Analyzer Cmeas ? Vbias Measures all Noise Classical (symmetric) Quantum (asymmetric)

Cooper-pair Box Spectrum: Electrostatic and Quasi-particle Odd: single q.p. Even: no q.p. kT/4Ec 2e 2e-periodic Cooper-pair Staircase 2e -0.5 0.5

Cooper-Pair Resonance Spectroscopy Cg Vapp Vapp=Vg+Vacsinwt 1 38 GHz 35 GHz 0.5 1