ChalmersNEC

REVIEW OF SOLID STATE QUANTUM BIT CIRCUITS

Two strategies

single particle statesin semiconductor structures

 global quantum statesof superconducting Josephson circuits

(A) Kane’s proposal : nuclear spins of P impurities in Si

(B) Electrons in quantum dots

(C) Propagating states: flying qubits NIST

SL SR

eeL

w

QHE edge states:LPA (ENS Paris)

B1) Charge: NTTB2) Spin: TU Delft, Harvard,…

U. Of New South Wales

TU DelftSchoelkopf et al, Yale

Quantronics

From charge states to phase states

First demonstration of coherent oscillations in a double dotd

Coherent charge oscillations in a double dot(NTT, Hayashi et al. 2003)

T2 (charge qubit) ~ 1 ns

But charge too much coupled to the environment ! spin expected better

L

R

LR

LR

• 1-qubit control:• magnetic (ESR)• electric (modulate effective g-factor)

• 2-qubit coupling: exchange interaction between 2 dots

• Read-out through charge

SL SR

B2) Spin qubits

Initial ideas: Loss & DiVincenzo (1998)

EZ = gBB

EZ = gBB

BZ

J(t) J(t)

Expts: TU Delft , Harvard

b) 2e spin qubit in a double dot (NEW)

charge readout Two electron spin qubit

GDGD

GSRGSRGSLGSL

Harvard

Charge sensor

0(1,1) (1,1) mS T with QPC

Harvard U. C. Marcus team,

# e indot

Nature, june 2005,& Petta et al., in prep.

Bloch sphere in (1,1) S - T0 subspace

(1,1)S

(1,1)S (0,2)S

(0,2)S

ε

2t

(1,1)T0

Measuring Spin Dephasing (T2)

(electrostatic energy difference)

Move from (0,2)S to (1,1) s & let evolve

(1,1)S

(1,1)S (0,2)S

(0,2)S

ε

2t

(1,1)T0

dephasing causes failure to return to (0,2)

Measuring Spin Dephasing (T2)

spin T2 ~ 10 ns

BZeeman

BNuclear

BTotal

Short coherence time : 10 ns due to nuclei

But coherence restored byspin echo

experiment

tflip

tflip

pattern still observed at long times: coherence time TE =1.2 s

model

0 40 ns

40 ns0

Electron spin resonance in a double dot

Gates ~ 30 nm gold

Dielectric ~ 100nm calixerene

Stripline ~ 400nm gold

team L. Kowenhouven& L. Vandersypen, TU Delft

B0

Expected AC current ~ 1mA Expected AC field ~ 1mT

ESR detection via Pauli spin blockade

Flip right spin: ↑↑ ↑↓ = S11 + T0 S02

Flip left spin: ↑↑ ↓↑ = -S11 - T0 S02

Flip both spins: ↑↑ (T+ + 2 T0 + T- )/2 S02

Advantages: - low frequencies (B0 > 20mT, f > 100MHz. )

- not sensitive to electric fields (unlike single-dot ESR)

- no confusion with ESR in leads

hfac = g B0

T-

T0

T+

S

Bext

Ene

rgy

31jan06_20

Pulsed ESR scheme

Rabi oscillations

pA

F. Koppens et al.

Way 1

single particle statesin semiconductor structures

 global quantum statesof superconducting Josephson circuits

A)Kane’s proposal : nuclear spins of P impurities in Si

B) electrons in quantum dots (charge or spin)

C)Propagating states: flying qubits

Phase qubit

SL SR

eeL

w

Flux qubit

C P Box

Why superconductivity ?Why different flavors?

Solid State Qubits: way 2

Charge-flux qubit

C P Box

Way 2

NIST NEC, ChalmersTU Delft

Josephson qubits come in different flavors

Yale (Schoelkopf)

(N)

N

12 N

flux

1~2N

phase chargecharge-phase

Single Cooper pair boxes

Saclay, Yale (Devoret)

Energy spectrum of an isolated electrode

Superconducting stateNon superconducting

state

N

+

2S

singlet ground state

Superconducting helps making qubits

2 2

The Josephson junction

N

01

01

01 0/1

U1U1

Building blocks for quantum bit circuits

?

