quantum dynamics of josephson junction based circuits
TRANSCRIPT
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Quantum dynamics in nano Josephson junctions
CNRS – Université Joseph Fourier Institut Néel- LP2MC
GRENOBLE
Permanent:Wiebke GuichardOlivier BuissonFrank HekkingLaurent LévyBernard Pannetier
PhD:Thomas WeisslEtienne DumurIoan PopFlorent LecocqAurélien FayRapaël LéoneNicolas DidierJulien ClaudonFranck Balestro
Post-doc:Alexey FeofanovIulian MateiZhihui Peng Emile HoskinsonH. JirariAlex Zazunov
Scientific collaborations: PTB Braunschweig ( Germany)LTL Helsinki (Finland)Rutgers (USA)
Projects: ANR QUNATJO
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Quantum Nano-Electronics
Quantum properties and phase coherence in
electronic nano-circuits
Model systemCoherent transportInterference effects
On chip quantum experimentsArtificial atoms and molecules
Single molecule
New technologiesNanofabrication
Lithography processBottom-upOptics
MicrowavesLow noise electronics
Low temperature…
New materialsNanotubes
Nanowires (Si,GaN…) Graphene
New nano-devices
New functionalitiesSET
Quantum bitHigh sensitive detectors
Current standardMicrorefrigerartion
Ultimately small circuits
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Quantum experiments
Realizing quantum experiments in solid state system
Building « artificial atoms » or probing a « single atom »
Strong coupling to environment(thermal noise, charge or magnetic fluctuations, microscopic defects,…)
Fast manipulations(strong coupling with external field)
Strong and very strong coupling between qubits
Original properties compared to atoms:
Long term: - quantum information processingShort term: - high sensitive detectors
- model system for quantum nano-electronics or nano-photonics- experimental quantum demonstrators
- Manipulate the quantum states- Measure the quantum states- Long coherence time
Objectives: physics of these original quantum systems
Superconducting quantum circuits
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Introduction to superconducting qubits
Multi-levels artificial atom- current-biased Josephson junction and dc SQUID- quantum measurements- quantum dynamics in a multilevel quantum system- quantum or classical description- optimal control- decoherence processes
Two-degrees of freedom artificial atom- inductive dc SQUID- spectroscopy measurements- strong non-linear coupling- coherent oscillations
Multi-degrees of freedom system- Josephson junction chains- quantum phase slip- charging effects
Outline
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Introduction of the superconducting state
In many metals, T smaller than a critical temperature :the conduction electrons condenses to electron pairs : Cooper pairs
All the pairs forms a Macroscopic Quantum State: |�G>
|�G> is a very stable ground statebecause no excitations below the gap
�
2e
e
|�G> is analog to the coherent state of a laser beamthe phase is very well defined
Persistent current without any decay!
To perform quantum experiments, we need excited levels
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Basic building blocks: Josephson junction
� � ieQ 2, ���
Josephson relations:
Small Josephson junction: two energy scales
�
�cos)/ˆ(ˆ 2JC EeQEH �
eQ 2���
CIc
Electrical scheme
CeE
eIE
C
cJ
2
)2/(2
�
� �
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Scientific context
Macroscopic quantum effects…
… and quantized energy levels(microwave frequency regime)
Voss et al, PRL (1981)
Devoret et al, PRL (1985)
Martinis et al, PRL (1985)
- Superconducting qubit
Nakamura et al, Nature (1999)
�MQT
An artificial atom controlled by electronics signals
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Defense1th 2011
Engineering quantum mechanics
Vion et al, Science (2002) Wallraff et al, Nature (2004)Pashkin et al, Nature (2003)
- Coherence- Optimal point
- Readout- Memory- Bus
- Coupling- Gates- Algorithm
Neelley et al, Nature (2010)Sillanpaa et al, Nature (2007)DiCarlo et al, Nature (2010)
Scientific context
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Motivations
qubit 1 qubit 2
- artificial atom with two energy level(only one degree of freedom!)
- two coupled qubits
- two level system coupled to high Q cavity
qubit 1cavity
Is it possible to build and control artificial atom: - with multi-energy levels?- with two degree of freedom?
