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Non-linear driving and Entanglement of a quantum bit with a quantum readout
Irinel Chiorescu
Delft University of Technology
Quantum Transport group Flux-qubit teamProf. J.E. MooijProf. Kees Harmans
technical staff
students
visitorsYasunobu Nakamura (NEC Japan, 2001-2002)
Kouichi Semba (NTT Japan, 2002-2003)
postdocsPatrice BertetIrinel Chiorescu
PhD studentsAlexander ter HaarAdrian LupascuJelle Plantenberg
collaborationsNTT, NEC, MIT, TU Delft (theory), U Munich
acknowledgementsFOM (NL) , IST (EU) , ARO (US)
Outline
basics about the flux-qubit
qubit initialization, operation & readout
Rabi oscillations, Ramsey fringes
present status
- extreme stability during qubit operation
- strong microwave driving
multi-photon induced coherent oscillations
experimental demonstration of entanglement
quantum bit quantum readout (squid)
conclusions
3 Josephson-junctions Quantum Bit
superconducting loop, with 3 Josephson junctions
2 are identical and the 3rd is smaller (
Josephson Potential:
J.E. Mooij et al, Science, 285, 1036 (1999)
U=EJI
u = U/EJ
u = 2 + - cos1 - cos2 - cos(2 - 1 + 2f)
1 = (1 - 2)/2 , 2 = (1 + 2)/2
u = 2(1 - cos1 cos2) + 2sin2(1 - f)
Josephson potential - phase space
Tin
Tout
Tout
=0.8, f=0.5
2 wells separated by a barrier
for f=0.5, symmetric barrier
Flux Qubit – two level systemC. van der Wal et al, Science, 290, 773 (2000)
see also, J. Friedman et al, Nature, 406, 43 (2000)
Exact diagonalisation: two levels at the bottom of the spectra
Two wells separated by a barrier
Persistent currents of opposite direction | and |
SQUID critical current qubit persistent current
Microwave induced excitation level structure
0.5
Coherent oscillations
Rabi oscillations
microwave excitation with frequency and amplitude A
coherent rotations with Rabi A
Bloch sphere
|>=|>+|>
|g>
|e>
= E
Rabi AMW pulse
A
Magnetic resonance with a single, macroscopic quasi-spin
Qubit operated at the magic pointHamiltonian and eigenstatesH = -/2 z – /2 x
tan2 = / |0 = cos | + sin ||1 = -sin | + cos |
|0
|1
|
|
|
|
|0
|1
Initialization, = 0|Q = |0 = (| +|)/2
Operation , = 0|Q = |0 + |1
Readout , > 0|Q = |0 + |1
|Q
MW pulse ON(rotating frame)MW pulse OFF
(lab frame)
|Q
<x> = ||2 - ||2
Switching event measurements
Device
qubit merged with the SQUID
strong coupling L
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: Isw0 < Ib < Isw1
I pulse~30ns rise/fall time
t
Single shot resolution
2.7 2.8 2.9 3.00
20
40
60
80
100 ground state
excited stateswitc
hing
pro
babi
lity
(%)
pulse height @ AW generator (V)
(in an ideal sample)
Sample
EJ/EC = 34.65EC = 7.36 GHz = 0.8
= 3.4 GHzIp = 330 nA
large junctionsIc = 2 A
strong couplingL=10 pH
shunt capacitanceC = 10 pF
bias lineRb = 150
voltage lineRv = 1 k
Cavity, wiring
Qubit spectroscopy
-0.005 0.000 0.0050
5
10
15
0
1
2
0.46 0.48 0.50 0.52 0.54-40
-20
0
20
40
F (
GH
z)
ext / 0
= 3.4 GHz
0.008
(Isw
- I b
g) /
I ctr
(%
)
16 GHz
16 GHz
Ene
rgy
(GH
z)
total flux (0)
Rabi: pulse scheme
RF line: one microwave pulse with varying length
bias line: Ib pulse
time
trigger
MWpulse Ib pulse
read-outoperation
voltage line: detection of the switching pulse
Rabi coherent oscillations
FLarmor = 6.6 GHz
decay time 150 ns
0 10 20 30 40 50 60 70 80 90 100
40
60
80
40
60
80
40
60
80
0.0 0.5 1.0 1.5 2.00.0
0.1
0.2
0.3
0.4
0.5
0.6
pulse length (ns)
A = -12 dBm
A = -6 dBm
A = 0 dBm
Rab
i fre
qu
ency
(G
Hz)
MW amplitude
10^(A/20) (a.u.)