?

Basics of the Josephson junction

ˆN

N N N N

R Lθ = θ -θ 0,2

N=Q/2e θ,N i

RθLθ

single degreeof freedom

θ =N

iˆ

N N+1 ie

1

2N

N

N

ie

2

0

1

2NN

ied

ˆJJ J

N

N N+1 N+H = - = -1 c sE

E2

N o 2Jt

hΔ=

8e RE

Josephson Hamiltonian :

Phase qubits

J 0J

2

b

qH = - cos - I

C+

2E

: extended phase conjugated of q on CJ

b 0arcsin /II

I 0 >I 1 >I 2 >

12

31 10

i

i

0 0 50.7

0.8

0.9

1

pU

10

2132

pn

nE

E /

1

tilted washboard potential

V

ac

dc

Ib

1 2

A flux quantum bit : the three junction loop

EJEJ

EJ

k

i

0.5 0

E

Icirc

0

-1

0

1

2t

0.5 /o

+Ip

-Ip

0k

i

x=0/2

2D potential:

(0.5<<1)

Mooij et al. (TU Delft), (1999)

Single Cooper pair boxes

Vg

CgCJ

N θ

The first ‘working’  qubit

0 0.5 1 1.5 2 2.5 31

0

1

2

3

4EjEc1.1

Ng

En

erg

y (E

C)

0E

1E

2E

3E

JE 1

0

qubit

BJE >> k T

CJ/EE =1.1

energy spectrum

Josephson Phase Qubitsat UCSB

UC Santa Barbara

Team of John Martinis

Phase qubits

23

2/3

000 /1

24II

IU

C

4/1

0

2/1

0

0

122

III

p

I RCLJ

Lifetime of state |1>

0 50.7

0.8

0.9

1

pU

10

2132

pn

nE

E /

1

RC

Up

U()<V> = 0

<V> pulse(state measurement)

I0

= 0/2I0cos nonlinear inductor

I cos I 0j V ) (1/L J

0 sin I I

LJ

2

V 0

1: Tunable well (with I)2: Transitions non-degenerate3: Tunneling from top wells4: Lifetime from R

E0

E1

E2

0

1

2n

n+11000~

better phase qubit : rf SQUID with dc SQUID readout

J. Martinis team (2003-2005)NIST & SB

Is

I

low noise bias qubit

"sample and hold" readout

U()

~50

00 s

tate

s

“0”

“1”

… fastdeca

y

1

SQUID fluxS

witc

hin

g c

urre

nt

0

Is

I

time

QubitCycle

Qubit Op Meas Amp Reset flux

Measure p1

I 0 >I 1 >

2 2e

0J

xt1 qH = - cos + - 2 +

2L CE

2

T2 : 10-50 ns (?)

• State PreparationWait t > 1/ for decay to |0>

• Qubit logic with current bias

• State Measurement: U(Idc+Ip) Fast single shot – high fidelity

Josephson-Junction Qubit

|0>

|1>

I = Idc + Idc(t) + Iwc(t)cos10t + Iws(t)sin10t phase

pote

ntia

l

)2( z wsywcx IIH Idc(t)

pulse height of Ip

Pro

b. T

unn

el

|0> : no tunnel

|1> : tunnel 96%

|0>|1>

dt

e 01

Apply ~3ns Gaussian pulse

1.0

0.8

0.6

0.4

0.2

0.00.80.70.60.50.40.30.2

IC Fabrication

I

IsIwave

100m

Qubit

Al junction process& optical lithography

via junction

SiNxAl

AlAl

Al2O3 substrate

(old design)

Z

X,Y readout

Qubit Fidelity Tests (2006)Rabi:

Ramsey:

Echo:

T1:

Pro

bab

ility

1 s

tate

Large Visibility! T1 = 110 ns, T ~ 85 ns

~90% visibility

(slightly detuned)

(no detuning)

