1. NTT Basic Research Laboratories, NTT Corporation, Japan
Superconducting Flux Qubit as a Macroscopic Artificial Atom
Hideaki Takayanagi
NTT Basic Research Laboratories, NTT Corporation, Japan
NTT物性科学基礎研究所
髙 柳 英 明
Outline
Quantum Information Research at NTT
Fux Qubit
Single-Shot Measurement
Multi-Photon Absorption
Rabi Oscillation
Conclusions

2. QIT Project in NTT Basic Research Laboratories
Head: H. Takayanagi
About 20 researchers participate to the project
which consists of five sub-projects.
Four qubit-research projects and
a quantum cryptography one.

3. Solid-State Qubits Four Kinds of Qubit Coupled QDs
(artificial molecule)
Exciton in QDs
SQUID
Single-shot measurement
Multi-photon absorption
Rabi oscillation
Rabi oscillation
Quantum gate operation
cooled atom
Atom Chip

4. Quantum cryptography with a single photon
電気光学
変調器
Amp
Gene-
rator
時間間隔
解析器
Alice
Bob
Helium
Cryostat
Quantum
dot
lens
Pin-hole
Lens
Single-mode
fiber
Grating
Space filter
Beam
Splitter
Half-wavelength
¼ wavelength
50%-50%
Detector 1
Detector 2
Detector 3
Detector 4
waveguide
Counter
Photon 0
Mirror
Attenuator
Titanium-Sapphire Laser
Testing
Nature, 420 (2002) 762

5. Josephson persistent current Qubit
J. E. Mooij et al.,Science 285, 1036 (1999).
aEJ
, g3
Phase difference
Fqubit = f F0 B +
2 f p = g 1 2 3 q
f = Fqubit / F0 EJ
, g1 EJ
, g2
Josephson Energy :
cos
( 2 a - + = EJ U
) )
cos ( g 1 2
2 f p j p
=0.6 U
 =0.8
 =1.0 j p m j m
f = Fqubit / F0 = 0.5 g 2 ) ( 2 1 j = p + g m - g 1

6. Schematic qubit energy spectrum15
100 10 5 D
Energy (GHz)
Energy (GHz) -5
-100
-10
0.4
0.5
0.6 F / F
0.49
0.50
0.51
qubit
Fqubit / F0 80 .
eV, 44 30
1055 = a m C J E

7. Three-Josephson-junction Loop: Description
EJ ; C
ext= f 0 3 1 2
0< <1.0
Josephson Energy (1 junction):
Coupling energy (1 junction):
EC = e2/ 2C
Flux quantization:
Josephson Energy:
Our system consists in 3 JJ in series. 2 J are similar, the third one is smaller. It differs from a factor alpha, which should be between 0.5 and 1. Thus if EJ is the NRG of the 2 similar J, the NRG of the last one is alpha EJ. A magnetic flux =f0 penetrates the loop. f is called frustration.
The arrows define the direction of the phases.
Flux quantization gives:
Using this equation and the NRG for one J, we find the JE of the loop.
This energy is periodic in 1 and 2 with period 2.
J.E. Mooij､et al (1999)

8. Three-Josephson-junction Loop: Energy Diagram
f=0.5
2 minima in each unit cell.
We introduce two new phase variables m and p defined by these equations.
This graph is the NRG diagram of the qubit as a function of m and p.
We note that there are 2 minima in each unit cell.
Top View

9. Three-Josephson-junction Loop:  Dependence of the Potential
=0.6
 =0.8
 =1.0
Now let’s see the dependence of the potential on the parameters alpha and f.
We note that When  increases, the barrier height between the two minima in one unit cell increases while the barrier height between a minima of one cell and a minima of the next cell decreases. Note if alpha is less that 0.5, there is only one minima in each cell (for faille m and faille p equal 0 modulo 2 paille)
If  increases, the barrier height :
increases between the two minima of one unit cell
decreases between the minima of adjacent cells