Up to now in superconducting systems:
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Motivations
10µm
Ib
I� 10µm
Multi-level quantum system
Two-degrees offreedom
Multi-degrees offreedom
New physics…
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Introduction to superconducting qubits
Multi-levels artificial atom- current-biased Josephson junction and dc SQUID- quantum measurements- quantum dynamics in a multilevel quantum system- quantum or classical description- optimal control- decoherence processes
Two-degrees of freedom artificial atom- inductive dc SQUID- spectroscopy measurements- strong non-linear coupling- coherent oscillations
Multi-degrees of freedom system- Josephson junction chains- quantum phase slip- charging effects
Outline
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Driven anharmonic oscillator
H(t) � P 2
2m 12 m p
2 X 2 fext cos(2��t)XHarmonic oscillator:
�aX 3 � bX 4By adding anharmonic terms
New physics appear which was extensively studied
Classical mechanics: - Landau&Lifchitz- modification of the reonance peak- bi stability (used as amplifier Siddiqqi 04, Ithier 05)- parametric amplifiers
Quantum mechanics: many theoretical studies (Dykman88, Milburn86, Enzer97, Katz07, etc..)
Quantum dynamics in an anharmonic quantum oscillator
The quantum particle follows a motion very close to theclassical one
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Current-biased Josephson junction
���
����
����
��
����
��
��� �
���
���
�� sin
22200
2
20
c
pc
III
dtd
RdtdC
��
ddU )(
�« mass » m
Current conservation:
�sincb IIRV
dtdVC ��
V
Ib
CId Is
�
U
�)cos()( ��� Cb IIU ���
Ic Josephson relations:R
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In the quantum limit…
� � ieQ 2, ���
p (Ib)
�U (Ib)
U(�)
�quantum tunneling
level quantization
Quantum anharmonic oscillator!
EJ>>EC
Josephson potentialCharging energy
)ˆ)/(ˆ(cos)/ˆ(ˆ 2 �� cbJC IIEeQEH ��
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Quantum experiments with a dc-SQUID(J. Claudon et al PRB07)
|0>|1>
|2>
�b(t)
Ib
�
Imeasure (t)
dc-SQUID: Ic(�b)
Deep well with quantized states
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(J. Claudon et al PRB07)
|0>|1>
|2>
�b(t)
Ib
�
Imeasure (t)
dc-SQUID: Ic(�b)
Deep well with quantized states
Microwave flux: �(t)excitation
An external driving force!
Quantum state manipulation
MW
�,�1
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p (Ib,�b )
�U (Ib,�b)
-3.0
-1.5
0
1.5
3.0
-500 -250 0 250 500
Ic
I b(�
A)
Vs (�V)
A nano-second flux pulse reduces the barrier
Hysteretic junction: escape leads to voltage
�0
J. Claudon, et al, PRL04, PRB2007
�b(t)
Ib
�
Imeasure (t)
dc-SQUID: Ic(�b)
Quantum measurements with a dc-SQUID
Pro
babi
lity
ofes
cape
Nanosecond flux pulse amplitude (�0)
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Quantum measurements
p (Ib,�b )
�U (Ib,�b)
-3.0
-1.5
0
1.5
3.0
-500 -250 0 250 500
Ic
I b(�
A)
Vs (�V)
A nano-second flux pulse reduces the barrier
Hysteretic junction: escape leads to voltage
�0
�1
�2
Pro
babi
lity
ofes
cape
Nanosecond flux pulse amplitude (�0)
�b(t)
Ib
�
Imeasure (t)
J. Claudon, et al, PRL04, PRB2007
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Experimental set-up
• MW manipulation• fast measurements
• courant bias• voltage state of SQUID
Vout
IbI
Le
Filtered linesDC-200kHzbas bruit
(on-chip)
VMW
�(t)
�bCe
25 mK
Broadband lineDC-20 GHz
Ze( )
50 �
15μm
Aluminium circuitNanoFab
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Introduction to superconducting qubits
Multi-levels artificial atom- current-biased Josephson junction and dc SQUID- quantum measurements- quantum dynamics in a multilevel quantum system- quantum or classical description- optimal control- decoherence processes
Two-degrees of freedom artificial atom- inductive dc SQUID- spectroscopy measurements- strong non-linear coupling- coherent oscillations
Multi-degrees of freedom system- Josephson junction chains- quantum phase slip- charging effects
Outline
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Ib = 2.222 �A, �b = -0.117 �0
J. Claudon, A. Fay, L.P. Lévy, and O. Buisson (PRB2008)Spectroscopy measurements
� 01
(GH
z)
7
8
9
10
11
12�b� -0.117 �0
�b� -0.368 �0
0.5 1 1.5 2 2.5Ib (�A)
|0�
|1�
�
SQUID parametersI0 =1.242 �AC0= 560 fFLS= 280 pH
8 8.1 8.2 8.3 8.4 8.52
4
6
8
Pes
c(%
)
�01
� (GHz)
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measurement�MW=�01
• Flux-pulse sequence:
Coherent oscillations in a dc SQUID(J. Claudon F. Balestro, F. Hekking, and O. Buisson, PRL 2004)
P = +6 dBm����� = 260 MHz
�RLO = 208 MHz
Tmw (ns)
Pes
c
�RLO = 66 MHz
�RLO = 122 MHz
P = 0 dBm����� = 130 MHz
P = -6 dBm����� = 65 MHz
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0 20 40 60 800.2
0.4
0.6
0.8
Pes
cP
esc
E/h (GHz)
0204060
0
5
12
7 levels
�01
• Anharmonic oscillator:
TMW: tunable
Anharmonicity:�01��12 = 160 MHz
Are Rabi oscillations?