SUPERCONDUCTING FLUX QUBITS

Group of Prof. Hans MooijTU DELFT

team Y. Nakamura NEC

team K. Semba NTT

tunneling: Tin = m exp{ -0.64(EJ/EC)½}

Tout = 1.6 m exp{ -1.5(EJ/EC)½}

Tin

Tout

1 (

2

()

magnetic flux 0.5 o

1 2

persistent-current quantum bit

EJEJ

EJ

barrier scales with EJ,depends on

effective massscales with junction capacitance C

<1 to suppressinfluence of charge noise

for =0.8

1 2

flux qubits : the three junction loop

EJEJ

EJ

k

i

0.5 0

E

Icirc

0

-1

0

1

2t

0.5 /o

+Ip

-Ip

0k

i

x=0/2

2D potential:

(0.5<<1)

Mooij et al. (TU Delft), (1999)

SQUID readoutof the flux qubit(readout # 1)

Switching measurement(Ic 200 nA)

w on resonance

o

switchingcurrent

o-0.5

x10-3

f GHz

Van der Wal et al., Science 290, 773 (2000)

also SUNY (Friedman, Lukens et al.)

First spectroscopy of flux qubits

Irinel Chiorescu and Yasu Nakamura (NEC, Delft)

pH 5 pH 5

A 20 cI

exI

qqI ,

0~

05.0~

Science 299, 1869 (2003)

Dephasing: T2Ramsey, T2echo measurement (sample5)

~ 4ns

t/2

readout pulse

~2ns

t/2

~2ns

t

correspond to detuning

readout pulse

Ramsey interference

spin echo

NEC

Observation of vacuum Rabi oscillationsin a flux qubit coupled to a SQUID resonator

VSQUID

Isw

I

|↓>

|↑>

DC SQUID detector:

VSQUID = 0 qubit state |↑> ≠ 0 qubit state |↓>

qubitcircqubit

n Id

dE,

ISQUID~Mqub,SQIcirc,qubit

ISQUID Ithreshold

6.7μm

6.3μm

MIcirc,qubit

SQUID

qubit

K. Semba et al., NTT

|e0

|e1

|g1|g0

readout readout qubit state qubit state

excite qubit excite qubit by aby a-pulse-pulse

1 → 21 → 2 3 ⇔ 43 ⇔ 4

shift qubit shift qubit adiabaticallyadiabatically

|e0

|e1

|g1|g0

|e0

|e1

|g1|g0

2 → 32 → 3

|g0

|g １

|e0

4

shift qubit shift qubit adiabaticallyadiabatically

I I qubitqubit, , LC-oscillator LC-oscillator >> 　　Vacuum Rabi : measurement schemeVacuum Rabi : measurement scheme

Vacuum Rabi oscillationsVacuum Rabi oscillationsDirect evidence of level quantization in a 0.1 mm large

superconducting macroscopic ＬＣ -circuit

J. Johansson et al., cond-mat/0510457 → to appear in J. Johansson et al., cond-mat/0510457 → to appear in Phys. Rev. Lett.Phys. Rev. Lett. (2006). (2006).

Quantization of Rabi periodQuantization of Rabi period

7.0

1

)2/(1/ |e2

|g2

√２

|e0

|g1|g0

|e1

/

J. Johansson et al., cond-mat/0510457 → to appear in J. Johansson et al., cond-mat/0510457 → to appear in Phys. Rev. Lett.Phys. Rev. Lett. (2006). (2006).

Readout of single Cooper pair boxes

Vg

CgCJ

N θ

Hamiltonian and energy spectrum

Vg

CgCJ

2 characteristic energies:

2

C

g J

2e

2E

C C

=

2Jt

h

8eE

R

θ,N i1 degree of freedom: 1 knob: or /g g gN = C V (2e)gV

Nθ

ˆ ˆˆ cos2gC JEH = ( -E N N ) - Hamiltonian:

ˆg

N

J2C

EE N N N N+1 NH = ( -N ) -

2N N+1

0 0.5 1 1.5 2 2.5 31

0

1

2

3

4EjEc1.1

Ng

En

erg

y (E

C)

0E

1E

2E

3E

JE 1

0

qubit

BJE >> k T

CJ/EE =1.1

energy spectrum

0 0.5 1

0

1

0 0.5 1

0

1

EjEc1.1

Ene

rgy

(EC)

