Quantum Dynamics of a Superconducting Flux-Qubit

Quantum coherent oscillations in a
superconducting flux-qubit
Irinel Chiorescu
K. Harmans
J.E. Mooij
Yasunobu Nakamura
(NEC & TU Delft)
Delft University of Technology

Quantum Transport group
Prof. J.E. Mooij
Kees Harmans
Flux-qubit team
visitors
Yasunobu Nakamura (NEC Japan, 2001-2002)
Kouichi Semba (NTT Japan, 2002-2003)
PhD students
Alexander ter Haar
Adrian Lupacu
Jelle Plantenberg
postdocs
Patrice Bertet
Irinel Chiorescu
students
technical staff
collaborations
MIT, TU Delft (theory), U Munich, CEA-Saclay
acknowledgements
FOM (NL) , IST (EU) , ARO (US)

Outline
introduction
initialization, operation, readout
sample
results
conclusion
future plans

3 Josephson-junctions Quantum Bit
J.E. Mooij et al, Science, 285, 1036 (1999)
superconducting loop, with 3
Josephson junctions
2 are identical and the 3rd is
smaller (α=0.6 − 0.8)
Josephson Potential:
U=ΣE J
I
u = U/E J
u = 2 + α - cosγ 1 - cosγ 2 - αcos(γ 2 - γ 1 + 2πf)
φ 1 = (γ 1 - γ 2 )/2 , φ 2 = (γ 1 + γ 2 )/2
u = 2(1 - cosφ 1 cosφ 2 ) + 2αsin 2 (φ 1 - πf)

Josephson potential - phase space
2 wells separated by a barrier
for f=0.5, symmetric barrier
α=0.8, f=0.5
T in
T out
T out

Josephson potential - spin space
u = 2 + α - cosγ 1 - cosγ 2 - αcos(γ 2 - γ 1 + 2πf)
u = 2(1 - cosφ 1 cosφ 2 ) + 2αsin 2 (φ 1 - πf)
s x = cosφ 1 cosφ 2 , s y = cosφ 1 sinφ 2 , s z = sin φ 1
u = 2(1 ± s x ) + 2α(1 - s z2 ) +
2α[(1 - 2s z2 )cos 2 πf ± s z √1-s z2 sin(2πf)]

Flux Qubit – two level system
C. van der Wal et al, Science, 290, 773 (2000)
Exact diagonalisation: two
levels at the bottom of the
spectra
Two wells separated by a
barrier
0.5
Persistent currents of
opposite direction |↑ and |↓
SQUID critical current qubit
→ persistent current
Microwave induced
excitation → level structure
see also, J. Friedman et al, Nature, 406, 43 (2000)

Coherent oscillations
Magnetic resonance with a single, macroscopic quasi-spin
Bloch sphere
|Ψ>=α|↑>+β|↓>
Rabi oscillations
microwave excitation with
frequency ω and amplitude A
coherent rotations with Ω Rabi ∝ A
A
MW pulse
ω = ∆E
Ω Rabi ∝ A
|e>
|g>

Qubit operated at the magic point
|↑
Hamiltonian and eigenstates
|1
H = -ε/2 σ z – ∆/2 σ x
tan2θ = ∆/ ε
|0 = cosθ |↑ + sinθ |↓
|0
|1 = -sinθ |↑ + cosθ |↓ |↓
shift
|↓
|↑
Initialization, ε = 0
|Q = |0 = (|↑ +|↓)/√2
Operation , ε = 0
|Q = α|0 + β|1
Readout , ε > 0
|Q = α|0 + β|1
|Q
|1
|0
MW pulse ON
(rotating frame)
= |α| 2 - |β| 2
|Q
MW pulse OFF
(lab frame)

Switching event measurements
Device
qubit merged with the SQUID
strong coupling L
I pulse
~30ns rise/fall time
shift
t
Readout
bias current to switch the SQUID
ramping generates the shift
(preserving the qubit information)
switching current depends on
qubit state (spin up or down)
pulse height: I sw0 < I b < I sw1
see also the quantronium: Science, 296, 886 (2002)

Automatic shift of γ Q and switching
qubit merged with the SQUID
big junctions
strong coupling L
large circulating
currents
bias current generates
a shift in γ qubit
switching occurs far
from degeneracy

Sample
E J
/E C
= 34.65
E C
= 7.36 GHz
α = 0.8
∆ = 3.4 GHz
I p
= 330 nA
large junctions
I c
= 2 µA
strong coupling
L=10 pH
shunt capacitance
C = 10 pF
bias line
R b
= 150 Ω
voltage line
R v
= 1 kΩ

