Manipulating Q-bits in condensed matter

Frank Hekking

Université Joseph Fourier& Institut Universitaire de France

Laboratoire de Physique et Modélisationdes Milieux Condensés

Maison des Magistères Jean PerrinCNRS-Grenoble, France

International Meeting on Mesoscopic Physics with Matter and WavesOrsay, March 21-22, 2005

CRTBT-CNRS-GrenobleJ. Claudon (PhD)A. Fay (PhD)F. Balestro (Post-doc Delft UJF)O. Buisson

LCMI-CNRS-GrenobleW. GuichardL. Lévy

LP2MC-UJF-CNRS-GrenobleA. Ratchov (PhD)F. FaureF. Hekking

G R E N O B L E 1

UNIVERSITEJOSEPH FOURIERSCIENCES. TECHNOLOGIE. MEDECINE

Low Temperature Laboratory-HelsinkiJ. Kivioja (PhD)T.E. Nieminen (undergraduate student)J. Pekola

Collaborating teams

Quantum mechanics with Rydberg atoms

Four

vitesse

Préparationcirculaires

Détection

Sourceclassique

Cavité

0 30 60 900.0

0.2

0.4

0.6

0.8

1.0Ω

0 = 47 kHz

temps ( µs)

Taux

de

tran

sfer

t de

|+>

ver

s |-

>

Towards artifical atoms in the solid state

solid state physics fabrication of integrated circuits

Search for appropriate two-level systems: - semiconductor quantum dots- superconducting nanocircuits with Josephson junctions

Single qubit… … integrated multi-qubit system

(Delft flux qubit)

Charge qubit: Cooper pair box (Nakamura et al , Nature 1999)

)C(C2/eE gJ2

c +=Charging energy :

Josephson energy: EJ = Φ0I c / 2π

0 ⇒ 12 cos(EJt / h) 0e + i sin(EJt /h) 2e( )

Manipulation: picosecond gate-pulse:

T1~10ns et T2~2ns

0 1000

20

40

0.0 0.1 0.2 0.3 0.4 0.5 0.6

30

35

40

0.0 0.5 1.030

35

40

45

S

witc

hing

pro

babi

lity

p (%

)S

witc

hing

pro

babi

lity

p (%

)A

B

Microwave pulse duration τ (µs) Nominal Uµw

(µV)

Uµw

τ

Time between pulses ∆t (µs)

∆t R

abi f

requ

ency

(M

Hz)

Improved superconducting qubit schemes

(Vion et al , Science 2002; Collin et al. cond-mat 2004)

T1~1800ns et T2~500ns T1~900ns et T2~150ns to 20ns

(Chiorescu et al. Science 2003)Quantronium Flux qubit

Quantum oscillations in two coupled charge qubits

(Pashkin et al , Nature 2003)

Basic building block: Josephson junction

21 ϕϕϕ −=

ϕ1 ϕ2

+

+++

−−−−

Equation of motion

Current-biased circuit

ϕ

U

ωp

Tilted washboard potential

average slope ~ Ix/Ic

damped motion:Q = ωp RC

Hysteretic Josephson junction

-0.4 -0.2 0.0 0.2 0.4

-9

-6

-3

0

3

6

9

I (nA

)

V (mV)

N

U(ϕ)

ϕ

ωp

N-state

S-state

Hysteresis- little damping: RCωp>1- heavy particle: C not too small

Quantum effects in small Josephson junctions

U(ϕ) |0⟩|1⟩

|2⟩|3⟩

ϕ

shape of potential:ωp(Ix)

average slope: Ix/Ic

Dynamics of junction governed by Hamiltonian

Quantum limit: replace variables by non-commuting operators

quantum tunnelling

level quantisation (anharmonic system) (Martinis et al., PRL 1985)

(Devoret et al., PRL 1985)

∆U (Ix)

Quantum effects- little damping: RCωp>1- low energies < ħωp

Current-biased dc-SQUID: a tunable quantum system

|0⟩|1⟩

|2⟩

MW

ωp(Ix, φdc )

∆U (Ix, φdc )

φdc

Ixφ(t)

JJ1 JJ2

Ix, φdc form of potential

Microwaves φ(t) inducetransitions between levels

Anharmonic system: f01 ~ 7 à 12 GHzf01-f12 ~ 30 à 200 MHz

ˆ H e = hω p ⋅ ˆ P 2 + ˆ X 2( )− hω pσ ⋅ ˆ X 3

+ hω pα RF (t) ˆ X

10µm

- deep well with quantised states: quantum dynamics- shallow well with tunnelling: quantum measurement

0

0.2

0.4

0.6

0.8

1

4.1 4.2 4.3 4.4 4.5 4.6

Esca

pe p

roba

bilit

y

Ib (µA)

41 mK150 mK227 mK392 mK

∆t=50µs

Escape from the ground statein a dc-SQUID (1)

Mesuring escape probability

Dependance on bias current

MQT

TA

⎢1⟩

⎢0⟩

Shape of histogramescape mechanism

Analysis of width ∆I(between 10% and 90 %)

∆I

escape

t

Ib

<V>

∆t=50 µs

(Franck Balestro, Thesis)

(Balestro et al, PRL 2003)

20

30

40

50

60

70

80

40 60 80100 300

TheoryExperiment

∆I (nA

)

Temperature (mK)

ΦDC

/Φ0 = -0.247

MQT

|0⟩

TA

|0⟩

Escape from the ground state in a dc-SQUID (2)

(Single junction: Devoret et al., PRL 1985)

Increasing sensitivity of escape measurement

Best resolution when both EJ and EC are small. But how far can we go?

