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ComeniusUniversity
Sisyphus cooling and pumping of linear oscillator by superconducting qubit
A. Izmalkov, S.H.W. van der Ploeg, Th. Wagner, E. I’lichev, H.-G. Meyer
Institute for Physical High Technology, Germany
M. GrajcarComenius University, Slovakia
A. Fedorov, A. Shnirman, Gerd Schön, Institut für Theoretische Festkörperphysik UniversitätKarlsruhe, Germany
S.N. Shevchenko, A.N. Omelyanchouk,B.Verkin Institute for Low Temperature Physics and Engineering,Kharkov, Ukraine
S. Ashhab, J.R. Johansson, A. Zagoskin and Franco Nori,The Institute of Physical and Chemical Research (RIKEN), Wako-shi, Japan
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ComeniusUniversity
Outline
1. Superconducting flux qubit2. Adiabatic measurement of the qubit in
the ground state3. Spectroscopic measurement 4. Sisyphus cooling and pumping5. Lower limit on the achievable
temperature
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ComeniusUniversity
Single-junction interferometer (RF-SQUID)
-0.4 -0.2 0.0 0.2 0.4
-1.0
-0.5
0.0
0.5
1.0
I/Ic
f=x/
-0.4 -0.2 0.0 0.2 0.4
-1.0
-0.5
0.0
0.5
1.0
I/Ic
f=x/
-0.4 -0.2 0.0 0.2 0.4
-1.0
-0.5
0.0
0.5
1.0
I/Ic
f=x/
)sin()(
2
2
sin-f2
sin
00
0
c
cJ
L
cx
II
LIL
L
LI
Classical two level System!
Or in normalized Units:
xx x
0
1
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ComeniusUniversity
Classical picture
JE
U
-1
0
1
2
3
4
-1
0
1
2
3
4
-1
0
1
2
3
4
Particle with mass ~ CJ in potential:
f
minU 01
2)2(2
1cos f
E
U
J
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ComeniusUniversity
-1
0
1
2
3
4
Quantum Picture
JE
U
If CJ is small enough tunneling between both wells becomes possible and therefore the degeneracy is lifted. So we need Small Josephson Junctions with EJ/EC~10-100
f
-1
0
1
2
3
4
-1
0
1
2
3
4
-1
0
1
2
3
4
-1
0
1
2
3
4
0 1
10 minU
10
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ComeniusUniversity
Persistent current (flux) qubit – analogue of ammonia molecule
rightleft
U/E
J
e
h
20
B 710
m
eB 2
n0
B
N
H
H
H++
+
Superconducting persistent current qubit – oscillation of a magnetic dipole moment (magnetic flux), Ammonia molecule – oscillation of an electric dipole moment(f=24 GHz)
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ComeniusUniversity
Size problem and solution
dL 0
)μA(
μm250
2 0
0
cc IId
x1
2
0
cLI
For quantum behavior
Typical parameters for aluminum technology :
276
22
A/m10-10
F/m104
c
s
j
cm][ 10 44 aa
E
E
C
J
EJ/EC~10-100
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Solution of the size problem
‚Size‘ problem solved in 70´sT. Yamashita et al., J. Appl. Phys. 50, 3547 (1979)
This idea was dusted off by J.E. Mooij et al., Science 285, 1036, 1999x
x
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Hamiltonian. Energy surface.
.)21( ,2)2/(
, ,2/)( ,2/)(
,)22cos(coscos2,,
,,,22
20
,,2121
22
0
MMCM
iP
fEfU
fUM
P
M
PH
J
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
/2
/2
1,CI 2,CI
CI
-0.2
-0.1
0
0.1
0.2
-0.4-0.2
00.2
0.41.5
2
2.5
3
/2/2
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Tunneling amplitude
2)2/1)(cos(2)( jEU
cj EES /2/)14(2
220
4(2 1)expJE S
h g
ЕС=5 GHz, g=EJ/EC=66, ЕJ=330 GHz.
jE
E00
2
2
1
2
1arccos
rightleft
U/E
J
-0 0
0.85 0.86 0.87 0.88 0.89 0.9 0.901 0.902 0.905 0.91 0.92
GHz
20 13 8.45 5.44 3.49 2.24 2.14 2.05 1.79 1.43 0.92
E0
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ComeniusUniversity
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
(fx) (GHz)
Adiabatic measurement away from degeneracy point
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ComeniusUniversity
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
(fx) (GHz)
Adiabatic measurement at degeneracy point
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Lagrangian of the qubit-resonator system
Expanding into Taylor series up to the second order term
2
-
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Quantum approach
zrx
T
bbbb
HHHH
)(
int0
LCT
Ib
LT
Φi
At the degeneracy point
0, HAThe sufficient condiction for Quantum Nondemolition Measurements
is satisfied. 0, Hx
xq
rx
q
r
Wkbb
Wk
H
21
22
2
2qq
q
ILW
No perturbation of the measured observable [V.B. Braginsky and F.Ya. Khalili, Quantum Measurement, (Cambridge University Press, Cambridge, 1992].
pr ILk
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Impedance Measurement,classical resonator
Φ
0.0 0.4 0.8 1.2 1.6 2.0-2
-1
0
1
2
, rad
VT
LTL CT
Ib
M
Ya. S. Greenberg et al., PRB 66, 214525 (2002)DC-Squid Josephson Inductance: A. Lupascu et al., PRL 93, 177006 (2004).
