1. SPEC, CEA Saclay (France),
Josephson qubits
P. Bertet
SPEC, CEA Saclay (France),
Quantronics group

2. Outline Lecture 1: Basics of superconducting qubits
Lecture 2: Qubit readout and circuit quantum electrodynamics
Readout by a linear resonator
Nonlinear resonators for high-fidelity readout
Resonant qubit-resonator coupling:
quantum state engineering and tomography
Lecture 3: 2-qubit gates and quantum processor architectures

3. Fabrication techniques
small junctions e-beam lithography
1) e-beam patterning
2) development
3) first evaporation
4) oxidation
5) second evap.
6) lift-off
7) electrical test O2
Al/Al2O3/Al junctions
PMMA
PMMA-MAA
SiO2
small junctions Multi angle shadow evaporation
I.3) Decoherence

4. QUANTRONIUM (Saclay group)
gate
160 x160 nm
I.3) Decoherence

5. FLUX-QUBIT (Delft group)
I.3) Decoherence

6. TRANSMON QUBIT (Saclay group)
40mm
2mm
I.3) Decoherence

7. The ideal qubit readout
relax.
p=|b|2
p=|a|2 a b +
tmeas << T1 1 ? 1 1 1 1 1
a|0>+b|1> a b + ? 1
Hi-Fi
Quantum Non Demolishing
(QND)
BUT ….HOW ???
SURPRISING DIFFICULT AND INTERESTING QUESTION FOR SUPERCONDUCTING QUBITS
II.1) Linear resonator

8. The readout problem 1) Readout should be FAST : for high fidelity ( )
Ideally,
2) Readout should be NON-INVASIVE
Unwanted transition caused by readout process errors
(but full dephasing can’t be avoided !!!)
3) Readout should be COMPLETELY OFF during quantum state preparation
(avoid backaction)
II.1) Linear resonator

9. Readout by a linear resonator
1D CPW resonator
Superconducting
artificial atom
R. Schoelkopf, 2004
A. Blais et al., Phys. Rev. A 69, (2004)
A. Walraff et al., Nature 431, 162 (2004)
I. Chiorescu et al., Nature 431, 159 (2004)
Modern readout methods by coupling to a resonator
(CIRCUIT QUANTUM ELECTRODYNAMICS)
II.1) Linear resonator

10. Physical realization L=3.2cm, fn=n 1.8GHz 3mm Coupling capacitor Cc
Typical lateral dimensions :
10mm
- 1-dimensional mode
- Very confined :
- Large voltage quantum fluctuations
- Quality factor easily tuned by designing Cc
II.1) Linear resonator

11. CPB coupled to a CPW resonator
A. Blais et al., PRA 69, (2004) Cg
Vext
2-level approximation + Rotating Wave Approximation
Jaynes-Cummings
Hamiltonian
II.1) Linear resonator

12. Strong coupling regime with superconducting qubits
GEOMETRICAL dependence of g
Easily tuned by circuit design
Can be made very large !
(Typically : 0 – 200MHz)
Strong coupling condition naturally fulfilled
with superconducting circuits
(Q=100 enough for strong coupling !!)
II.1) Linear resonator

13. The Jaynes-Cummings model
couples only level doublets
|g,2>
|e,1>
|g,n+1> , |e,n>
|g,1>
|e,0>
Exact diagonalization possible
|g,0>
Restriction of HJ-C to
|g,n+1> , |e,n>
Note : |g,0> state is left unchanged by HJ-C with Eg,0=-d/2
II.1) Linear resonator

14. The Jaynes-Cummings model
couples only level doublets
|g,2>
|e,1>
|g,n+1> , |e,n>
|g,1>
|e,0>
Exact diagonalization possible
|g,0>
Coupled states
II.1) Linear resonator

15. The Jaynes-Cummings model
II.1) Linear resonator

16. The Jaynes-Cummings model
II.1) Linear resonator

17. Two interesting limits
RESONANT REGIME (d=0)
II.1) Linear resonator

18. DISPERSIVE REGIME (|d|>>g) DISPERSIVE REGIME (|d|>>g)
Two interesting limits
RESONANT REGIME (d=0)
DISPERSIVE REGIME (|d|>>g)
DISPERSIVE REGIME (|d|>>g)
II.1) Linear resonator

19. QUBIT STATE READOUT QUBIT STATE READOUT
Two interesting limits
QUANTUM STATE ENGINEERING
QUBIT STATE READOUT
QUBIT STATE READOUT
RESONANT REGIME (d=0)
DISPERSIVE REGIME (|d|>>g)
DISPERSIVE REGIME (|d|>>g)
II.1) Linear resonator

20. The Jaynes-Cummings model : dispersive interaction
2) Light shift of the qubit transition in the presence of n photons
Field in the resonator causes qubit frequency shift
and decoherence
1) Qubit state-dependent shift of the cavity frequency
Cavity can probe the qubit state non-destructively
with
the dispersive coupling constant
II.1) Linear resonator

