Leonardo DiCarlo Leonardo DiCarlo Superconducting quantum circuits: - - PowerPoint PPT Presentation
Leonardo DiCarlo Leonardo DiCarlo Superconducting quantum circuits: Superconducting quantum circuits: The transmon qubit The transmon qubit Natures quantum bits true qubits effective qubits energy = 1 e = 1 = 0 E 01 g = 0
single atom
g =
1 e =
true qubits effective qubits
single spin-1/2
= 1 ¯ =
artificial atoms built from circuits
Image credit: W.D. Oliver & P.B. Welander, MRS Bulletin 38, 816 (2013)
01
E energy
Image credit: A. Bruno
100 µm
Josephson inductance Capacitance Island 1 Island 2
Theory of the transmon: J. Koch et al., Phys Rev. A 76, 042319 (2007)
F
Q
++ ++
Inductive term
2 2 LC
ˆ ˆ ˆ 2 2 Q H C L F + = Hamiltonian:
Capacitive term Electrode 2 (reference) Electrode 1
ˆ ˆ ,Q i é ù F = ë û ! ˆ F ˆ Q
= Flux through the inductor. = Charge on capacitor plate.
Canonically conjugate variables:
2 2 LC
ˆ ˆ ˆ 2 2 Q H L C F + = ˆ ˆ ,Q i é ù F = ë û ! ˆ ˆ X F « ˆ ˆ Q P « C m « 1 L k «
1 k m LC w = «
2 2 SHO
ˆ ˆ ˆ 2 2 kX P H m + = ˆ ˆ , X P i é ù = ë û !
† LC
1 ˆ ˆ ˆ 2 H a a w æ ö + ç ÷ è ø ! =
† r r
ˆ ˆ , 1 a a é ù = ë û
zpf zpf
ˆ ˆ ˆ Q a i Q æ ö F =
÷ ç ÷ F è ø
† zpf zpf
ˆ ˆ ˆ Q a i Q æ ö F = + ç ÷ ç ÷ F è ø
zpf
2 Z F = !
zpf
2 / Z Q = !
1 L Z L C C w w = = =
E
1 2 4 5 6 3 7
w !
F Correspondence: Solve using ladder operators:
S I S superconductor- insulator- superconductor tunnel junction
h e F =
The Josephson junction
V = F &
I
+
c
I I p æ ö F = ç ÷ F è ø flux quantum Josephson potential
2
J
E
2 J J
+ const 2 E L F »
c
2 L I p F =
For small flux:
F
Josephson inductance
c
2 I E p F =
Josephson Energy
stored J 1 cos 2
E p æ ö æ ö F =
÷ ç ÷ ç ÷ F è ø è ø
c
2 I e R p D =
Transmon energy spectrum
E
1 2 3 4
F
Two-junction transmon
10 µm
100 µm
200 nm
Al Al Si substrate Si AlOx Superconductor-Insulator-Superconductor junction
ext
Flux control of transmon frequency
Flux sweetspot Schreier et al., PRB (2008)
I
ext
F
Flux bias (a.u.) Qubit transition frequency (GHz)
( )
( )
J ext C C
8 / E E E h F