Ng

Ng

kN

0N

1N

CJ/EE =1.1

expectation value of the box charge:

(measurement of the quantum state)

ˆ ˆˆ cos2gC JEH = ( -E N N ) -

ˆ1ˆC

gg

N N=H

-E

+2 N

Readout through the charge

Capacitive coupling to a Single Electron Transistor

gV

N

V

I N

( )gq e0 1/2 1

en

V. Bouchiat et al.Quantronics (1996)

-1.5 0 0.5 1 1.5-2

-1

0

1

2

T = 2 0 m K

<N

0>

Ng

Expérience

Théorie

Sans effet Josephson

0 1

20

N NE

EJ/EC=0.1

theory

experiment

Theory with no Josephson effect

too slow ...

A box with a continuous readout Nakamura, Pashkin &Tsai (NEC,1999)

N

V

2e- or 0

gV

I N

Continuous measurement by

energy relaxation

1n

0n

Ng0.5

DC pulses from Ng=0.25 to 0.5 with duration t

Pulsedurationt (ps)

200 400 600

5

(I pA)

0

First Rabi oscillations

Short coherence time : a few ns2000: WHY??

Open ports Decoherence

write

0

Readout

1?1

0U1

Decoherence Sources

The main difficulty

Readout port... lets noise in

1

0

AXFluctuating

environment

A -meter

0 0A 1 1A

01n

( )X tddetuning :

Readout + environment

0101( ) ( )

Xt X t

ndn d

¶¶

=

DEPHASING

[ ] 011 1 0 0A A hXn

- =¶¶

signal (if A measured) :

dephasing and readout closely related !

... and noise dephases...

2

0 1 ei

012 ( ') ) '( tt dtj p n= ò

( )

tTi te e jjD

-

=

2 11 1 0 0( ))(0A A XSTj p -

-=

if

constant

at low f eq.

( )

r

XS w

... and depolarises

Fermi Golden Rule

w

w

G

=

= -

G +

01

2

2

2

01

2

10 1

1( )0

(

1

)X

X

R

E

A

SA

S

XS

ww+ 01

relaxationw- 01 0

dephasingexcitation

to summarize:

( ) - -= +G GG 1 11 R RET

Solving the noise/readout dilemma:connect only at readout time

01 0Xn¶

=¶

Operate qubit at a stationary point:

For readout:

-Move away adiabatically at:

-better: stay there, and apply an ac drive(to be shown later on)

01 0Xn¶

¹¶

gate 160 x160 nm

A qubit ‘protected’ from decoherence:the quantronium

A general strategy now applied to different Josephson qubits

Vion et al, Science 2002; Esteve&Vion, cond-mat 2005

UU

0δ = /

δ

ˆ cos cosˆ ˆ 2

g JCE NH (2

N E- ) -

The quantronium: 1) a split Cooper pair box

2 knobs : g gN C U/2e

2 energies:

2

Cisland

EC

2e

2 =

2Jt

h

8eE

R

State dependent persistent currents

k

0 0

ˆ1 H 1 Ei

i

ˆθ,N i1 d° of freedom

N

U

-1€€€€€2

0

1€€€€€2 0

1€€€€€2

1

-0.25

0

0.25

0.5

-1€€€€€2

0

1€€€€€2

2) protected from dephasing

EJ=0.86 kBKEC=0.68 kBK

1

0

-1€€€€€2

0

1€€€€€2 0

1€€€€€2

1

0

5

10

15

20

-1€€€€€2

0

1€€€€€2

01(

GH

z)Ng

h01

ene

rgy

(kBK

)

Ng

CJE E

01 0Xn¶

=¶

Optima working points exist in many qubits

-1€€€€€2

0

1€€€€€2 0

1€€€€€2

1

0

5

10

15

20

-1€€€€€2

0

1€€€€€2

I0 Ib

V=0or

V0

U

Ng

3) with a readout junctionfirst readout of

persistent currents with dc switching

switching

no switching

0 1 2 3 4 5 6

200

400

600

800

1000

bia

s c

urr

en

t (n

A)

time (µs)

0

20

40

60

80

100

ou

tpu

t vo

lta

ge

(µ

V)

RF

am

plitu

de

(a

.u.)