10. Three-Josephson-junction Loop: Flux Dependence of the States
Quantum ground state |0>
Classical states
Quantum first excited state |1>
<Iq>/Ip
E0 (1) E
Level splitting
/0
Classical states = persistent currents of opposite sign. Degenerated at f = 0.5
Quantum tunnelling “anti-crossing”
Symmetric and antisymmetric superposition of the macroscopic persistent currents
Let’s see the consequences on the states.
When the applied flux  is close to 0.50, the Josephson energy becomes a double potential well.
The classical states in each well correspond to persistent currents of opposite sign. Classically they are degenerated at half quantum flux.
These two classical states are coupled via quantum tunnelling through the barrier. quantum tunnelling leads to an “anti-crossing” with symmetric and antisymmetric superposition of the macroscopic persistent currents.
This figure shows the energy of the classical states (in dashed lines) and of the ground state and the first excited state as a function of the applied flux faille. We can note the level splitting due to quantum tunnelling.
The bottom plot shows the quantum mechanical expectation value of the persistent current in the loop, for the ground state and the first excited state, plotted in unit of IP.
This insert shows the potential when f is smaller than 0.5, equal to 0.5 and greater than 0.5. The horizontal axis is the Josephson phase coordinate, faille m. This is a schematic of the position of the energy levels.

11. Mutual inductance M ~ 7 pH
Sample Fabrication
e-beam lithography
Shadow evaporation
Lift-off
Josephson junctions
Al / Al2O3 / Al
Junction area
SQUID : 0.1 x 0.08 m2
Qubit : 0.1 x L m2,
( a = 0.8 ) L = 2 ~ 0.2
Loop size
SQUID ~ 7 x 7 m2
Qubit ~ 5 x 5 m2
Mutual inductance M ~ 7 pH
Qubit and a detector dc-SQUID
NTT Atsugi

12. suspended-bridge & shadow evapolation
e-beam lithography
suspended-bridge & shadow evapolation

13. Sample and Cavity Cavity To mixing chamber Microwave line Thermometer
DC measurement
To mixing chamber
Microwave line
Thermometer
Vm line
Ibias line
NTT Atsugi
Samples
A loop
Cavity

14. 1 2 3 4 5 Twisted Constantan wire 100  Sample box
RF line
DC measurement 1 2 3 4 5
10 nF
HP 20dB
connectors
R.T.
2.4mm connectors
4.2K
Through capacitor
Flexible coaxial cable
HP 10dB
1.2K
attenuator
0.8K
resistance
0.4K
10mK
Heat anchor
for outer shield
Twisted
Constantan wire
100 
No on-chip capacitor and resistor
No on-chip control line
Change twisted wires to thin coaxial cables to introduce dc-pulse
200 
200 
200 
Sample
box
Loop antenna
~ 1mm above the sample

15. Readout through a dc-SQUID
DC measurement
Readout through a dc-SQUID Vm
I b
qubit
Record each switching
when Vm = Vth~ 30 mV
as a function of Isw
Sweep Ib ( 140 Hz )
Tilt SQUID potential
I b
~ 100 nA
Isw
Isw
Isw
4~6 nA
Isw t
70 ~ 100 μsec Vm
~ 7 ms
Vth(~30μV) t

16. Readout with a dc-SQUID
DC measurement
Readout with a dc-SQUID
Current is swept
I(V) curve
Isw(/ 0) curve
Magnetic field is swept
The current flowing in the SQUID is swept upwards. We measure the voltage to obtain the I-V characteristic of the SQUID. This graph is an example of I(V) curve. From this curve we deduce the value of the switching current.
The applied magnetic field is also swept and thus we can plot the switching current of the SQUID as a function of the magnetic field or as a function of the filling. Several I-V curves are measured for each value of the magnetic field.
This graph is an example of such a curve. The unit of the horizontal axis is the filling of the qubit. Data are averaged.
Theoretical calculation leads to this equation. The switching current is a periodic function of the magnetic flux. Its period is equal to the flux quantum 0 . The results we obtain are in agreement with this equation apart from some rounding at the bottom.
Furthermore, we observe additional steps when the filling fo the qubit is equal to 0.5.

17. Qubit step in the SQUID Isw
DC measurement
Qubit step in the SQUID Isw
Qubit switches its current sign
Flux in SQUID changes through M
SQUID Isw changes
Step on the
Isw(/ 0) M
dc-SQUID
Qubit Φ
Why do we observe these steps ?
At half quantum flux, the qubit switches its current sign, for example from clockwise to anticlockwise. Thus the flux in the SQUID changes by mutual inductance, leading to a change of the switching current. As a consequence, a step appears on the curve. This graph shows an example of step.
Fqubit / F0