swit
chin
g p
rob
abili
ty (
%)
I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, J.E. Mooij, Science, 299, 1869 (2003)
Fast oscillations
0 50 100 150 200 250 300 350 400 450 50020
30
40
50
60
70
80
90
100
0 10 20 3030
60
90
500 510 520 530
58
60
62
Sw
itch
ing
pro
bab
ility
(%
)
RF pulse length (ns)
Psw
(%
)
RF pulse length (ns)
Psw
(%
)
RF pulse length (ns)
Ramsey interference
Ramsey: two /2 pulses with varying time in between
time
trigger
Ib pulse
read-outoperation
/2freerun
/2
Ramsey fringes
0 5 10 15 20 25 3030
60
90
310 MHz
PS
W (
%)
time between two /2 pulses (ns)
detuning
0 MHz
FL = 5.61 GHz
Ramsey interference
Ramsey: decoherence time 20 ns
FL = 5.7 GHz, dF= 220 MHz, TRamsey: 4.5 ns
0 5 10 15 20 25 30
50
60
70
80
PS
W
(%)
/2 /2
distance between two pulses (ns)
Relaxation measurements
one pulse and read-out pulse delayed
time
trigger
Ib pulse
read-outoperation
delay time
0 1 2 3 4 5 6 7 8 9 1030
40
50
60
70
80
90
100
switc
hing
pro
babi
lity
(%)
delay time (s)
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 = 870 ns
Sample (2003)
heat sinkqp traps quasi-particle traps
strong coupling with the MW line
heat sinks on the current and voltage lines
current injection: high frequency noise ground via the shunt capacitance
Ib
V
Spectroscopy level repulsion
5.866 GHz persistent current
272 nA
0.500 0.502 0.504 0.5060
2
4
6
8
10
12
spectroscopy peaks fit: E
J/E
C=30.834
EC=7.281 GHz
=0.76
Lar
mo
r fr
equ
ency
(G
Hz)
/0
= 5.866 GHzIq = 272 nA
spectroscopy peaks: Q – qubit –plasma frequency 2.91GHz Q+/- – sidebands 2-, 3-photon peaks
1 2 3 4 5 6 7 8 9 10 11 12
15
20
25
30
35
40
45
50
swit
chin
g p
rob
abili
ty (
%)
frequency (GHz)
Q
Q-3
Q+3
(Q+3) /2
Q/2
Q/3
3
0.000 0.002 0.004 0.0060
2
4
6
8
10
12
Q/2
(Q + )/2
Q -
Q +
Res
on
ant
freq
uen
cies
(G
Hz)
/0
Q
Rabi oscillations at the magic point low coherence time, but extreme stability of the qubit energy levels
0 5 10 15 20 25 30 35 40 45 50
20
25
30
35
40
distance between pulses
Ramsey with pulses (Hadamard)
Rabi oscillations: Fmw
= + FRabi
swit
chin
g p
rob
abili
ty (
%)
pulse length (ns)
Rabi oscillations: Fmw
=
Hadamard gate
Ramsey fringes at the magic point coherence time ~15-20 ns
5 10 15 20 25 30 35 40
5.6
5.7
5.8
5.9
6.0
6.1
distance between two /2 pulses (ns)
Fre
qu
ency
(G
Hz)
24.00
31.00
38.00
45.00
52.00P
sw (%)
= 5.856 GHz