21. Dispersive readout of a transmon: principle
|0> or |1> ??
II.1) Linear resonator

22. Dispersive readout of a transmon: principle
|0> or |1> ??
w=wc
II.1) Linear resonator

23. Dispersive readout of a transmon: principle
|0> or |1> ??
|a0>
w=wc
II.1) Linear resonator

24. Dispersive readout of a transmon: principle
|0> or |1> ??
|a0>
w=wc p
|0>
|1> -p
wd/wc
II.1) Linear resonator

25. Dispersive readout of a transmon: principle
|0> or |1> ??
|a0>
L.O
w=wc or
??? p
|0>
|1> -p
wd/wc
II.1) Linear resonator

26. Typical implementation (Saclay)
5 mm
(f0=6.5GHz)
80mm
Q=700
40mm
2mm
(optical+e-beam
lithography)
II.1) Linear resonator

27. Typical setup (Saclay)MW
meas
20dB
50MHz
Fast
Digitizer
A(t)‏ I
MW drive LO
(t)‏ Q
COIL Vc
G=56dB dB
300K
G=40dB
20dB
TN=2.5K
50 4K
DC-8
GHz
30dB
600mK
1.4-20
GHz
4-8
GHz
18mK
20dB
50
II.1) Linear resonator

28. Signature for strong coupling:
Observation of the vacuum Rabi splitting with electrical circuits
(courtesy of S. Girvin)
Signature for strong coupling:
Placing a single resonant atom inside the cavity
leads to splitting of transmission peak
2008
vacuum Rabi splitting
atom
off-resonance
observed in:
cavity QED R.J. Thompson et al., PRL 68, 1132 (1992)
I. Schuster et al. Nature Physics 4, (2008)
circuit QED
A. Wallraff et al., Nature 431, 162 (2004)
quantum dot systems
J.P. Reithmaier et al., Nature 432, 197 (2004)
T. Yoshie et al., Nature 432, 200 (2004)
on resonance
II.1) Linear resonator
A. Wallraff et al., Nature 431, 162 (2004)

29. Qubit spectroscopy with dispersive readout
MW drv
Pump TLS
MW meas
Probe resonator
phase  
/c
II.1) Linear resonator

30. Typical spectroscopy of a transmon + cavity circuit
II.1) Linear resonator

31. Rabi oscillations measured with dispersive readout
Variable-length
drive
MW drv
x 10000
 Ensemble averaging
Projective measurement
MW meas
T2R=316 ns Y X
II.1) Linear resonator

32. Dispersive readout : the signal-to-noise issue
|0> or |1> ??
|a0>
Ideal amplifier
L.O
w=wc or
??? p
|0>
|1> -p
wd/wc
II.1) Linear resonator

33. Dispersive readout : the signal-to-noise issue
|0> or |1> ??
Real amplifier
TN=5K
L.O
|a0>
w=wc or
??? No discrimination in 1 shot p
|0>
|1> -p
wd/wc
II.1) Linear resonator

34. Dispersive readout : the signal-to-noise issue
QUANTUM-
LIMITED AMPLIFIER ??
|a1>
|0> or |1> ??
|a0>
Real amplifier
TN=5K
L.O
w=wc or
in one single-shot ?? p
|0>
|1> -p
wd/wc
II.1) Linear resonator

35. How to build an amplifier with minimal noise ???
signal out
pump
signal in
Nonlinear resonator
l/4
l/4
Junction causes Kerr non-linearity
Resonator can behave as parametric amplifier
K. Lehnert group
M. Devoret group
I. Siddiqi group
II.2) Nonlinear resonator

36. A nonlinear resonator as quantum-limited amplifier
max
M. J. Hatridge, R. Vijay, D. H. Slichter, J. Clarke and I. Siddiqi,
Phys. Rev. B 83, (2011)
II.2) Nonlinear resonator
(courtesy I. Siddiqi)

37. A nonlinear resonator as quantum-limited amplifier
Small signal
Saturated
linecut.
SINE WAVES FLIPPED OR UN FLIPPED TO GO UP OR DOWN
-128 dB for 1 photon power - 1dB compression at -130 in other system, but here not a problem. want faithful representation of analog signal. animate to add qubit as digital (small signal digital). then run as saturated. binary signals? if not a two state signal, lose info in saturated regime. since qubit is binary, we don’t care
(courtesy I. Siddiqi)
II.2) Nonlinear resonator

38. Signal-to-noise enhancement by a paramp
M. Castellanos-Beltran, K. Lehnert, APL (2007)
(quantum limit on how good an amplifier can be : Caves theorem)
Actually reached in several experiments : quantum limited measurement
II.2) Nonlinear resonator