18. Parameter dependence of the qubit step
( D, Ej, Ec )
SQUID I
Lqubit
LSQUID
Qubit
Josephson junctions : Al / Al2O3 / Al
Junction area : SQUID 0.2 x 0.2 m2
qubit 0.2 x L m2,
L=0.3, 0.5, 1.0
Now I present you the measurements we have performed.
a SQUID is used to measure the flux of the qubit. It is fabricated around the qubit loop, on the same layer.
This SEM picture also shows the system qubit ( the inner loop) and Squid (the outer loop).
The superconductor we use is Al. The insulator is Al oxide.
The geometrical parameters are indicated here : square micrometer….
Loop size : Lqubit = 5.1, 9.7, 19.0
(m) LSQUID = 6.3, 10.9, 20.2

19. Number of tunneled pair n
Two energy scale Ec, EJ
energy
energy
Pair tunneling
superconductor -p p
superconductor
Number of tunneled pair n
Phase difference
Tunnel barrier
H = Ec - EJ cos g - Iex g [n,g]=i
Josephson energy : EJ
charging energy : Ec =(2ne)2/(2CJ )
kBT << EJ << Ec < D →　charge qubit
　　　　　 kBT << Ec << EJ < D →　phase、flux qubit

20. D kBT~25mK Qubit energy splitting QB# 6 Junction area = 0.2 m2 QB# 5
Loop size : Lqubit = 9.7 m LSQUID = 10.9 m
QB# 5
Junction area = 0.1 m2
Loop size : Lqubit = 9.7 m LSQUID = 10.9 m
QB# 4
Junction area = 0.06 m2
Loop size : Lqubit = 9.7 m LSQUID = 10.9 m
( D ~ 2MHz << kBT )
( D ~ 0.4GHz ~ kBT )
( D ~ 2GHz > kBT )
Fqubit / F0
Fqubit / F0
Fqubit / F0
QB# 3
Junction area = 0.2 m2
Loop size : Lqubit = 5.1 m LSQUID = 6.3 m
QB# 8
Junction area = 0.1 m2
Loop size : Lqubit = 19.0 m LSQUID = 20.2 m
QB# 7
Junction area = 0.06 m2
Loop size : Lqubit = 19.0 m LSQUID = 20.2 m
Fqubit / F0
Fqubit / F0
Fqubit / F0 D
kBT~25mK
Qubit energy splitting

21. Calculated qubit energy level
D=0.4 GHz
D=2 GHz
D=2 MHz
Ej=544 GHz
Ec=1.6 GHz
Ej/Ec=338
Ej=280 GHz
Ec=3.2 GHz
Ej/Ec=87
Ej=130 GHz
Ec=5.4 GHz
Ej/Ec=24

22. Optimal operation point for SQUID
Qubit signals appear
at half-integer
Sensitivity of dc-SQUID
depends on magnetic fields
We can achieve excellent
resolution at f = 1.5
↓
↑

23. Spectroscopy D = 2.6 GHz 0.001 F0 M 2.4 GHz after averaging
EJ = 312 GHz, EC = 3.8, a = 0.7
D = 2.6 GHz
after averaging
w/o averaging
0.001 F0 M 2.4 GHz

24. Qubit signals at different SQUID modulation
DC measurement
Qubit signals at different SQUID modulation
S/N depends on SQUID Isw
design
qubit and SQUID
to be crossed
at small Isw
|>
|>
|>
|>
T = 25 mK

25. f= <Iq>/Ip E0 (1) E /0 Level splitting Classical states
Quantum ground state |0>
Classical states
Quantum first excited state |1>
<Iq>/Ip
E0 (1) E
Level splitting
/0 f=

26. Boltzman Distribution
Quantum ground state |0>
Classical states
Quantum first excited state |1>
<Iq>/Ip
E0 (1) E
Level splitting
/0

27. Schematic qubit energy spectrum15
100 10 5 D
Energy (GHz)
Energy (GHz) -5
-100
-10
0.4
0.5
0.6 F / F
0.49
0.50
0.51
qubit
Fqubit / F0 80 .
eV, 44 30
1055 = a m C J E

28. Spectroscopy D excited state ground state DC measurement
Pulse measurement D
excited state
ground state

29. Readout without averaging
DC measurement
Single shot measurement
into { l0>, l1> } bases
The <Iq> step shape does not change.
Only the population changes.
Fqubit / F0

30. Close-up of Isw, T=25 mK Fqubit / F0 Histogram is well separated !
DC measurement
Histogram is well separated !
counts
counts
Fqubit / F0 f
f =
0.001 F0 M 2.4 GHz

31. ( canonical ensemble average )
Readout after averaging
DC measurement
Expected Current
( canonical ensemble average )
Fqubit / F0