Coherence time at the magic point coherence time ~20 ns (mostly limited by the relaxation time)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
70
75
80
85
Ib=2.841A Ib=2.976A Ib=2.565A
swit
chin
g p
rob
abili
ty (
%)
delay between two /2 pulses (microseconds) when optimizing the readout ~120 ns
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
5.5
6.0
6.5
7.0
7.5
Larm
or freq
uen
cy (GH
z)
-0/2 (m
0)
10
20
30
40
50
60
70
80
90
100
an
d r (
ns)
Spectroscopy
spectroscopy peaks: Q – qubit –plasma frequency 2.91GHz Q+/- – sidebands 2-, 3-photon peaks
1 2 3 4 5 6 7 8 9 10 11 12
15
20
25
30
35
40
45
50
swit
chin
g p
rob
abili
ty (
%)
frequency (GHz)
Q
Q-3
Q+3
(Q+3) /2
Q/2
Q/3
3
0.000 0.002 0.004 0.0060
2
4
6
8
10
12
Q/2
(Q + )/2
Q -
Q +
Res
on
ant
freq
uen
cies
(G
Hz)
/0
Q
Multi-photon processes
20
25
30
35
40
45
50
55
60
20
25
30
35
40
45
50
55
60
0 5 10 15 20 25 30 35 4020
25
30
35
40
45
50
55
60
20
25
30
35
40
45
50
55
60
20
25
30
35
40
45
50
55
60
0 5 10 15 20 25 30 35 4020
25
30
35
40
45
50
55
60
Fmw
=3.62 GHz
A = -15 dBm
A = -17 dBm
TWO-PHOTON
A = -19 dBm
A = -14 dBm
Fmw
=7.16GHz
A = -18 dBm
pulse length (ns)
Sw
itch
ing
pro
bab
ility
(%
)ONE-PHOTON
A = -22 dBm
Multi-photon processes
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0
0.5
1.0
1.5
2.0
2.5
3.0
one-photon Rabi frequency
J1(b10A/20) with b=0.92 ,=5.344 GHzO
ne-
ph
oto
n R
abi fr
equ
ency
(G
Hz)
10A/20 (a.u.)
0.00 0.02 0.04 0.06 0.08 0.10 0.120.00
0.05
0.10
0.15
0.20
0.25
0.30
two-photon Rabi frequency
J2(b10A/20) with =5.344 GHz, b=5.15
Tw
o-p
ho
ton
Rab
i fre
qu
en
cy (
GH
z)
10A/20 (a.u.)
Rabi frequency: n=Jn(mw/FL)
can be renormalized
by noise ( < ) ~
power calibration (check the b fit parameter)
Coherent rotations in the non-linear regime
several peaks in the Fourier transform of the oscillations
Rabi frequencies higher than the Larmor frequency
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.00
1
2
3
4
5
6
7
Rab
i fre
qu
ency
(F
FT
) (G
Hz)
10^(A/20)
=5.03 GHzb=1.41
J1(b10A/20)
0 2 4 6 8 10 12 14 16 18 20 22 24 260
2
4
6
8
10
12
14
Pea
ks in
FF
T o
f th
e R
abi o
scill
atio
ns
(GH
z)
1 (GHz)
~12.25 GHz
x=0.1 GHz
0
0
Numerical simulationsH/h=0z/2+xx/2+(1xcost)/2
Qubit entangled with a quantum readout
QUBIT, two-level system SQUID, harmonic oscillator
hp
hFL
microwave field
MIqIcirc
.
.
.
|0, |1 |0, |1, ..., |N
|00
|10 |11
|01
|12
|02
...