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« Rotating wave » approximation
H* : Time independent Hamiltonian eigenvector eigenvalues
Rabi oscillations (I)
XZ thtH ����� )2cos(221)(ˆ
101 ��� �
Two level system:
�|0>
|1>?
2/)10(
2/)10(
1121*
1
1021*
0
����
����
�
�
e
e
0)0(
)()(ˆ)(
�
�
�
�� ttHtdtdi�
���
����
�� �
�
00ˆ2
2*1
1
H
temporal evolution ?)(t�
0)0(
)(ˆ)(
*
***
�
�
�
�� tHtdtdi�
Rotating referential
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Rabi oscillations (II)
Two level system:
�01|0>
|1>?
))cos(1()())cos(1()(
121
1
121
0
ttpttp
���
��
))exp(()(
)()0(*11
*02
1*
*1
*02
1*
etiet
ee
���
�
�
�
0 2���1t
1
0
|0> |1> |0>
)10(2
1 i
The duration of the microwave controls the state
�1
E*
�1
�0
�1
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Rabi oscillations of a two level system
� �1
0
50
100
150
200
250
0 50 100 150 200 250 300
MW amplitude : �1/2� (MHz)
�R
LO(M
Hz)
�01��12 = 160 MHz
Rabi
Strong deviation compare to Rabi prediction!
We must take into account the multi-level dynamics
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Multilevel dynamics in an harmonic oscillator
��H(t) � 1
2 � p (P 2 X 2) ��� P X 3 � 2��1 cos(2��t)X
From a coherent source ( � p),the quantum state is descibed by the coherent states
Its energy grows as : ))2/(( 212
1 tE p �!�" �
Continuous growth! No oscillations!!
Harmonic oscillator with equidistant energy levels
When its energy is very large,its dynamics describe very well the classical motion
No way to build arbitrary states and explainsoscillations
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0
200
400
600
800
0 100 200 300
« Rotating wave » approximation
Eigenvalues
Time independent Hamiltonian
#�$!
#�%!
#��!
#��!
#�&!
�����'(MHz)�
(MH
z)
Multilevel dynamics
|0(t=0+)>=a0#�$!a�'#��!a�#��!a%#�%!
low amplitude: large amplitude:
��H(t) � 1
2 � p (P 2 X 2) ��� P X 3 � 2��1 cos(2��t)X
0(t � 0) � ( �0 �1 ) / 2
Rabi oscillations
Initial state:
Multi-level dynamics
�R�R,1
�R,2
�R,3
�R,4
� 01�
� 12
Time independent Hamiltonian
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quantum theorywith ���01
• Low excitation power :two level description
Cross-over between two and multi-level
• Intermediate power :multi- level description
�01��12 �1
Cross- over condition:
Anharmonicity MW amplitude
� �1
0
50
100
150
200
250
0 50 100 150 200 250 300
MW amplitude : �1/2� (MHz)
2 3 4
�R
LO(M
Hz)
# involved levels
�01��12 = 160 MHz
Rabi
p2<12%
(J. Claudon,A. Zazunov, F. Hekking, and O. Buisson, arXiv:0709.3787)
RabiT ,2
1
coherent superpositionof excited states
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Classical description?
Integrate equations of motion
(J. Claudon A. Zazunov, FH, and O. Buisson, PRB 2008)
(A. Ratchov, PhD-thesis 2005Gronbech-Jensen 2006)
p (Ib)�)
fmw (u.a.)
*'fmw2/3
Transient regime
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Classical or quantum description?
Classical model Quantum model
(J. Claudon A. Zazunov, F. Hekking, and O. Buisson, PRB 2008)
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(J. Claudon A. Zazunov, F. Hekking, and O. Buisson, PRB 2008)
������+$'MHz �������+$'MHz ������,�$'MHz
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Introduction to superconducting qubits
Multi-levels artificial atom- current-biased Josephson junction and dc SQUID- quantum measurements- quantum dynamics in a multilevel quantum system- quantum or classical description- optimal control- decoherence processes
Two-degrees of freedom artificial atom- inductive dc SQUID- spectroscopy measurements- strong non-linear coupling- coherent oscillations
Multi-degrees of freedom system- Josephson junction chains- quantum phase slip- charging effects
Outline