0.0 0.4 0.8 1.2 1.6 2.00
2
4
6
8
10
Am
plit
ude
TTT CL
1
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ComeniusUniversity
Response of resonator
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
(R
ad)
/0
EJ/Ec103 EJ/Ec<102
=0.8
EJ/Ec<102
=0.9
0.86 0.88 0.9 0.901 0.902 0.905 0.91 0.92
GHz 13 5.44 2.24 2.14 2.05 1.79 1.43 0.92
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Resonant frequency of the resonator
Y. Greenberg et al., PRB 66214525 (2002).
Fitting parameters
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Sisyphus work
As a punishment from the gods for his trickery, Sisyphus was compelled to roll a huge rock up a steep hill, but before he reached the top of the hill, the rock always escaped him and he had to begin again.
Greek mythology
Titian (1549) artist vision of Sisyphus work
Physical realization: For atomsD. J. Wineland, J. Dalibard and C. Cohen-Tannouji, J.Opt. Soc. B9, 3242 (1992).
For qubit Grajcar et al., arXiv:0708.0665Nature Physics 4, 612-616 (2008).
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ComeniusUniversity
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
(fx) (GHz)
Sisyphus cooling
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ComeniusUniversity
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
(fx) (GHz)
Sisyphus pumping
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ComeniusUniversity
Adiabatic vs. spectroscopic measurement
Solid line is theoretical curve for Parameters determined from adiabatic measurement
0.000 0.005 0.010 0.0152
4
6
8
10
12
14
16
18
20
f [G
Hz]
dc
(0)
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ComeniusUniversity
Strong microwave driving at fmw=4.5 GHz
Weak driving
Transition from weak to strong driving
dc (0)
M. Sillanpää et al., PRL 96, 187002 (2006)
W.D. Oliver et al.,SCIENCE 310, 1653(2005)
Strong driving
A. Izmalkov et al., PRL 101, 017003 (2008)
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ComeniusUniversity
Landau-Zener interferometry
S.N. Shevchenko et al. Phys. Rev. B 78, 174527 (2008)
A.V. Shytov, D.A. Ivanov, and M.V. Feigel’man, Eur. Phys. J. B 36, 263 (2003).
E
a2
2
1
2
1
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ComeniusUniversity
More rigorous treatment of Sisyphus cooling/pumping
A. Fedorov, A. Shnirman, Gerd Schön
fmw=14 GHz
M. Grajcar et al., Nature Physics 4, 612-616 (2008).
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ComeniusUniversity
Sisyphus and sideband cooling limit
q
osc
q T
T
0
200
oscqq
TT
M. Grajcar, A. Ashhab, J.R. Johansson, F. NoriPhys. Rev. B 78, 035406 (2008)
2q
qT
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ComeniusUniversity
Conclusions
1. Superconducting flux qubits are well described by two-level (pseudospin) Hamiltonian
2. Experimental data obtained from adiabatic and spectroscopic measurement are consistent and fully agree with the quantum-mechanical predictions to the experimental accuracy.
3. The qubit can be used as an artificial atom for Sisyphus cooling of a low frequency oscillator (electrical, nanomechanical, etc.)
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Spectroscopy of the system of two coupled flux qubits.
Without microwave driving fmw= 14 GHz
fmw= 18 GHz fmw= 21 GHz
A. Izmalkov et al., PRL 101, 017003 (2008)
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ComeniusUniversity
Nanomechanical oscillators
Neik et al., Nature 443, 193 - 196 (2006)
Nanobridge from IPHT Jena
Prepared for measurement at temprature below1 mK in ulra low temp. lab in Košice
I. Martin, A. Shnirman, Lin Tian, P. ZollerGround state cooling of mechanical resonators Phys. Rev. B 69, 125339 (2004)
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Quantum metamaterials
Design of high efficiency microwave photon detector for GHz range
G. Romero et al., Microwave Photon Detector in Circuit QED, arXiv:0811.3909v1
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Anti-Ferromagnetic and Ferromagnetic Coupling
AFM
FM
Iq2=-10 µA
Iq3=0
Iq4=-250 µA