1. REVIEW OF SOLID STATE QUANTUM BIT CIRCUITS
Two strategies
single particle states
in semiconductor structures
 global quantum states
of superconducting Josephson circuits
(A) Kane’s proposal : nuclear spins of P impurities in Si
Chalmers
NEC
Quantronics
U. Of New
South Wales
TU Delft
(B) Electrons in quantum dots
Schoelkopf et al, Yale SL SR
B1) Charge: NTT
B2) Spin: TU Delft, Harvard,…
(C) Propagating states: flying qubits
NIST e
QHE edge states:
LPA (ENS Paris)
From charge states to phase states

2. First demonstration of coherent oscillations in a double dotd
Coherent charge oscillations in a double dot
(NTT, Hayashi et al. 2003)
|R>
|L> + |R>
|L> - |R>
|L>
T2 (charge qubit) ~ 1 ns
Result tells you two things:
Coherent manipulation of electronic states in quantum dots is possible! So; quantum computing is feasible…
Coherence time of charge states is very short (ns); spin states should be much better!
But charge too much coupled to the environment !
spin expected better

3. Expts: TU Delft , Harvard
B2) Spin qubits SL SR BZ
Initial ideas: Loss & DiVincenzo (1998)
1-qubit control:
magnetic (ESR)
electric (modulate effective g-factor)
2-qubit coupling:
exchange interaction between 2 dots
Read-out through charge
Expts: TU Delft , Harvard

4. GD GSR GSL b) 2e spin qubit in a double dot (NEW)
Harvard U.
C. Marcus team, GD
GSR
GSL
Harvard
Charge
sensor
Nature, june 2005,
& Petta et al., in prep.
Two electron spin qubit
charge readout
with QPC
# e in
dot

5. Bloch sphere in (1,1) S - T0 subspace

6. Measuring Spin Dephasing (T2)ε
(electrostatic energy difference)
Move from (0,2)S to (1,1) s & let evolve

7. dephasing causes failure to return to (0,2)
Measuring Spin Dephasing (T2)
(1,1)S
(0,2)S ε 2t
(1,1)T0
dephasing causes failure
to return to (0,2)
spin T2 ~ 10 ns

8. Short coherence time : 10 ns due to nucleiB
Zeeman
Nuclear
Total

9. But coherence restored by spin echo e e tflip tflip model experiment
tflip
40 ns
40 ns
tflip
pattern still observed at long times: coherence time TE =1.2 ms

10. team L. Kowenhouven& L. Vandersypen, TU Delft
Electron spin resonance in a double dot B0
Gates ~ 30 nm gold
Dielectric ~ 100nm calixerene
Stripline ~ 400nm gold
Expected AC current ~ 1mA
Expected AC field ~ 1mT
team L. Kowenhouven& L. Vandersypen, TU Delft

11. ESR detection via Pauli spin blockade
Bext
Energy
31jan06_20
hfac = g m B0
Flip right spin: ↑↑  ↑↓ = S11 + T  S02
Flip left spin: ↑↑  ↓↑ = -S11 - T  S02
Flip both spins: ↑↑  (T+ + 2 T0 + T- )/2  S02
Advantages: - low frequencies (B0 > 20mT, f > 100MHz. )
- not sensitive to electric fields (unlike single-dot ESR)
- no confusion with ESR in leads

12. Pulsed ESR scheme

13. Rabi oscillations
F. Koppens et al. pA

14. Solid State Qubits: way 2
single particle states
in semiconductor structures
 global quantum states
of superconducting Josephson circuits
A)Kane’s proposal : nuclear spins of P impurities in Si
Phase qubit
Flux qubit
B) electrons in quantum dots (charge or spin)
C P Box
Charge-flux qubit SL SR
C P Box
C)Propagating states: flying qubits e
Why superconductivity ?
Why different flavors?

