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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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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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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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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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-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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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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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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Pseudospin Hamiltonian

IC,

IC,

IC

(0.5<<1)

x

1 umE

a2

2

1

2

1

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Flux qubit coupled to oscillator

Φi

VTLT

L

CT

Ib

M

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-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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-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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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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-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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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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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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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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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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Spectral density of the voltage noise of the tank

fmw=8 GHz

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Tank circuit coupled to mechanical oscillator

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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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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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Ground state energy modulation

222

4

+

m=

m= -1/2 m= 1/2

-

-

+

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Sisyphus cooling

222

4

4

zv

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Design for spectroscopic measurement

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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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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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Four qubit sample

MicrographLayout

q1

q2

q3

q4

Iq2

Iq3Iq1

Ib4 A1

A2

A3

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Anti-Ferromagnetic and Ferromagnetic Coupling

AFM

FM

Iq2=-10 µA

Iq3=0

Iq4=-250 µA

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Theoretical fits. Phys. Rev. Lett. 96, 047006 (2006)

Experiment Theory

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Psedo-spin Hamiltonian