1. Superconducting qubits
Overview of Solid state qubits
Superconducting qubits
Detailed example: The Cooper pair box
Manipulation methods
Read-out methods
Relaxation and dephasing
Single qubits: Experimental status

2. 1. Overview of solid state qubits

3. Solid State Qubits

4. Energy scales Atoms and ions Solid state systems NMR-systems
Single system
Identical systems possible
∆E=> ~500 THz
optical frequencies
High temperature
Hard to scale
Solid state systems
Single system
System taylored
∆E=> ~10 GHz
microwave frequencies
Low temperature 20mK
Potentially scalable
NMR-systems
Ensemble system
Identical systems possible
∆E=> ~100 MHz
radio frequencies
High temperature
Impossible to scale ?

5. Things to keep in mind
These is a macroscopic system, they typically contains 109 atoms
Thus coherence is hard to keep, relatively short decoherence times
Nanolithography makes the system (relatively) easy to scale
Experimental research on solid state qubits started late compared to other types of qubits

6. Requirements for read-out systems
Two different strategies: Single quantum systems quantum limited detectors Ensembles of systems normal detectors
Photons Photon detectors (hard for IR and mw)
Magnetic flux SQUIDs
Charge Single Electron Transistors
Single spin (Convert to charge)

7. 2. Superconducting qubits

8. Advantages and drawbacks of superconducting systems
Energy gap Protects against low energy exitations
Good detectors SQUIDs, SETs
Nano lithography Relatively easy scaling
Drawbacks
”Large systems” Relatively short decoherence times
Cooling The low energies require cooling to <<1K

9. Superconducting qubit characteristics
Which degree of freedom charge, flux, phase
True 2-level system quasi 2-level system
Representation of one qubit single system
Manipulation Microwave pulses, rectangular pulses
Level splitting ~10GHz
Type of read-out Squid, SET, dispersive
Back action of read out Single shot possible
Operation time 100 ps to 1ns
Decoherence time 4 µs obtained
Scalability potentially good
Coupling between qubits static, tunable, via cavity

10. Carge versus phase/flux
Fluxoid Quantisation in a superconducting ring
Charge Quantisation on a metalic island
(Josephson)
junction
Josephson
junction
Flux Charge
Inductance Capacitance
Josephson coupling energy EJ Charging energy EC
Current Voltage
Conductance Resistance

11. Superconducting qubits
charge qubit
EJ ~ 0.1 EC
Quantronium
EJ ~ EC
flux qubit
EJ ~ 30 EC
phase qubit
EJ ~ EC
NEC, Chalmers, Yale
Saclay, Yale
Delft, NTT
NIST/Santa Barbara
Q and j are conjugate variables which obey the commutation relation:
Q well defined
 well defined

12. The NEC qubit I1〉 I0〉 superposition state
0.5 1
1.5 -1
Probe
Tunnel
junction
SQUID
loop
Box
Gate
Single
Cooper-pair tunneling
Reservoir
I0〉
I1〉
superposition state
Y. Nakamura et al., Nature 398, 786 (1999).

13. The quantronium, charge-phase (Saclay)
T2≈0.5 µs
D. Vion et al., Science 296, 286 (2002)

14. Coupling a qubit to a resonator (Yale)
Jaynes-Cummings Hamiltonian
Vacuum Rabi splitting, A. Wallraff, Nature (2004)

15. Persistent-current qubit (Delft)
flux qubit with three junctions & small geometric loop inductance
0.5 
Icirc E
F/Fo 2D
+Ip
-Ip F
H = hsz + Dsx
with h=(F/Fo-0.5) FoIp
Ibias
Science 285, 1036 (1999)

16. The phase qubit (NIST) Large size ~100µm
McDermott et al. Science 307, 1299 (2005)

17. 3. Detailed example: The Cooper pair box

18. The Single Cooper-pair box (SCB)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
-0.2
E / E C n g E J
Charge states
degenerate
∆ > EC > EJ > T
2.5K ~1.5K ~0.5K mK
Likharev and Zorin LT17 (84)