1

0discrimination

t

Rabi precession

1.0 2.0 3.0 4.0

-50

0

50

100

16 GHz

mic

row

ave

outp

ut v

olta

ge (

mV

)

time (ns)

0 50 100 150 200 250

30

40

50

pulse duration (ns)

Rabi oscillationsswitchingprobability (%)

μwA cos( ) t

Note: visibility : <40%

1

µw

X

Y

0

Effectivefield

rotationRabiURF

0

readout

Optimal point for other qubits?

the flux qubit

Nakamura, Ciorescu,Bertet, Mooij et al. Delft, 2003-2004

pH 5 pH 5

A 20 cI

exI

qqI ,

0~

05.0~

0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54

-50

-48

-46

-44

-42

shift

read-outpoint

operation point

E /

EC

total flux (0)

shift in practice

operating point readout point

coupling to SQUID

Ib=0

Ib=-0.2

Ib=-0.4

Optimal point for other qubits? the flux qubit

2005: T2=400 ns

pH 5 pH 5

A 20 cI

exI

qqI ,

0~

05.0~

Chiorescu et al, TU Delft 2003Nakamura et al., NEC 2005

projection fidelity ?

1

0

Fluctuatingenvironment

A -meter

0 0A 1 1A

idealQuantum

measurement:1 1

0 0

Readout: 1

Readout: 0

errors: wrong answer & projection error

510 515 520 525 530 535 5400

20

40

60

80

100

d ifference

ground state pulse

M agic pointN

g = 1/2

switc

hin

g p

rob

ab

ility

(%

)

b ias current am plitude (nA)

Switching readout fidelity ?

40% contrast (only)

0 200 400 600

-200.0

0.0

200.0

400.0 µw pulse

plateau

15mK

I b (

nA

)time (ns)

-300 -200 -100 0 100 200 3000.30

0.35

0.40

0.45

0.50

100 ns long plateau

Ip (nA)

Sw

itc

hin

g P

rob

ab

ility

0 200 400 600 800 1000 1200 1400 1600 1800 20000.25

0.30

0.35

0.40

0.45

0.50

T1=730 ns

T1=59 ns

Sw

itc

hin

g p

rob

ab

ility

plateau duration (ns)

Relaxation during readout ramp !

partly explains readout fidelity

T1=730ns

T1=60ns

switchingdc pulse

simple, but:

rep rate limited by quasiparticlesfidelity <1 due to relaxation qubit reset : NOT QND

tV

t

dc readout

U

Switching readout resets the qubit

resets the qubit

Towards QND readout ‘at’ optimal point

flux qubit : charge qubit :

SQUID inductance quantum capacitance

Chalmers, Helsinki

charge-phase qubit :

readout junction inductance

Quantum capacitance

C/C

J

0

1

TU Delft Yale, Saclay

0

1

PULSE IN

PULSE OUT

U

“RF” pulse

dynamics in anharmonic potential

more complex, but:

-better fidelity ?-no reset: possibly QND

switching

dc pulse

simple, but:

-fidelity 40%-qubit reset : NOT QND

t

U

rf readout (M. Devoret, Yale)

dc versus ac readout in the quantronium

M. Devoret team at Yale I. Siddiqi et al., (2004)

µW Pulse IN

QuBitcontrol

0 1

0

1

OUT

-1€€€€€2

0

1€€€€€2 0

1€€€€€2

1

0

5

10

15

20

-1€€€€€2

0

1€€€€€2

Towards non destructive readout at optimal point with an AC drive

UJop

timal

P

1

001n

Similar dispersive methods developed for other qubits

M. Devoret team at Yale I. Siddiqi et al., (2004)

µW Pulse IN

QuBitcontrol

0 1

0

1

OUT

UJop

timal

P

1

001n 01

180°

-180°

am

plitu

de

µW drive amplitude

µW

pha

se

State dependent bifurcation

The Josephson Bifurcation Amplifier

Enhanced

300 K

Quantronium from Yale

Quantronium + JBA SETUP

4 K

0.6 K

30 mK

1.3-2GHz

MS

-20dB

-30dB0 100 200 300 400

0

1

2

3

4

VM (

V)

time (ns)