32. Experimental setup SLP-1.9 RF line R.T. 4.2K RFin : ２ attenuators
Pulse measurement
Experimental setup 1 2 3 4 5
SLP-1.9
RF line
R.T.
4.2K
RFin : ２ attenuators
RFout : terminator
+ attenuator
DC : LP filter + Meander filter
Flexible coaxial cable
HP 10dB
1.2K
0.8K
0.4K
I -
I +
V +
V -
29mK
Weinschell
10dB
Thin coaxial cable
f 0.33 mm
Meander
filters
Sample
cavity
RF in
RF in
Terminator
50 W
Sample cavity
On-chip strip line

33. Multi-photon transition between
superposition of macroscopic quantum states ｰ
( ) /√2　1st excited state ＋
( ) /√2　ground state 3 3 2 2 1 1 3 1 2 2 1 3

34. Multi-photon spectroscopy
Multi-photon transition
Multi-photon spectroscopy
SQUID readout -2 -1 1 2
d I SW
(nA)
1.504
1.502
1.500
1.498
1.496 F
qubit /
RF : 3.8 GHz
-10 dBm １ 3
D=0.86GHz
1-photon
2 -photon 2 1 -1 -2
d I SW
(nA)
1.504
1.502
1.500
1.498
1.496 F
qubit /
RF : 3.8 GHz
0 dBm １ 3

35. Multiphoton absorption at 9.1 GHz
RF Power dependence
triple
double
single
off
off
off
PRF =
- 21 dBm
0 dBm
9.6 dBm
13.2 dBm
10 dBm
12 dBm
12 dBm

36. Peak width vs MW intensity
Multi-photon transition
Peak width vs MW intensity
Bloch Kinetic Equation
9100MHz
----- (3)
(4)

37. Pulse measurement scheme
repetition: 3kHz ( ms)
180 ns
~1μs
SQUID switch
Ib DC pulse
Non-switching
resonant microwave
time MW
discrimination of the switching event
Vout +
Vout -
V th
I bias
Fext
Non-switching
events
Switching
events
Ibias +
Ibias -
SQUID
Non-switching event Switching event

38. Relaxation time T1 Probability [%] D Delay Time [ m s] data exp-fit T
Pulse measurement
030304_1 (1,2)FQB2
Relaxation time T1 15
9.1 GHz 1 ms pulse
1st excited sate 10 55 50 45 30 25
Probability
[%] 1 2
data MW 5 D
exp-fit
Energy (GHz) T
= 1.6 m s 1 -5 40 35
Ground state
-10
0.49
0.50
0.51
Fqubit / F0
500 ns
3 ms
Delay Time [ m s]
1 ms
delay time
Ib pulse height V, Trailing height ratio 0.6

39. Quantum Oscillation : Rabi oscillations
Pulse measurement
Quantum Oscillation : Rabi oscillations
11.4 GHz
150 ns
Dephasing time ~ 30 ns
600 s
pulse width ( ns )
switching probability ( % )
Resonant MW
pulse width
Trailing height ratio 0.7
MW amplitude (a.u.)
Rabi frequency ( MHz )
NTT Atsugi

40. Summary Future plan Spectroscopy of MQ artificial 2-level system
Qubit readout without averaging (DC)
Multi-photon transition between superposed MQ states
Coherent quantum oscillation ( Rabi oscillation )
T1 ~1.6 ms, T2 Rabi ~ 30 ns
Future plan
Ramsey, Spin echo
Two qubit fabrication and operation
MQC with single shot resolution

41. collaborators NTT Basic Research Laboratories Hirotaka Tanaka
                                                     
collaborators
NTT Basic Research Laboratories
Hirotaka Tanaka
Shiro Saito
Hayato Nakano
Frank Deppe
Takayoshi Meno
Kouich Semba
Tokyo Institute of Technology
Masahito Ueda
Yokohama National University
Yoshihiro Shimazu
Tomoo Yokoyama
Tokyo Science University
Takuya Mouri
Tatsuya Kutsuzawa

42. エネルギー固有状態をone-shot measurementで見た。
の時、
を測っている。
を測っているのではない。
これを測ると､
0.5
と のsuperpositionは、生きている。
と の間のsuperpositionは死んでいる。

43. 0.5
time domain で真ん中に出る理由
Qubitの磁場の量子力学的平均値を取っているから
Qubitの磁場はszのはず（projection)。
を使って､time-dependentなSchrödinger
方程式を解き､SQUIDのswitching current
を求めると、EJ/ECが小さくなると、ピークは
１つ、反対にEJ/ECが大きくなると、ピークは
２つになる。
ピーク１つ
ピーク２つ
0.5