(in tunnelling regime)

100 120 140

0.0

0.2

0.4

0.6

0.8

1.0

Pes

cape

Pulse height / nA

MAX dP/dI

MQT & TAT=20 mK

Measured histograms at various temperatures and flux biases

IC = 200 nACJ = 100 fF

IC,SQUID = IC,SQUID(Φ)

(Kivioja et al, 2005)

Red lines: MQT/TA

Phase diagram of Josephson junctions with intermediate coupling (Kivioja et al, 2005)

-escape sets in at lower bias current-subsequent barrier tops close in energy

damping retraps phase particle

phase diffusion

(overdamped regime: Vion et al., PRL 1996)

Phase diffusion

V ~ 0

Including dissipation in analysing the escape characteristics

- Blue line: MQT + TA- Red line: full level dynamics calculation

-Dissipation barrier: Γk = 0, unless

(Larkin&Ovchinnikov, 1987)

(Kivioja et al, 2005)

Escape and measurement of quantum states in a SQUID

Measurement time ∆T:

Constraint: ∆T shorter than relaxation (50 – 100 ns)

use of antenna

Escape via tunnelling:strong contrast between levels(Γ1 = 800. Γ0)

If ∆T satisfies:

then:

Occupation of |1 ⟩selectively measured

φdc

Ixφ(t)JJ1 JJ2

|0⟩|1⟩

|2⟩|3⟩

Γ1

Γ0

Quantum dynamics: typical experiment

t

Ix(t)

φdc

1

φ(t)

2

φdc

Ix

φφdc+ φnano

Ic

3

2

Measurement principle

Preparation:shape of potential

1 3

initial state: |0⟩

MW

adiabatic deformation (tm=1ns) :occupation of excited states

3

manipulation :(deep well)

2φnano

Spectroscopy

Resonance |0 ⟩ - |1 ⟩

excitation measurement

|0⟩

|1⟩MW

mesurementfMW: tunable

• Flux-pulse sequence:

7

8

9

10

11

12

13

4,4 4,6 4,8 5 5,2

f 01 (G

Hz)

Ip (µA)

φ/φ0=0.095

0

1

2

3

11 11.2 11.4 11.6 11.8

P esc (%

)

ν (GHz)

180 MHz

(J. Claudon et al, PRL 2004)

TMW: fixed

TMW: tunable

measurementfMW=f01

• Flux-pulse sequence:

Measurement of coherent oscillations

Characteristic damping time = 14 ns

0

0,2

0,4

0,6

0,8

1

0 5 10 15 20 25 30 35

Pec

h

TRF

(ns)

ARF

= 0.501 U.A.

TR= 3.2 ns

0,2

0,3

0,4

0,5

0,6

0,7

0 5 10 15 20 25 30 35

Pec

h

ARF

= 0.126 U.A.

TR= 9.5 ns

T=30mK

(J. Claudon et al, PRL 2004)

Rabi oscillations?

0

200

400

600

800

1000

0 0,5 1 1,5 2 2,5

fr (M

Hz)

ARF

(U.A.)

Rabi theory

Strong deviation from behaviour of two-level system

Role of MW amplitude: multi-level oscillations

0,2

0,4

0,6

0,8

0 10 20

Pech

|0⟩

Coherent superposition of excited states

(J. Claudon et al, PRL 2004)

0 0,25 0,50

5

10population

état

Multi-level theory

competitionanharmonicity MW amplitude

16 levels in well,how many participate in oscillations?

0

250

500

750

1000

0 0,002 0,004 0,006

f r (M

Hz)

φRF

/φ0

Rab

i fre

quen

cy(M

Hz)

MW amplitude ΦMW/Φ0

2 levels

|0⟩, |1⟩ and |2⟩

|0⟩, |1⟩, … |10⟩ Pes

cape

Conclusions

Preparing quantum states |1⟩, |2⟩, |3⟩ ...

Decoherence in the SQUID

Towards two-level limit

Outlook

0,2

0,4

0,6

0,8

0 30 60 90 120

Prob

abili

té d

'éch

appe

men

t

Durée du pulse MW (ns)

Coherent multi-level oscillations

Quantum tunnelling and level quantisation in a dc-SQUID at different values of Φdc

Escape phenomena for intermediateJosphson coupling: novel phase diagram

φdcIx

φ(t)JJ1 JJ2

New developments…

|0⟩|1⟩

|2⟩|3⟩

Multi-level systemTwo-level system

|-⟩

|+⟩

Transistor

JJ1

JJ2

Gate

(Aurélien Fay and Wiebke Guichard)

…towards cavity QED? (Buisson&Hekking, 2001)

Four

vitesse

Préparationcirculaires

Détection

Sourceclassique

Cavité?

« cavity »

2 postdoc openings(theory & experiment)

0 30 60 900.0

0.2

0.4

0.6

0.8

1.0Ω

0 = 47 kHz

temps ( µs)

Taux

de

tran

sfer

t de

|+>

ver

s |-

>

« atom »

?