39. Qubit and amplifier at 30 mK
INPUT
DRIVE
OUTPUT
The experimental implementation of our circuit QED plus paramp architecture looks like this. Readout photons enter from the input and are directed through a circulator into the cavity, where they INTERACT with the qubit and acquire a phase shift that depends on the qubit state. The readout photons, now carrying info about the qubit state in their phase, then REFLECT out of the cavity and pass through several circulators, which isolate the qubit from the strong pump used to bias the parametric amplifier. The readout photons combine with this pump in a directional coupler and are amplified by the paramp. The amplified signal reflects out and is directed to the ouput port, where it is further amplified, mixed down to zero frequency, and digitized.
We have done this with a variety of qubit and paramp samples. The data I will show in this talk come from two different qubit samples (and two different paramps). Our first sample didn’t have great coherence properties, but the latest sample has shown a factor of 4 improvement in T1. We operate where the dispersive shift is comparable to the cavity linewidth, so that the qubit-dependent phase shift is substantial.
(courtesy I. Siddiqi)
II.2) Nonlinear resonator

40. Individual measurement traces
readout off
readout on
In our measurement protocol, we PREPARE the qubit state with a microwave pulse and then turn on the readout to measure the qubit. Let’s FOLLOW THREE different individual time traces to see what happens. Here the phase shift signal has been CONVERTED TO A VOLTAGE SIGNAL through HOMODYNE detection. The white region denotes that the readout is off, while the grey region shows when the readout is on. We prepare the qubit state here. Once we turn the readout on, the traces SEPARATE AND MOVE TO SOME NEW VALUE depending on the qubit state. In this case, the red and green traces were prepared in the excited state using a pi pulse, while the blue trace was prepared in the ground state with a 2 pi pulse. We can choose a DISCRIMINATION THRESHOLD and say that signals above are in the excited state, and signals below are in the ground state. Note that the our signal-to-noise ratio is good enough so that we can readily determine whether our signals is above or below this threshold. We know that the qubit has a finite lifetime, and so it will eventually decay back to the ground state. However, if we continue to measure the qubit, we force it to be in an eigenstate, either 0 or 1. This means as we monitor the qubit, it will decay to the ground state with a SUDDEN JUMP, and THIS IS in fact WHAT WE SEE. The jump is a STOCHASTIC process, so it happens at a different time on each trace. If we extract the jump times from many traces, we find they are exponentially distributed with a time constant that is equal to the independently measured T1 of the qubit, as we would expect. This is the first time such quantum jumps of a superconducting qubit have been observed.
R. Vijay, D.H. Slichter, and I. Siddiqi, PRL 106, (2011)
(courtesy I. Siddiqi)
II.2) Nonlinear resonator

41. Bivalued histograms Single-shot discrimination of qubit state
The dots in the middle that rise above the gaussian fits represent about 3% of the total number of traces; these are the ones in transition.
Single-shot discrimination of qubit state
(courtesy I. Siddiqi)
II.2) Nonlinear resonator

42. used as threshold detector
Other strategy : sample-and-hold detector integrated with qubit
l/4
pump
Nonlinear resonator
used as threshold detector
II.2) Nonlinear resonator

43. Other strategy : sample-and-hold detector integrated with qubit
Kerr-nonlinear resonator
l/4
l/4
pump L H
BISTABILITY FOR
Pd /Pc = 0.2
0.5
1.0
1.8 -
II.2) Nonlinear resonator

44. wd The Cavity Josephson Bifurcation Amplifier (CJBA) Pd H state
M. Devoret group, Yale
JBA: I. Siddiqi et al., PRL (2004)
CJBA: M. Metcalfe et al, PRB (2007)
MW drive : Pd(t) , wd
jin
Non linear resonator
jout Pd
Switching from L to H : BIFURCATION
Stochastic process governed by
thermal or quantum noise.
M.I. Dykman and M.A. Krivoglaz, JETP 77, 60 (1979)
M.I. Dykman and V.N. Smelyanskiy, JETP 67, 1769 (1988) H L Pd
H state
Bistable
region wd
L state wc
II.2) Nonlinear resonator 44

45. wd The Cavity Josephson Bifurcation Amplifier (CJBA) Pd H state
M. Devoret group, Yale
JBA: I. Siddiqi et al., PRL (2004)
CJBA: M. Metcalfe et al, PRB (2007)
MW drive : Pd(t) , wd
jin
Non linear resonator
jout H Pd Pd
H state
Bistable
region L wd
L state wc
II.2) Nonlinear resonator 45

46. SINGLE-SHOT QUBIT READOUT
Readout of transmon with CJBA
MW drive : Pd(t) , wd
jin
Non linear resonator
jout
qubit in |0> or |1> Pd Pd
H state
SINGLE-SHOT QUBIT READOUT
|1> wd
L state
wc|1>
wc|0>
|0>
II.2) Nonlinear resonator 46

47. Rabi oscillations visibility
h01
h12
tp,12 Dt
Pswitch (%)
TRabi=500ns
Dt (µs)
Mallet et al.,
Nature Physics (2009)
Single-shot 93% contrast Rabi oscillations
See also
A. Lupascu et al., Nature Phys. (2007)
II.2) Nonlinear resonator