Constructing Non-Linear Quantum Electronic Circuits...Constructing Non-Linear Quantum Electronic...
Transcript of Constructing Non-Linear Quantum Electronic Circuits...Constructing Non-Linear Quantum Electronic...
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Constructing Non-Linear Quantum Electronic Circuits
Review: M. H. Devoret, A. Wallraff and J. M. Martinis, condmat/0411172 (2004)
circuit elements:
Josesphson junction:a non-dissipative nonlinear element (inductor)
anharmonic oscillator: non-linear energy level spectrum:
electronicartificial atom
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A Classification of Josephson Junction Based Qubits
current bias flux biascharge bias
Common options of bias (control) circuits:
phase qubit flux qubitcharge qubit(Cooper Pair Box, Transmon)
How is the control circuit important?
How to make use in of Jospehson junctions in a qubit?
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The Cooper Pair Box Qubit
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A Charge Qubit: The Cooper Pair Box
discrete charge on island:
continuous gate charge:
total box capacitance
Hamiltonian:
electrostatic part:
magnetic part:
charging energy
Josephson energy
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completeness
orthogonality
eigenvalues, eigenfunctions
Hamilton Operator of the Cooper Pair Box
basis transformation
Hamiltonian:
commutation relation:
charge number operator:
phase basis:
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Solving the Cooper Pair Box HamiltonianHamilton operator in the charge basis N :
solutions in the charge basis:
Hamilton operator in the phase basis δ :
transformation of the number operator:
solutions in the phase basis:
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energy level diagram for EJ=0:
• energy bands are formed
• bands are periodic in Ng
energy bands for finite EJ
• Josephson coupling lifts degeneracy
• EJ scales level separation at charge degeneracy
Energy Levels
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Charge and Phase Wave Functions (EJ << EC)
courtesy CEA Saclay
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Charge and Phase Wave Functions (EJ ~ EC)
courtesy CEA Saclay
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Tuning the Josephson Energysplit Cooper pair box in perpendicular field
SQUID modulation of Josephson energy
J. Clarke, Proc. IEEE 77, 1208 (1989)
consider two state approximation
�𝐻𝐻 = 𝐸𝐸𝐶𝐶( �𝑁𝑁 − 𝑁𝑁𝑔𝑔)2−𝐸𝐸𝐽𝐽,𝑚𝑚𝑚𝑚𝑚𝑚 cos 𝜋𝜋𝜙𝜙𝑒𝑒𝑚𝑚𝑒𝑒𝜙𝜙0
cos �̂�𝛿
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Two-State Approximation
Shnirman et al., Phys. Rev. Lett. 79, 2371 (1997)
Restricting to a two-charge Hilbert space:
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A Variant of the Cooper Pair Box
J. Koch et al., Phys. Rev. A 76, 042319 (2007)J. Schreier et al., Phys. Rev. B 77, 180502 (2008)
5 µm
a Cooper pair box with a small charging energy
circuit diagram:
standard CPB: Transmon qubit:
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The Transmon: A Charge Noise Insensitive Qubit
J. Koch et al., Phys. Rev. A 76, 042319 (2007)
Cooper pair box energy levels: Transmon energy levels:
dispersion: relative anharmonicity:
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Control of Coupling to Electromagnetic Environment
Decoupling schemes using non-resonant impedance transformers …
coupling to environment (bias wires):
… or resonant impedance transformers
control spontaneous emission by circuit design
decoherence due to energy relaxationstimulated by the vacuum fluctuations of the environment (spontaneous emission)
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Realizations of Superconducting Artificial AtomsNECChalmersJPLYale
SaclayYale
DelftNTTIPHT
NISTSanta-BarbaraMaryland
YaleETHZ
NISTSanta-BarbaraMaryland
DelftIPHTNEC
YaleNIST
review: J. Clarke and F. WilhelmNature 453, 1031 (2008)
'artificial molecules' -- coupled superconducting qubits
'artificial atoms‘ -- single superconducting qubits
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Realizations of Harmonic Oscillators
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Superconducting Harmonic Oscillators
• typical inductor: L = 1 nH
• a wire in vacuum has inductance ~ 1 nH/mm
• typical capacitor: C = 1 pF
• a capacitor with plate size 10 µm x 10 µm and dielectric AlOx (ε = 10) of thickness 10 nm has a capacitance C ~ 1 pF
• resonance frequency
LC
a simple electronic circuit:
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inductor L
+qφ
-q
Realization of H.O.: Lumped Element Resonator
capacitor C
currents andmagnetic fields
charges andelectric fields
a harmonic oscillator
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Types of Superconducting Harmonic Oscillators
planar transmission line resonator:
A. Wallraff et al., Nature 431, 162 (2004)
3D cavity:
H. Paik et al., PRL 107, 240501 (2011)
I. Chiorescu et al., Nature 431, 159 (2004)
weakly nonlinear junction:Z. Kim et al., PRL 106, 120501 (2011)
lumped element resonator:
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Realization of H.O.: Transmission Line Resonator
• coplanar waveguide resonator• close to resonance: equivalent to lumped element LC resonator
distributed resonator:
ground
signal
couplingcapacitor gap
M. Goeppl et al., Coplanar Waveguide Resonatorsfor Circuit QED, Journal of Applied Physics 104, 113904 (2008)
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1 mm
Realization of Transmission Line Resonator
Si + + --
E B
cross-section of transm. line (TEM mode):
measuring the resonator:
photon lifetime (quality factor) controlled by coupling capacitors Cin/out
coplanar waveguide:
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Resonator Quality Factor and Photon Lifetime
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Controlling the Photon Life Time
photon lifetime (quality factor)controlled by coupling capacitor Cin/out
1 mm
100µm
100µm
100µm
100µm
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Quality Factor Measurement
ext. load ext. load
=
M. Goeppl et al., J. Appl. Phys. 104, 113904 (2008)