19. The Single Cooper-pair box (SCB)
Which degree of freedom Charge
Representation of qubit single circuit
Levels multi but uses only the two lowest
Manipulation mw- or rectangular- pulses
Type of read-out Single Electron Transistor, Josephson junction,
dispersive
Back action of read out single shot possible, very low for dispersive RO
Operation time 100 ps
Decoherence time 1µs
Scalability Yes
Coupling capacitive, or via resonator

20. How does the SCB work as a Qubit
Analogy to a single spin in a magnetic field
Shnirman, Makhlin, Schön, PRL, RMP, Nature

21. Eigenstates and Eigenvalues
E/4EC
<n> ng
Eigen values for two state system
Eigen states
The Coulomb
staircase
Expectation value for n

22. Energies, and the optimal point
Energy levels
E/4EC
Optimal point
<n>
The Coulomb
staircase
Level splitting
ng=CgVg/2e
At the optimal point the system is insensitive (to first order) to fluctuations in the control parameters ng and 

23. Representation of the Qubit
The basic building block of a quantum computer is called a Qubit
Any two level system which acts quantum mechanically (having quantum coherence) can in principle be used as a Qubit.
The Bloch Sphere
Ground state
Exited state

24. 4. Manipulation methods

25. Manipulation with rectangular pulses
t<0 Starting at ng0
t=0 Go to ng0+∆ng
t=∆t Go back to ng0
The left sphere with two adjacent pure charge states at the north and south poles corresponds to a CPB with EJ /EC << 1, which is driven to the charge degeneracy point with a fast dc gate pulse.
Nakamura et al. Nature (99)

26. Microwave pulses (NMR-style)
The energy eigenstates at the poles described within the rotating wave approximation. The spin is represented by a thin arrow whereas fields are represented by bold arrows. The dotted lines show the spin trajectory, starting from the ground state.
NMR-like Control of a Quantum Bit Superconducting Circuit
E. Collin, et al. Phys. Rev. Lett., 93, 15 (2004).

27. Comparing the two methods
Microwave pulses
Slower, typical π pulse
Timing is easier, NMR techniques can be used
Smaller amplitude and monocromatc
More gentle on the environment
Works at the optimal point
Rectangular pulses
Faster, typical π pulse
More accurate timing required
Large amplitude and wide frequency content
Shakes up the environment
Can not stay at optimal point

28. 5. Read-out methods

29. Read-out with a probing junction (NEC)
Josephson-quasiparticle cycle
Fulton et al. PRL ’89
-2e -e
+ probe
Cooper-pair box
detect the state as current
initialize the system to
Repeted measurement gives a current

30. The NEC qubit I1〉 I0〉 superposition state
0.5 1
1.5 -1
Probe
Tunnel
junction
SQUID
loop
Box
Gate
Single
Cooper-pair tunneling
Reservoir
I0〉
I1〉
superposition state
Y. Nakamura et al., Nature 398, 786 (1999).

31. Read-out with SET sample and hold (NEC)
Manipulation
Trap quasi particles
Measure trap with SET
Single shot, but still fairly slow
Measurement circuit is electrostatically decoupled from the qubit
Final states are read out after termination coherent state manipulation

32. Switching SQUID readout scheme (Delft)
pulsed bias current
~ 5 ns rise/fall time
tmeas~5 ns, ttrail~500 ns
time 
quantum operations
reference trigger I V
Switching
each readout only two possible outputs pulse height adjusted for ~50% switching
SQUID switched to gap voltage probability measured with ~ 5000 readouts
SQUID still at V=0 2 . 3 4 5 6 7 1 8 9
switching probability (%) p u l s e h i g t @ A W n r a o ( V )
pulse height 
Room temp. output signal
time 
Vthr V

