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