frank hekking · 2007. 8. 21. · damped motion: q = ωp rc. hysteretic ... rabi oscillations? 0...
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![Page 1: Frank Hekking · 2007. 8. 21. · damped motion: Q = ωp RC. Hysteretic ... Rabi oscillations? 0 200 400 600 800 1000 0 0,5 1 1,5 2 2,5 fr (MHz) A RF (U.A.) Rabi theory Strong deviation](https://reader036.vdocuments.site/reader036/viewer/2022081410/6092784ffe157f6349534f06/html5/thumbnails/1.jpg)
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
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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
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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 |-
>
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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)
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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
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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
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Quantum oscillations in two coupled charge qubits
(Pashkin et al , Nature 2003)
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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
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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
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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
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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
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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)
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(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)
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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
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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
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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
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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)
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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
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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
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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
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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)
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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
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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
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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
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New developments…
|0⟩|1⟩
|2⟩|3⟩
Multi-level systemTwo-level system
|-⟩
|+⟩
Transistor
JJ1
JJ2
Gate
(Aurélien Fay and Wiebke Guichard)
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…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 »
?