15. Josephson qubits come in different flavorsY ( N ) 1 2
>> 1 ~ 2 N
phase
charge-phase
charge
flux
NIST
TU Delft
NEC, Chalmers
Saclay, Yale (Devoret)
Yale (Schoelkopf)
Single Cooper pair boxes

16. N S Superconducting helps making qubits +
Energy spectrum of an isolated electrode
Superconducting state
Non superconducting
state S N
singlet ground state +
The Josephson junction

17. Building blocks for quantum bit circuits1
0/1 U1 N ? ?

18. Basics of the Josephson junction
single degree
of freedom
Josephson Hamiltonian :

19. V Phase qubits Ib tilted washboard potential ) ( d w10 w21 w32
d : extended phase
conjugated of q on CJ d V 5
0.7
0.8
0.9 1
w10
w21
w32 ( ) p n E w h / - + Ib dc
I 2 >
I 1 > ac
I 0 >
tilted washboard potential

20. A flux quantum bit : the three junction loop E
Icirc 2t
0.5
F/Fo
+Ip
-Ip
aEJ
(0.5<a<1) g1 g2 EJ EJ
 0.5 F0
Fx=F0/2
2D potential:
Mooij et al. (TU Delft), (1999)

21. Single Cooper pair boxes
The first ‘working’  qubit

22. energy spectrum
Energy (EC)
qubit Ng

23. Josephson Phase Qubits at UCSB
UC Santa Barbara
Team of John Martinis

24. Lifetime of state |1>
Phase qubits
= F0/2pI0cos d
nonlinear inductor I d
cos j = V )
(1/L J
sin LJ 2 p F I R I0 LJ C
U(d) 5
0.7
0.8
0.9 1
w10
w21
w32 ( ) p n E w h / - + E0 E1 E2 G0 G1 G2 Gn
Gn+1
1000 ~
<V> = 0
g10 wp DU
<V> pulse
(state measurement)
Many ways to use non-linearity – for our qubit:
1) Non-linearity turns parabola to cubic -- Parameterized by dU & wp
2) QM : wavefunctions like H.O. – E levels now NOT equally spaced
3) Get tunneling, with ratio
deep levels – no tunneling , Use top levels to tunnel, part of measuring process
4) Lifetime of deep levels given by dissipation, parameterized by R [ 2 3 ] / 1 4 I U - F = D p C [ ] 4 / 1 2 I p - ÷ ø ö ç è æ F = w
1: Tunable well (with I)
2: Transitions non-degenerate
3: Tunneling from top wells
4: Lifetime from R
g10 @ 1 RC
Lifetime of state |1>

25. better phase qubit : rf SQUID with dc SQUID readout
U(d)
~5000 states
“0”
“1” …
fast
decay
1 F0
SQUID flux
Switching
current
10 mA Is If
time
Qubit
Cycle
Qubit Op Meas Amp Reset flux
Measure p1
I 0 >
I 1 >
low noise bias
qubit Is If
"sample and hold" readout
J. Martinis team ( )
NIST & SB
T2 : ns (?)

26. Josephson-Junction Qubit
State Preparation
Wait t > 1/g10 for decay to |0>
Qubit logic with current bias
State Measurement: DU(Idc+dIp)
Fast single shot – high fidelity
potential
|1>
|0>
I = Idc + dIdc(t) + Imwc(t)cosw10t + Imws(t)sinw10t
phase H s
dIdc(t) + s I + s I ( 2 ) z x
mwc y
mws
Apply ~3ns
Gaussian pulse
1.0
0.8
0.6
0.4
0.2
0.0
0.7
0.5
0.3
|1>
|0>
Deep well
Preparation
Logic by DC and AC current
Measurement by tunneling out of cubic well
2 ways to measure
|1> : tunnel
Prob. Tunnel
96%
|0> : no tunnel
pulse height of dIp

27. IC Fabrication Qubit Imwave Is If Al junction process
X,Y
readout
Imwave Is Z
100mm
(old design) If
Made with standard fabrication techniques: metalization and etch, use insulator crossovers for complex circuits
Point out structures
via
junction
Al junction process
& optical lithography Al Al
SiNx Al
Al2O3 substrate

