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

Nature’s quantum bits true qubits effective qubits energy ¯ = 1 e = 1 ­ = 0 E 01 g = 0 single spin-1/2 single atom Manmade quantum bits artificial atoms built from circuits Image credit: W.D. Oliver & P.B. Welander, MRS Bulletin 38 , 816 (2013)

Transmon qubits embedded in a planar circuit 100 µ m Image credit: A. Bruno

The Transmon qubit Island 1 Josephson Capacitance inductance Island 2 Theory of the transmon: J. Koch et al. , Phys Rev. A 76 , 042319 (2007)

The quantized LC oscillator Hamiltonian: ˆ ˆ Electrode 1 F 2 2 Q ˆ + H = F LC 2 C 2 L Q ++ ++ Capacitive term Inductive term C L - - - - Canonically conjugate variables: ˆ F Electrode 2 = Flux through the inductor. (reference) ˆ Q = Charge on capacitor plate. é ˆ , Q ˆ ù F = ! i ë û M. Devoret, Les Houches Session LXIII (1995)

Correspondence with simple harmonic oscillator ˆ ˆ ˆ ˆ 2 2 kX P F + 2 2 Q ˆ ˆ + H = H = SHO LC 2 2 m 2 L 2 C é ˆ ù é ˆ ˆ ù = ˆ , Q F = X P , i ! i ! ë û ë û 1 ˆ ˆ F « « X L 1 k k Correspondence: w = « ˆ ˆ « m LC Q P « C m Solve using ladder operators: E æ ˆ ö ˆ F 7 Q F = 2 Z ! = - a ˆ ç i ÷ zpf ç ÷ F Q 6 è ø zpf zpf = Q 2 / Z ! zpf 5 æ ˆ ö ˆ F Q 1 L = + ˆ † ç ÷ a i = w = = Z L 4 ç ÷ F w Q C C è ø zpf zpf 3 2 w æ ö 1 ˆ † + ˆ ˆ é ù = H = ! a a ˆ ˆ † a a , 1 ç ÷ ë û 1 w LC ! è 2 ø r r 0 M. Devoret, Les Houches Session LXIII (1995) F

The Josephson junction h + æ ö F I F = = p superconductor- I I sin 2 S ç ÷ o c F 2 e è ø insulator- o I flux quantum superconductor V & V = F S tunnel junction - p D = I c 2 e R F æ ö æ ö F I Josephson = c o = - p E E E J 1 cos 2 ç ÷ ç ÷ ç ÷ J p stored Energy F 2 è ø è ø o Josephson potential 2 E J F 2 For small flux: » E + const F J 2 L F J o F Josephson inductance = L o J p 2 I c M. Devoret, Les Houches Session LXIII (1995)

Transmon energy spectrum E 4 3 2 1 0 F

Two-junction transmon Superconductor-Insulator-Superconductor junction AlOx 10 µ m Si Al F Si substrate ext Al 200 nm L C 100 µm

Flux control of transmon frequency Short-circuited transmission line Qubit transition frequency (GHz) Flux I Schreier et al., PRB (2008) sweetspot F ext ( ) ( ) F - 8 E E E / h J ext C C Flux bias (a.u.)