1. 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:

2. 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

3. 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.

4. 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?

5. 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

6. Cooper-pair Box Vg Vg

7. 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

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

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

10. 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)

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

12. Experiment Diagram

13. 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

14. Cooper-Pair Resonance SpectroscopyCg
Vapp
38 GHz
Vapp=Vg+Vacsinwt 1
w/2p=38 GHz
0.5 1

15. 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:

16. 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

17. 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

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

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

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

21. 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

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

23. 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

24. 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

25. 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

26. 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

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

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

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

30. 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

31. Cooper-Pair Resonance SpectroscopyCg
Vapp
Vapp=Vg+Vacsinwt 1
38 GHz
35 GHz
0.5 1