28. Qubit Fidelity Tests (2006)
~90% visibility t
Rabi:
Ramsey: t
(slightly detuned)
Probability 1 state
Echo: t
(no detuning) t
T1:
Observe coherent oscillations by procedure…
Uw produces transition 0 -> 1 -> 0 …
Best data, with 70% visibility
From this data we’ve now calibrated amplitude of drive, can define pi pulse
Large Visibility! T1 = 110 ns, Tf ~ 85 ns

29. SUPERCONDUCTING FLUX QUBITS
Group of Prof. Hans Mooij
TU DELFT
team Y. Nakamura
NEC
team K. Semba
NTT

30. persistent-current quantum bit
aEJ
g1 (p) g2
(p)
magnetic flux
 0.5 Fo g1 g2 EJ EJ
barrier scales with EJ,
depends on a
Tout
a<1 to suppress
influence of charge noise
effective mass
scales with
junction capacitance C
Tin
tunneling: Tin = wm exp{ -0.64(EJ/EC)½}
Tout = 1.6 wm exp{ -1.5(EJ/EC)½}
for a=0.8

31. flux qubits : the three junction loop E
Icirc 2t
0.5
F/Fo
+Ip
-Ip
aEJ
(0.5<a<1) g1 g2 EJ EJ
 0.5 F0
Fx=F0/2
2D potential:
Mooij et al. (TU Delft), (1999)

32. SQUID readout of the flux qubit (readout # 1) Switching measurement
(Ic 200 nA)
mw on resonance

33. First spectroscopy of flux qubits
switching
current f
GHz
x10-3
F/Fo
F/Fo-0.5
Van der Wal et al., Science 290, 773 (2000)
also SUNY (Friedman, Lukens et al.)

34. Irinel Chiorescu and Yasu Nakamura (NEC, Delft)
Science 299, 1869 (2003)

35. Dephasing: T2Ramsey, T2echo measurement (sample5)
NEC
Dephasing: T2Ramsey, T2echo measurement (sample5)
correspond to detuning
Ramsey interference
p/2~2ns
p/2
readout pulse t
spin echo
p/2~2ns
p ~ 4ns
p/2
readout pulse
t/2
t/2

36. Observation of vacuum Rabi oscillations
in a flux qubit coupled to a SQUID resonator
K. Semba et al., NTT
DISQUID~Mqub,SQIcirc,qubit
SQUID
qubit
6.7μm
VSQUID
Isw I
|↓>
|↑>
Icirc,qubit M
6.3μm
DISQUID
Ithreshold
DC SQUID detector:
VSQUID = 0  qubit state |↑>
≠ 0  qubit state |↓>

37. Vacuum Rabi : measurement scheme
|g1
|g0 p
I qubit, LC-oscillator >　
|e0
|e1
|g1
|g0
|e1
|e0
|g1
2 → 3
|g0
excite qubit
by a p-pulse
shift qubit
adiabatically
shift qubit
adiabatically
readout
qubit state
3 ⇔ 4
1 → 2
|e0 4
|g１
|g0

38. Vacuum Rabi oscillations
Direct evidence of level quantization in a 0.1 mm large
superconducting macroscopic ＬＣ-circuit
J. Johansson et al., cond-mat/ → to appear in Phys. Rev. Lett. (2006).

39. Quantization of Rabi period
|g2
√ W ２
|e0
|g1
|g0 W
|e1
J. Johansson et al., cond-mat/ → to appear in Phys. Rev. Lett. (2006).

40. Readout of single Cooper pair boxes

41. Hamiltonian and energy spectrum
2 characteristic energies:
1 degree of freedom:
1 knob: or
Hamiltonian:

42. energy spectrum
Energy (EC)
qubit Ng

43. Readout through the charge
Energy (EC) Ng
expectation value
of the box charge:
(measurement of the quantum state)

44. Capacitive coupling to a Single Electron Transistor
EJ/EC=0.1
theory
experiment
Theory with no Josephson effect
too slow ...
V. Bouchiat et al.
Quantronics (1996) V
1/2 1

45. Short coherence time : a few ns
A box with a continuous readout
Nakamura, Pashkin
&Tsai (NEC,1999) Ng
0.5 V
2e- or 0
Continuous
measurement by
energy relaxation
Pulse durationDt (ps)
200
400
600 5
First Rabi oscillations
DC pulses from Ng=0.25 to 0.5
with duration Dt
Short coherence time : a few ns
2000: WHY??

