What's super about superconducting

qubits?

Jens Koch

Departments of Physics and Applied Physics, Yale University

Chalmers University of Technology,Feb. 2009

Outline

Introduction

Superconducting qubits

► overview, challenges► circuit quantization► the Cooper pair box

Transmon qubit

► from the CPB to the transmon► advantages of the transmon► experimental confirmation

Circuit QED with the transmon: examples

next lecture:

charge qubit - Chalmers

phase qubit - UCSB

flux qubit - Delft

state

state

Quantum Bits and all that jazz

2-level quantum system (two distinct states )

can exist in an infinite number of physical states intermediate between and .

superpositionof

AND

quantum cryptographyN. Gisin et al., RMP 74, 145 (2002)

computational speedupP.W. Shor, SIAM J. Comp. 26, 1484 (1997)

fundamental questions

What makes quantum information more powerful than classical information?

Entanglement – how to create it? How to quantify it?

Mechanisms of decoherence?

Measurement theory, evolution under continuous measurement …

2-level systems

Nature provides a fewtrue 2-level systems:

Polarization of electromagnetic waves

(→ linear optics quantum computing)

Spin-1/2 systems,e.g. electron (→ Loss-DiVincenzo proposal) nuclei (→ NMR)

artificial atoms: superconducting qubits, quantum dots (→ cavity QED, → circuit QED…)

2-level systems

……

Requirements:• anharmonicity

• long-lived states• good coupling to EM field• preparation, trapping etc.

Using multi-level systemsas 2-level systems

e.g. atoms and molecules (→ cavity QED, → trapped ions → liquid-state NMR)

C. Schönenberger

R. Schoelkopf

The crux of designing qubits

environment environment

control measurementprotection against

decoherence

qubit

►need good coupling! ►need to be uncoupled!

Relaxation and dephasing

relaxation – time scale T1 dephasing – time scale T2

qubit

transition

► phase randomization► random switching

► fast parameter changes: sudden approx, transitions

► slow parameter changes: adiabatic approx, energy modulation

Bringing the into electrical circuitsIdea of superconducting qubits:

Electrical circuits can behave quantum mechanically!

What's good about circuits?

• Circuits are like LEGOs! a few elementary building blocks, gazillions of possibilities!

Idea of superconducting qubits: Electrical circuits can behave quantum mechanically!

Bringing the into electrical circuits

What's good about circuits?

• Circuits are like LEGOs! a few elementary building blocks, gazillions of possibilities!

• Chip fabrication:

well-established techniques

hope: possibility of scaling

Idea of superconducting qubits: Electrical circuits can behave quantum mechanically!

Bringing the into electrical circuits

E

2~ 1 meV

superconductor

superconducting gap

“forest” of states

Why use superconductors?

Wanted:

► electrical circuit as artificial atom

► atom should not spontaneously lose energy

► anharmonic spectrum

Superconductor

► dissipationless!

► provides nonlinearity via Josephson effect

► can use dirty materials for superconductors

Building Quantum Electrical Circuits

Two-level system: fake spin 1/2

circuit elements

ingredients:

• nonlinearities• low temperatures• small dissipation

SC qubits: macroscopic articifical atoms( )

Review: Josephson Tunneling

Tight binding model: hopping on a 1D lattice!

SC gap

normal state conductanceTunneling operator for Cooper pairs:

Josephson energy

• couple two superconductors via oxide layer → acts as tunneling barrier

• superconducting gap inhibits e- tunneling • Cooper pairs CAN tunnel!

► Josephson tunneling (2nd order with virtual intermediate state)

Review: Josephson Tunneling II

Tight binding model:

‘position’ ‘wave vector’ (compact!)

‘plane wave eigenstate’

… …

Diagonalization:

Junction capacitance: charging energy

+2en -2en

Transfer of Cooper pairs across junction

charging of SCs► junction also acts as capacitor!

with

quadratic in n

charging energy

Circuit quantization

Best reference that I know:(beware of a few typos though)

Circuit quantization – a quick survival guide

► Step 1: set up Lagrangian - determine the circuit's independent coordinates

branch

node

► use generalized node fluxes

as position variablesalso:

ideal current sources, ideal voltage sources,resistors

Circuit quantization – a quick survival guide

► Step 1: set up Lagrangian

capacitive energies inductive energies

► Step 2: Legendre transform Hamiltonian conjugate momenta: charges

Circuit quantization – a quick survival guide

► Final Step 3: canonical quantization

Canonical quantization makes NO statement about boundary conditions!Usually, assume

Works if each node is connected to an inductor ( confining potential).

This does NOT work if SC islands are present!

• charge transfer between island and rest of circuit: only whole Cooper pairs!

• canonical quantization is blind to the quantization of electric charge!

Circuit quantization – a quick survival guide

► Final Step 3: quantization in the presence of SC islands

island charge operator has discrete spectrum:

position

momentum?

Peierls: leads to contradiction -- phase operator is ill-defined!

charge basis

► is periodic!

Circuit quantization – a quick survival guide

► Final Step 3: quantization in the presence of SC islands

Have already defined charge operator

What about ?

► should define this in phase basis!

usually:

now:

► lives on circle!

Different types of SC qubits

chargequbit

fluxqubit

phasequbit

Reviews: Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001)M. H. Devoret, A. Wallraff and J. M. Martinis, cond-mat/0411172 (2004)J. Q. You and F. Nori, Phys. Today, Nov. 2005, 42J. Clarke, F. K. Wilhelm, Nature 453, 1031 (2008)

Nakamura et al., NEC LabsVion et al., SaclayDevoret et al., Schoelkopf et al., Yale,Delsing et al., Chalmers

Lukens et al., SUNYMooij et al., DelftOrlando et al., MITClarke, UC Berkeley

Martinis et al., UCSBSimmonds et al., NISTWellstood et al., U Maryland

NEC, Chalmers,Saclay, Yale

EJ = EC,

EJ =50EC

NIST,UCSB

TU Delft,UCB

EJ = 10,000EC

EJ = 40-100EC

► Nonlinearity from Josephson junctions

CPB Hamiltonian

charge basis:

phase basis:exact solution withMathieu functions

numerical diagonalization

3 parameters:

offset charge (tunable by gate)

Josephson energy

charging energy (fixed by geometry)

CPB as a charge qubit

Charge limit:

bigsmall perturbation

CPB as a charge qubitCharge limit:

bigsmall perturbation

Next lecture: from the charge regime to the transmon regime