Q

50

TN=2.5KG=40dB

G=40dB

ILO

demodulator

bifurcation

NO bifurcation

-6.8 -6.6 -6.4 -6.20.0

0.2

0.4

0.6

0.8

1.0

Bifu

rca

tio

n p

rob

ab

ility

readout µW input power (dB)

0 20 40 60 80 100 120 140 160

gate µW pulse duration (ns)

45-5

0%

Rabi oscillations with the JBA

Contrast : 50%

0

1

100ns 125ns

JBA pulse

(Saclay exprt)

1 0

100ns 125ns5n

s

20ns

40ns

JBA readout

10ns

gate

100ns

0

1

0

1

0

1

0

1

partially QND

initial finP( , , ral esult)

1

0 0

10

34%

100%

66%

0%

10

25%9%

30%36%

1

10

17%83%

10

0%0%

Notice: relaxation againpartly avoidableby tuning the qubit

0

1

Quantum Non Demolition ? read twice

initial A B

& correlations

Note: results for flux-qubit now available

Dispersive readout of the flux qubit

detection

Tdetection

Tplateau

switching

time

Iac

A. Lupascu et al.TU DELFT

Activation rates for different detuning values

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

0.5

1.0

1.5

2.0

2.5

3.0

Tplateau

=400 ns

Tplateau

=80 ns

I ac

2 (au

)

ln(a/)2/3

F = 775 MHzFres=822 MHz

Tk

U

attsw

Be

2

2

0 1B

acaa I

I

2/32

1

B

acdyn I

IuU

2033

4 pa RC

Iac,bifurcation2

slope=udyn/(kT)

Thy: M. Dykman

Rabi oscillations with optimal settings

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

Psw

Dt (ns)0 10 20 30 40 50 60 70 80

0

2

4

6

8

10

12

14

Psw

Dt (ns)

0.1 1

0.01

0.1

FR

abi (

GH

z)

Imw

(au)

Dt = length of MW pulse

Ramsey oscillations with optimal settings

0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

97 %

Psw

Dt (ns)

11 %

Rabi oscillation

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

Psw

tRamsey

(ns)

Ramsey: ge-mw= 69 MHz

Ramsey frequency vs detuning

4.50 4.55 4.60 4.650

10

20

30

40

50

60

70

80 F

Ramsey

|Fmw

-Fqubit

|

FR

amse

y (

MH

z)

Fmw

(GHz)

Relatively strong low frequency fluctuations visible in the drift of the Ramsey frequency.

QND data : analysis in progress

A Circuit Analog for Cavity QED2g = vacuum Rabi freq.

= cavity decay rate

= “transverse” decay rate

L = ~ 2.5 cm

Cooper-pair box “atom”10 m10 GHz in

out

transmissionline “cavity”

Blais, Huang, Wallraff, Girvin & RS, cond-mat/0402216; to appear in PRA

Cavity QED with a Cooper pair box: first dispersive readout

R. Schoelkopf, A. Wallraff, S. Girvin et al., Yale (2004)

Dispersive readout with out of resonance photons

Dressed Artificial Atom: Resonant Case

? T01 R

2g

/ R

T

2

1“vacuum Rabi splitting”

Rabi Oscillations of Qubit

Prf = 0 dB Prf = +6 dB

Prf = 18 dB

Coherence time measurementswith 2 pulse Ramsey sequence

CONCLUSION

Solid state qubits at work:

Semiconductor qubits recently demonstrated

Superconducting qubits:qubit control

single-shot readout

Decoherence, QND readout, couplingin next lectures

The work on

SPEC

ELECTQUANRONICSUM

GROUP

G. ITHIERE. COLLINN. BOULANTD. VION P. ORFILA P. SENATP. JOYEZP. MEESOND. ESTEVE

A. SHNIRMANG. SCHOENY. MAKHLINF. CHIARELLO

1

0

Fluctuatingenvironment

A -meter

0 0A 1 1A

01n

the Quantronium

1µm

boxqp

trap

dc gatedc gateµw

readout junction

2004

Thanks to

NEC