46. Open ports Decoherence
The main difficulty a b + ? 1 1 U1
Decoherence
Sources
write
Readout
Open ports Decoherence

47. Readout port... lets noise in1
Fluctuating
environment
A -meter
detuning :
Readout + environment
DEPHASING
signal
(if A measured) :
dephasing and readout closely related !

48. ... and noise dephases... y = 2 1 + e ij

49. ... and depolarises Fermi Golden Rule to summarize: dephasing
dephasing
excitation
relaxation

50. connect only at readout time
Solving the noise/readout dilemma:
connect only at readout time
Operate qubit at a stationary point:
For readout:
-Move away adiabatically at:
better: stay there, and apply an ac drive
(to be shown later on)

51. A general strategy now applied to different Josephson qubits
A qubit ‘protected’ from decoherence:
the quantronium
gate
160 x160 nm
A general strategy now applied to
different Josephson qubits
Vion et al, Science 2002; Esteve&Vion, cond-mat 2005

52. State dependent persistent currents
The quantronium: 1) a split Cooper pair box
2 knobs : F U
1 d° of freedom
2 energies:
State dependent persistent currents i

53. 2) protected from dephasing
hn01
energy (kBK)
n01(GHz)
d/2p
d/2p Ng Ng
EJ=0.86 kBK
EC=0.68 kBK
Optima working points exist in many qubits

54. 1 3) with a readout junction discrimination Ng first readout of
persistent currents
with dc switching
1: switching
0: no switching 1
discrimination
3) with a readout junction
d/2p Ng

55. c Rabi precession Rabi oscillations nµw readout rotation wRabi = aURF
switching
probability
(%)
nµw c X Y
Effective
field
Note: visibility : <40%

56. Switching readout fidelity ?
40% contrast (only)

57. partly explains readout fidelity
Relaxation during readout ramp !
T1=60ns
T1=730ns
partly explains readout fidelity

58. Optimal point for other qubits?
the flux qubit
Nakamura, Ciorescu,
Bertet, Mooij et al.
Delft,
shift in practice
operating point readout point
coupling to SQUID

59. Optimal point for other qubits? the flux qubit
Chiorescu et al, TU Delft 2003
Nakamura et al., NEC 2005
Ib=0
Ib=-0.2
Ib=-0.4
2005: T2=400 ns

60. projection fidelity ? Fluctuating environment 1
ideal
Quantum
measurement:
Readout: 1
Readout: 0
errors: wrong answer & projection error 1
Fluctuating
environment
A -meter

61. Switching readout fidelity ?
40% contrast (only)

62. partly explains readout fidelity
Relaxation during readout ramp !
T1=60ns
T1=730ns
partly explains readout fidelity

63. Switching readout resets the qubitdc
readout V d U
resets
the qubit
dc pulse
 switching
simple, but:
rep rate limited by quasiparticles
fidelity <1 due to relaxation
qubit reset : NOT QND

64. Towards QND readout ‘at’ optimal point
flux qubit :
charge qubit :
charge-phase qubit :
Quantum
capacitance
C/CJ j0 j1 1
SQUID inductance
readout junction inductance
quantum capacitance
TU Delft
Yale, Saclay
Chalmers, Helsinki

65. dc versus ac readout in the quantronium
PULSE IN
PULSE OUT
rf readout
(M. Devoret, Yale) d U U d
dc pulse
“RF” pulse
 switching
 d dynamics in anharmonic potential
simple, but:
-fidelity 40%
qubit reset : NOT QND
more complex, but:
-better fidelity ?
-no reset: possibly QND