Cavity, wiring

Qubit spectroscopy
Energy (GHz)
40
20
0
-20
-40
0.46 0.48 0.50 0.52 0.54
total flux (Φ 0 )
(I sw - I bg ) / I ctr (%)
F (GHz)
2
1
0
15
10
5
0
16 GHz
16 GHz
0.008 π
∆ = 3.4 GHz
-0.005 0.000 0.005
∆Φ ext / Φ 0

artificial + automatic shift
MW line: DC pulse added to the MW pulse (16GHz)
DC pulse
DC pulse
0.0316 V
0.585 V
1.202 V
3.8 V
- 0.224 V
- 4.7 V
0.004 π
0.0045 π

Spectroscopy around the ∆ gap
DC shift
16
14
12
F (GHz)
10
8
6
4
2
0
-0.005 0.000 0.005
∆Φ ext / Φ 0

Rabi: pulse scheme
RF line: one microwave pulse with varying length
bias line: Ib pulse
voltage line: detection of the switching pulse

Rabi coherent oscillations
decay time ≈ 150 ns
80
0.6
switching probability (%)
60
40
80
60
40
80
A = 0 dBm
A = -6 dBm
0.5
0.4
0.3
0.2
Rabi frequency (GHz)
60
0.1
F Larmor
= 6.6 GHz
40
A = -12 dBm
0.0
0 10 20 30 40 50 60 70 80 90 100 0.0 0.5 1.0 1.5 2.0
pulse length (ns)
MW amplitude
10^(A/20) (a.u.)
I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, J.E. Mooij, Science, 299, 1869 (2003)

Fast oscillations
100
90
Psw (%)
90
60
Switching probability (%)
80
70
60
50
40
30
Psw (%)
30
62
60
58
0 10 20 30
RF pulse length (ns)
RF pulse length (ns)
20
500 510 520 530
0 50 100 150 200 250 300 350 400 450 500
RF pulse length (ns)

Towards single-shot resolution
100
80
F=6.512 GHz
A=-3 dBm
switching probability (%)
60
40
20
0
N- number of averaged events
N=1
N=10
N=10000
0 10 20 30 40 50 60 70 80
RF pulse length (ns)

Ramsey interference
Ramsey: two π/2 pulses with varying time in between
π
π

Ramsey fringes
0 MHz
F L
= 5.61 GHz
detuning
P SW
(%)
90
60
30
0 5 10 15 20 25 30
time between two π/2 pulses (ns)
310 MHz

Ramsey interference
Ramsey: decoherence time T 2* ≈ 20 ns
80
P SW (%)
70
60
50
π/2 π/2
0 5 10 15 20 25 30
π/2
distance between two π/2 pulses (ns)
F L
= 5.7 GHz, dF= 220 MHz, TRamsey: 4.5 ns

Spin-echo experiments
spin-echo: two π/2 pulses and one π pulse in between
with varying position
π
π
π
switching probability (%)
70
60
50
π/2
π π/2
-25 -20 -15 -10 -5 0 5 10 15 20 25
position of π pulse (ns)
F L = 5.7 GHz, dF= 220 MHz, TRamsey: 4.5 ns, Tspin-echo: 2.3 ns

Signal decay in spin-echo
switching probability (%)
60
50
40
30
20
10
0
Detuning
F res
= 5.64 GHz
50MHz
100MHz
200MHz
F Rabi
= 297.6 MHz
0 20 40 60 80 100
distance between two π/2 pulses (ns)
spin-echo: max signal decay time T 2 ≈ 30 ns

Relaxation measurements
one π pulse and read-out pulse delayed
π
100
switching probability (%)
90
80
70
60
50
40
30
0 1 2 3 4 5 6 7 8 9 10
delay time (µs)
8.3 ns, A=-12dBm
6 ns, A=-9dBm
4.5 ns, A=-6dBm
3.2 ns, A=-3dBm
2.445 ns, A=0 dBm
exp fit of A=-12dBm
T 1
= 870 ns

Conclusion
we demonstrate coherent quantum
oscillations in a superconducting flux-qubit
spectroscopy
relaxation time ≈ 1 µs
Rabi oscillations with decay time ≈ 150 ns
Ramsey interference: decoherence time ≈ 20 ns
spin-echo experiment: decay time ≈ 30 ns