33. Switching Junction readout scheme (Saclay)

34. Read out with the RF-SET

35. A single Cooper-pair box qubit (Chalmers) integrated with an RF-SET Read-out system
A two level systen based on the charge states
|1> = One extra Cooper-pair in the box
|0> = No extra Cooper-pair in the box
Büttiker, PRB (86)
Bouchiat et al. Physica Scripta (99)
Nakamura et al., Nature (99)
Makhlin et al. Rev. Mod. Phys. (01)
Aassime, PD et al., PRL (01)
Vion et al. Nature (02)
∆ >> EC >> EJ(B) >> T
2.5K K K mK

36. The Single Electron Transistor

37. The Radio-Frequency Single Electron Transistor (RF-SET)
Very high speed: 137 MHz
R. Schoelkopf, et al. Science (98)
Charge sensitivity: ∂Q= 3.2 µe/√Hz
A. Aassime et al. APL 79, 4031 (2001)
Current Voltage
characteristics
Modulation of conductance
and reflection
Vdc
Vac

38. Performance of the RF-SET
Frequency domain:
∂Q=0.035erms, fg=2MHz
Time domain:
∆Q=0.2e, inset 0.05e
Results: Aassime et al.
Charge sensitivity: ∂Q= 3.2 µe/√Hz APL 79, 4031 (2001)
Energy sensitivity: ∂e = 4.8 h PRL 86, 3376 (2001)

39. Dispersive read-outs, parametric capacitance or inductance

40. Coupling a qubit to a resonator (Yale)
A. Wallraff, Nature (2004)

41. The Cooper-pair box as a Parametric Capacitance
We define an effective capacitance which contains two parts
At the degeneracypoint we get:
Büttiker, PRB (86)
Likharev Zorin, JLTP (85)
c.f. the parametric Josephson inductance

42. Quadrature measurements with the RF-SET
Disspative part, R
Reactive part, C (or L)
Cooper-pair transistor similar
to Cooper-pair box
High speed: 137 MHz
Schoelkopf, et al. Science (98)
Charge sensitivity: ∂Q= 3.2 µe/√Hz
Aassime et al. APL 79, 4031 (2001)

43. The Josephson Quasiparticel cycle: JQP
Phase response
DJQP
Drain
-3, -1
-1, 1
Source
0, 2 J1 J2
Cooper-pair
resonance
Quasi part
transition
One junction is
resonant part of the time
Quasi part
transition
Cooper-pair
resonance

44. The Double Josephson Quasiparticle cycle: DJQP
Drain jcn in
resonance
Source jcn in
Drain
-3, 1
Source
-1, 1
Drain
-1, 1
Source
0, 2
One junction is always resonant
but only one at the time
This results in an average CQ

45. Quantitative Comparison of the Quantum Capacitance
If the phase shift is small it can be approximated by:
Assuming equal capacitances we can calculate CQ for a two level system and compare with the data.
T=140 mK
From spectroscopy EJ/EC=0.12
Temperature adds to the FWHM

46. Spectroscopy on the quantum capacitance
By tuning only one junction into resonance we can excite the system to the exited state, which has a capacitance of the opposite sign
Junction 1
Junction 2
P= Probability to be in the exited state
EJ1=3.0 GHz
EJ2=2.8 GHz

47. Quantum Cap Qubit Cc Cin CT Cg Operate and readout at optimal point
No intrinsic dissipation
Tank circuit protects qubit by filtering environment
Lumped element version of Yale cavity experiment, Wallraff et al. Nature (04)

48. Measuring the charge on the box The Coulomb Staircase

49. The Coulomb Staircase Comparing the Normal and the Superconducting State
2e periodicity is achieved
if Ec is sufficiently small
<1K
Bouchiat et al. Physica Scripta (98)
Aumentado et al. PRL (04)
Gunnarsson et al. PRB (04)

50. The Coulomb staircase comparing the normal and the superconducting state
Small step occures due to quasi particle poisoning
Bouchiat et al. Physica Scripta (98)