FL p
Coherent oscillations of the coupled system qubit Larmor frequency 7.16 GHz
plasma frequency : 2.91 GHz
coupled system at 10.15 GHz
0 2 4 6 8 10 12 14 1622
24
26
28
30
32
34
36
38
40Rabi oscillationsF=10.15 GHz, A=-5dBm
qubit: FL=7.16 GHz
squid: pl=2.91 GHz
swit
chin
g p
rob
abil
ity
(%)
pulse length (ns)0 2 4 6 8 10 12 14 16
18
20
22
24
26Rabi oscillationsF=10.15 GHz, A=3dBm
swit
chin
g p
rob
abil
ity
(%)
pulse length (ns)
qubit: FL=7.16 GHz
squid: pl=2.91 GHz
|10 |11
|01|00
blue-side band
Blue-side band qubit Larmor frequency 6.43 GHz, plasma frequency : 2.91 GHz coupled system at 9.38 GHz
|10 |11
|01|00
0 10 20 30 40
15
20
25
30
35
40
coherent oscillations |01> |11>coherent oscillations |00> |10>
Rabi oscillations at FL=6.43 GHz
sw
itc
hin
g p
rob
ab
ilit
y (
%)
pulse length (ns)
either pulse or incoherentpopulation with a bright pulse
Red-side band qubit Larmor frequency 6.43 GHz
plasma frequency : 2.91 GHz
coupled system at 3.52 GHz
|10 |11
|01|003 4 5 6 7 8 9 10
15
20
25
30
FL = 6.43 GHz
+10 dB
blue-side band 9.38 GHz
swit
chin
g p
rob
abili
ty (
%)
MW frequency (GHz)
red-side band 3.52 GHz
0 5 10 15 20 25
20
22
24
26
28
30
swit
chin
g p
rob
abili
ty (
%)
pulse length (ns)
red-side band:coherent oscillations |01> <10|
0 10 20 30 40
15
20
25
30
35
40
Rabi oscillations at FL=6.43 GHzs
wit
ch
ing
pro
ba
bil
ity
(%
)
pulse length (ns)
pulse
after
after 2
Conclusion
entanglement of the qubit with its quantum readout
multi-photon induced coherent oscillations
very strong (non-linear) qubit driving, FRabi>FL
qubit operated at the “magic point”
extreme stability of the qubit operation
rel 1 s, Rabi 150 ns
Ramsey interference: decoherence time 20 ns
Switching curves
1.6 1.7 1.8 1.90
20
40
60
80
100
switc
hing
pro
babi
lity
(%)
pulse height @ AW generator [ V ]
B=4.55 GsF=5.63GHz
-pulse(3 ns)
1 s pulse
No RF
1.6 1.7 1.8 1.90
500
1000
1500
2000
dP/d
I b (%
/V)
pulse height @ AW generator (V)
No RF
1 s pulse
-pulse(3 ns)
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0-2000200400600800100012001400
4.75Gs
4.57Gs
4.60Gs
4.65Gs
4.70Gs
pulse height @ AW generator [V]
-2
-1
0
1
2
ground state
excited state
FWHM
0.005
0.008
(Isw
-Ib
g)
/ Ict
r (%
)
Spin-echo experiments
spin-echo: two /2 pulses and one pulse in between with varying position
time
trigger
Ib pulse
read-outoperation
/2 /2
FL = 5.7 GHz, dF= 220 MHz, TRamsey: 4.5 ns, Tspin-echo: 2.3 ns
-25 -20 -15 -10 -5 0 5 10 15 20 2550
60
70
/2/2
switc
hin
g p
rob
ab
ility
(%
)
position of pulse (ns)
Signal decay in spin-echo
spin-echo: max signal decay time T2 30 ns
0 20 40 60 80 1000
10
20
30
40
50
60
Fres
= 5.64 GHz
FRabi
= 297.6 MHz
switc
hing
pro
babi
lity
(%)
distance between two /2 pulses (ns)
50MHz 100MHz 200MHz
Detuning
Automatic shift of Q and switching
qubit merged with the SQUID
big junctions
strong coupling L
large circulating
currents
bias current generates
a shift in qubit
switching occurs far
from degeneracy
Linear MW fieldH/h=0z/2+xx/2+(1xcost)/2
for a rotating mw field, the Rabi frequency is 1 (one peak in the FFT of oscillations)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
pulse length (ns)
x-compy-compz-comp
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
pulse length (ns)
x-compy-compz-comp
0 3 6 9 12 15 18 21 24 27 30 330
5
10
15
20
25
30
35
Pea
ks in
FF
T o
f the
Rab
i osc
illat
ions
(G
Hz)
1 (GHz)
12.5J1(0.084
1)
H/h=0z/2+xx/2+(1xcost)/2
Symmetry point: 0=5.86 GHz, x=0
20
40
60
“usual” Rabi
~7.1GHz