Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard Measurements in QIP (Rabi, Ramsey,Spin-echo)

Francisco Kim & Matteo Marinelli

QSIT

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Overview

1 Motivation

2 IntroductionBloch vector representation

3 Standard measurements in QIPEnergy gap ~ω0

Rabi oscillationEnergy-relaxation time & Coherence timeRamsey fringesSpin echo

4 Example of measurements

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Motivation

How well is the Qubit isolated from the environment?

What is the lifetime of our Qubit?

What is the coherence time?Long coherence = Many operations

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Motivation

How well is the Qubit isolated from the environment?

What is the lifetime of our Qubit?

What is the coherence time?Long coherence = Many operations

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Motivation

How well is the Qubit isolated from the environment?

What is the lifetime of our Qubit?

What is the coherence time?Long coherence = Many operations

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Introduction

Bloch vector representation

Bloch vector representation

ρ =

(ρ00 ρ01

ρ10 ρ11

)

~σ =

12(ρ10 + ρ01)i2(ρ01 − ρ10)ρ11 − ρ00

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Introduction

Bloch vector representation

Bloch vector in a rotating field

H =~ω0

2σz +

~Ω0

2(cos(ωt)σx + sin(ωt)σy)

E(t) = E0 cosωt and Rabi frequency Ω0 =eDE0

~

Applying Heisenberg equation of motion, we get d~σdt = ~Ω× ~σ

~Ω = (Ω0 cos(ωt),Ω0 sin(ωt), ω0)

Going to the rotating frame : ~Ω = (Ω0, 0, ω0 − ω)

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Introduction

Bloch vector representation

Bloch vector in a rotating field

H =~ω0

2σz +

~Ω0

2(cos(ωt)σx + sin(ωt)σy)

E(t) = E0 cosωt and Rabi frequency Ω0 =eDE0

~

Applying Heisenberg equation of motion, we get d~σdt = ~Ω× ~σ

~Ω = (Ω0 cos(ωt),Ω0 sin(ωt), ω0)

Going to the rotating frame : ~Ω = (Ω0, 0, ω0 − ω)

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Energy gap ~ω0

First : estimation of ω0

Variation of the frequency in order to find ω0

Courtesy: D. Vion et al., “Rabi oscillations, Ramsey fringes and spin echoes in an electricalcircuit”, Fortschr. Phys. 51, (2003)

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Rabi oscillation

Rabi oscillation

Apply a resonant pulse for different pulse-durationand measure the population of the upper state

Animation : Rabi oscillation

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Rabi oscillation

visibility

∆t for the π2 pulse

proof that it is a two-level system and not an harmonicoscillator

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Rabi oscillation

visibility

∆t for the π2 pulse

proof that it is a two-level system and not an harmonicoscillator

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Rabi oscillation

visibility

∆t for the π2 pulse

proof that it is a two-level system and not an harmonicoscillator

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Rabi oscillation

visibility

∆t for the π2 pulse

proof that it is a two-level system and not an harmonicoscillator

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Energy-relaxation time & Coherence time

Energy-relaxation time T1 & Coherence time T2

Γ1 = 1T1

Longitudinal relaxation rate

Γ2 = Γ12 + Γϕ Transverse relaxation rate

Animation : Energy loss Dephasing

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Energy-relaxation time & Coherence time

Apply a π-pulse and measure the population of the upper statewith different waiting times

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Ramsey fringes

Ramsey fringes

Apply two phase coherent π2 pulses,

separated by a delay ∆tduring which the spin precesses freely around z

Vary the delay ∆t

The envelope of oscillation gives T2

Animation : Ramsey fringes

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Spin echo

Spin echo

Apply a π pulse in the middle to recover the dephase

Vary the delay between the two pulses =⇒ T2

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Standard measurements in QIP

Spin echo

Animation : Spin echoSpin echo - T2 measurement

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Example of measurements

Application of these measurements

Peculiarity : Quantum Nondemolition Measurement can beperformed

A. Wallraff et al.,“Approaching Unit Visibility for Control of a Superconducting Qubit with Dispersive

Readout”,Phys. Rev. Let. 95, 060501 (2005)

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Example of measurements

Nondemolition measurements of the upper state population applyingπ, 2π and 3π pulses

Phase shift related to occupation probability

φ|↓〉 = −35.3deg, φ|↑〉 = 35.3deg P|↓〉 = P|↑〉 = 12 at φ = 0

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Example of measurements

Approaching unit visibility

Standard Measurements in QIP (Rabi, Ramsey, Spin-echo)

Example of measurements

References

A. Wallraff et al., “Approaching Unit Visibility for Controlof a Superconducting Qubit with Dispersive Readout”,Phys. Rev. Let. 95, 060501 (2005)

D. Vion et al., “Rabi oscillations, Ramsey fringes and spinechoes in an electrical circuit”, Fortschr. Phys. 51 (2003)

J. Bylander et al., “Noise spectroscopy through dynamicaldecoupling with a superconducting flux qubit”, NaturePhysics 7 (2011)