51. What would you expect in the superconducting state
Using EC<1.0K pure 2e periodicity is obtained
Tuominen et al. PRL (93)
Lafarge et al. Nature (93)

52. Size of the odd step versus magnetic field
Lower Ec is better
∆ is suppressed by parallell magntic field
Faster suppression in the reservoir than in the box, due to film thickness
Reservoir 40 nm
Box 25 nm

53. Spectroscopy

54. Energy Levels of the Cooper-Pair Box1 2
0.5
1.5
2.5 3
3.5 4 Q
box
[e] E J
>E C
<E 1 2 3 4
E [Ec]
Gate Charge Q
[e]
Box Charge N
box
| 0 >
| 1 > n 01
n01

55. Spectroscopy of the Cooper-Pair Box1 2
0.5
1.5
2.5 3
3.5 4 Q
box
[e] E J
>E C
<E
Level splitting ∆E(ng)
B-field dependence of EJ
0.2
0.4
0.6
0.8 15 20 25 30 35 40
ng [e]
fHF [GHz]
data E C
=42.0GHz, E J
=20.2GHz

56. Relaxation and dephasing

57. Dephasing and mixing T1 T2
The qubit can be disturbed in two different ways.
Relaxation or mixing
The environment can exchange energy with the qubit, mixing the two states by stimulated emission or absorption. This has the characteristic time T1
Describes the diagonal elements in the density matrix
Fluctuations at resonance, S(w01)
Important during read-out
Dephasing
The environment can create loss of phase memory by smearing the energy levels, thus changing the phase velocity. This process requires no energy exchange, and it has the characteristic time T
Describes the decay of the off-diagonal elements in the density matrix
Fluctuations at low frequencies, S(0)
Important during “computation” T1 T2

58. Decoherence sources
What are the major decoherence sources in your system?
Active sources (stimulated) absorption and emision heat, noise,….
Passive sources (spontaneous) emission only quantum fluctuations
external degrees of freedom
photons, phonons, quasiparticles...
Can they be controlled ?
Cooling, shielding, filtering, tailoring the environment
Qubit
50Ω
Manipulation
Read-out
Material dependent
Microscopic fluctuators

59. The spin boson model
A single two level system (weakly) coupled to an environment described as a bath of harmonic oscillators.
The effect of the harmonic oscillators can be described as a fluctuating gate voltage, or a fluctuating ng

60. Decoherence rates
The slow longitudinal fluctuations leads to dephasing => 
The transversal fluctuations which are resonant with the levelsplitting causes mixing (relaxation) => 
The rates are directly proportional to the spectral densities
Negative frequencies
exitation
Positive frequencies
relaxation

61. Examples for spectral densities
1/f like charge fluctuations
Log S(f)
Shot noise from SET read-out
Environmental circuit impedance 50Ω
Log f [Hz]
If these are the only contributions
1/f noise will be important for dephasing
and environment will be important for relaxation (if SET is switched off)

62. Measurements of T1 and T2

63. Manipulation with dc-pulses
T ≈100 ns r
∆t≈100ps
trise≈30ps
The probability to find the qubit in the exited state oscillates as a function of ∆t.
The charge is measured continuously by the RF-SET
Difference between these two curves = excess charge ∆Qbox
c.f. Nakamura et al. (99)
t<0 Starting at ng0
t=0 Go to ng0+∆ng
t=∆t Go back to ng0

64. Oscillations at the charge degeneracy
Oscillation frequency = EJ/h
agrees well with EJ from spectroscopy.
Good news:
We observe oscillations in 6 samples
A high fidelity! >70%
Deviation from 1.0 e due to finite risetime (~30ps) of pulses, i.e. no missing amplitude 16 12 8 4
Spectroscopy
Coherent oscillation
perpendicular B-field
EJ [GHz]
Bad news:
T2 only ~10 ns

65. Flux qubit : Rabi oscillations
time
trigger
Ib pulse
read-out
operation
AMW
tMW
Chuorescu, Bertet . 2 4 6 8 1 3 5
Rabi frequency (MHz) R F a m p l i t u d e , A / ( )

66. Rabi and Ramsey in the Quantronium (Saclay)
Ramsey fringes
Vion, Esteve, et al. Science 2002

67. Extracting T2 from free precession oscillations
Note: relatively large visibility, here ~60%
Oscillation period agrees well with level splitting
T2 decreases rapidly as the gate charge is detuned from the degeneracy point.