66. Towards non destructive readout at optimal point with an AC drive
M. Devoret team at Yale
I. Siddiqi et al., (2004)
µW Pulse IN
QuBit
control d j0 j1
OUT 1 UJ
optimal P
Similar dispersive methods developed for other qubits

67. State dependent bifurcation
The Josephson Bifurcation Amplifier
M. Devoret team at Yale
I. Siddiqi et al., (2004)
µW Pulse IN
QuBit
control d j0 j1
OUT
180°
-180°
d amplitude
µW drive amplitude
µW phase j
State dependent bifurcation 1 UJ
optimal P
Enhanced

68. Quantronium + JBA SETUP
50 W
TN=2.5K
G=40dB I LO
demodulator
1.3-2
GHz M S
-20dB
-30dB
NO bifurcation
300 K
4 K
0.6 K
Quantronium from Yale
30 mK
bifurcation

69. Rabi oscillations with the JBAp
45-50%
100ns
125ns
JBA pulse
Contrast : 50%
(Saclay exprt)

70. Quantum Non Demolition ? read twice
gate
JBA readout
partially QND 1
34%
100%
66% 0%
25% 9%
30%
36%
17%
83%
Notice: relaxation again
partly avoidable
by tuning the qubit p
100ns
125ns
5ns
20ns
10ns
40ns
100ns A B 1 p
& correlations 1
Note: results for flux-qubit now available

71. Dispersive readout of the flux qubit
A. Lupascu et al.
TU DELFT

72. Activation rates for different detuning values
F = 775 MHz
Fres=822 MHz
Iac,bifurcation2
slope=udyn/(kT)
Thy: M. Dykman

73. Activation rates for different detuning values

74. Optimal qubit manipulation and readout
87%

75. Rabi oscillations with optimal settings
Dt = length of MW pulse

76. Ramsey oscillations with optimal settings
Rabi oscillation
Ramsey: wge-wmw= 69 MHz
Ramsey frequency vs detuning
Relatively strong low frequency fluctuations visible in the drift of the Ramsey frequency.
QND data : analysis in progress

77. Experiments in Cavity QED with Superconducting Circuits
Rob Schoelkopf
Depts. of Applied Physics & Physics
Yale University
expt.
Andreas Wallraff
David Schuster
Luigi Frunzio
theory
Steve Girvin
Alexandre Blais
Ren-Shou Huang
And discussions w/ J. Zmuidzinas & M. Devoret
Merci to D. Esteve & co. for assistance!
Packard
Foundation
Keck Foundation
Funding:

78. A Circuit Analog for Cavity QED
2g = vacuum Rabi freq.
k = cavity decay rate
g = “transverse” decay rate
out
L = l ~ 2.5 cm
transmission line “cavity”
10 mm
Cooper-pair box “atom”
10 GHz in
Blais, Huang, Wallraff, Girvin & RS, cond-mat/ ; to appear in PRA

79. Cavity QED with a Cooper pair box: first dispersive readout
R. Schoelkopf, A. Wallraff, S. Girvin
et al., Yale (2004)
Dispersive readout with
out of resonance photons

80. Dressed Artificial Atom: Resonant Case2g T 1
“vacuum Rabi splitting”

81. Rabi Oscillations of Qubit
Prf = +6 dB
Prf = 0 dB
Prf = 18 dB

82. Coherence time measurements with 2 pulse Ramsey sequence

83. Solid state qubits at work:
CONCLUSION
Solid state qubits at work:
Semiconductor qubits recently demonstrated
Superconducting qubits:
qubit control
single-shot readout
Decoherence, QND readout, coupling
in next lectures

84. 2004 Fluctuating environment 1 A -meter The work on the Quantronium
Quantronium
Fluctuating
environment dc
gate dc
gate µw qp
trap
box
A -meter
readout
junction
1µm
2004
G. ITHIER
E. COLLIN
N. BOULANT
D. VION
P. ORFILA
P. SENAT
P. JOYEZ
P. MEESON
D. ESTEVE
SPEC
A. SHNIRMAN
G. SCHOEN
Y. MAKHLIN
F. CHIARELLO

85. Thanks to
NEC