68. Measurements of T2 vs. Gate Charge ng
0.2
0.4
0.6
0.8 1
1.2 10 8 9
gate charge n g
[e]
EJ/h=3.6GHz
EJ/h=9.4GHz
T2-1
Double pulse data agrees with single pulse data
Q0 dependence  coupling to charge
T2 limited by relaxation at the degeneracy point
T2 limited by 1/f-niose away from the degeneracy point
Very similar to data from NEC and JPL
T. Duty et al., J. Low Temp. Phys. (04)

69. Determining a T1 that is smaller than Tmeas
The average charge <Qbox> depends both on T1 and TR t
Qbox(t) 1 2
500
1000
0.2
0.4
0.6
0.8 1 T R
[ns]
=72ns
=87ns
<Qbox> [e]
n0 depends on the pulse rise time

70. T1 Measurements vs ng and EJ provide info on S(w) and form of coupling10 20 30 40 50 60
0.2
0.4
0.6
0.8 1
gate charge ng [e] T
[ns] E J
5GHz
8GHz
9GHz
The dependence
indicates that the qubit is
coupling to charge.
We can extrapolate the measurement to the degeneracy point and compare with T2 measurements

71. Measured samples K. Bladh, T. Duty, D. Gunnarsson, P. Delsing
New Journal of Physics, 7, 180 (2005)
Focussed issue on: Solid State Quantum Information

72. Summarizing T1 and T2 T2 away from degeneracy point
T2 seems to be limited by charge noise whan the qubit is tuned away from Degeneracy point. The extracted value for the 1/f noise is almost an order of magnitude worse than standard values for SETs
T2 at degeneracy point
At degeneract T2 seems to be limited by relaxation, Best value 10ns. T1
T1 seems to be due to charge noise. Possibly due to the back ground charges

73. Back-action

74. The needed measurement time
The mixing time depends on the shot noise in the SET
Signal to noise ratio (SNR)

75. Spectral density of the voltage fluctuations of the SET island for our best SET
Assuming readout at ∆E=2.4K
G. Johansson et al.PRL 2001

76. Summarizing results for two samples
Assume k=1% , EJ=0.1K, ∆EAl=2.4K, ∆Enb=15K
and that the SET dominates the mixing
Sample 1 : I= 6.7 nA, ∂Q= 6.3µe/√Hz [A.Aassime et al. PRL 86, 3376 (2001)]
tm tmix SNR
Al-qubit 0.40 µs 8.6 µs 4.6
Nb-qubit 0.40 µs 1.9 ms 68
Sample 2: I= 8 nA, ∂Q= 3.2µe/√Hz [A.Aassime et al. APL 79, 4031 (2001)]
Al-qubit 0.10 µs 6.4 µs 8.0
Nb-qubit µs ms 114
Summarized in: K. Bladh et al. Physica Scripta T102, 167 (2002)

77. We find T1 short and independent of SET bias in 6 different samples.
150
100
T1 [ns]
DJQP bias
JQP bias 50
100
200
300
400
500
600
700
800
900 I
[pA]
bias

78. Summary Macroscopic systems that allow tailoring and scaling.
Energy gap protects against low energy excitations.
Different flavors depending on EJ/EC ratio
Optimal point important to avoid decoherence
T2/Top ≈ 1 µs / 1 ns = 1000
T1 and T2 can be estimated from spectral densities
Dispersive read-out schemes promising -> QND
Coupling to cavities allow coupling and cavity QED on-chip