Upload File

typical i-v curve of underdamped josephson · of cooper pair box hamiltonian: with equivalent...

  • Home
  • Documents
  • typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically
15
typical I-V curve of underdamped Josephson junctions: band diagram

Upload: others

Post on 10-Feb-2020

2 views

Category:

Documents


0 download

Report

TRANSCRIPT

Page 1: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

typical I-V curve of underdamped Josephson junctions:

band diagram

Page 2: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

:bias current dependence

:

damping dependent prefactor

:

calculated using WKB method ( )

:

neglecting non-linearity

Quantum Mechanics of a Macroscopic Variable: The Phase Difference of a Josephson JunctionJOHN CLARKE, ANDREW N. CLELAND, MICHEL H. DEVORET, DANIEL ESTEVE, and JOHN M. MARTINISScience 26 February 1988 239: 992-997 [DOI: 10.1126/science.239.4843.992] (in Articles) Abstract » References » PDF »

Macroscopic quantum effects in the current-biased Josephson junction M. H. Devoret, D. Esteve, C. Urbina, J. Martinis, A. Cleland, J. Clarkein Quantum tunneling in condensed media, North-Holland (1992)

Early Results (1980’s)

J. Clarke, J. Martinis, M. Devoret et al., Science 239, 992 (1988).

A.J. Leggett et al.,

Prog. Theor. Phys. Suppl. 69, 80 (1980),

Phys. Scr. T102, 69 (2002).

Page 3: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

The Current Biased Phase Qubitoperating a current biased Josephson junction as a superconducting qubit:

initialization:

wait for |1> to decay to |0>, e.g. by spontaneous emission at rate γ10

Read-Out Ideasmeasuring the state of a current biased phase qubit

pump and probe pulses:

- prepare state |1> (pump)

- drive ω21 transition (probe)

- observe tunneling out of |2>

|1>|2>

|o> |o>|1>

tipping pulse:

- prepare state |1>

- apply current pulse to suppress U0

- observe tunneling out of |1>

|1>|2>

|o>

tunneling:

- prepare state |1> (pump)

- wait (Γ1 ~ 103 Γ0)

- detect voltage

- |1> = voltage, |0> = no voltage

Page 4: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Bouchiat et al. Physica Scripta 176, 165 (1998)

Josephson energy:

Charging energy:

Gate charge:

A Charge Qubit: The Cooper Pair Box

Page 5: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

of Cooper pair box Hamiltonian:

with

Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically solving the Schrödinger equation for the charge ( )representation or analytically solving the Schrödinger equation for the phase ( ) representation.

solutions for :

• crossing points arecharge degeneracy points

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

Page 6: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Charge and Phase Wave Functions (Ej << Ec)

courtesy Saclay

Charge and Phase Wave Functions (Ej ~ Ec)

courtesy Saclay

Page 7: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Tuning the Josephson Energy

split Cooper pair box in perpendicular field

SQUID modulation of Josephson energy

J. Clarke, Proc. IEEE 77, 1208 (1989)

consider two state approximation

Two State Approximation

Shnirman et al. Phys. Rev. Lett. 79, 2371 (1997)

Page 8: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Cavity QED with Electronic Circuits

Cavity Quantum Electrodynamics

D. Walls, G. Milburn, Quantum Optics (Spinger-Verlag, Berlin, 1994)

Page 9: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Dressed States Energy Level Diagram

Atomic cavity quantum electrodynamics reviews:

J. Ye., H. J. Kimble, H. Katori, Science 320, 1734 (2008)

S. Haroche & J. Raimond, Exploring the Quantum, OUP Oxford (2006)

Cavity Quantum Electrodynamics (QED)

superconductor circuitsYale, Delft, NTT, ETHZ, NIST, …

alkali atomsMPQ, Caltech, ...

Rydberg atomsENS, MPQ, ...

semiconductor quantum dotsWurzburg, ETHZ, Stanford …

Page 10: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Review: J. M. Raimond, M. Brune, and S. Haroche

Rev. Mod. Phys. 73, 565 (2001)

P. Hyafil, ..., J. M. Raimond, and S. Haroche,

Phys. Rev. Lett. 93, 103001 (2004)

Vacuum Rabi Oscillations with Rydberg Atoms

Vacuum Rabi Mode Splitting with Alkali Atoms

R. J. Thompson, G. Rempe, & H. J. Kimble,

Phys. Rev. Lett. 68 1132 (1992)

A. Boca, ... , J. McKeever, & H. J. Kimble

Phys. Rev. Lett. 93, 233603 (2004)

Page 11: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Cavity QED with Superconducting Circuits

A. Blais, et al. , PRA 69, 062320 (2004)

A. Wallraff et al., Nature (London) 431, 162 (2004)

Circuit Quantum Electrodynamics

A. Blais et al., PRA 69, 062320 (2004)

elements

• the cavity: a superconducting 1D transmission line resonator with large vacuum field E0 and long photon life time 1/κ

• the artificial atom: a Cooper pair box with large EJ/EC

with large dipole moment d and long coherence time 1/γ

Page 12: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Vacuum Field in 1D Cavity

+ + --

E B

1 mm

Resonator Quality Factor and Photon Lifetime

Page 13: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Qubit/Photon Coupling in a Circuit

Circuit QED with One Photon

A. Wallraff, …, R. J. Schoelkopf, Nature (London) 431, 162 (2004)

Page 14: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Resonant Vacuum Rabi Mode Splitting …

first demonstration: A. Wallraff, … and R. J. Schoelkopf, Nature (London) 431, 162 (2004)

this data: J. Fink et al., Nature (London) 454, 315 (2008)

How to Measure Single Microwave Photons

Page 15: typical I-V curve of underdamped Josephson · of Cooper pair box Hamiltonian: with Equivalent solution to the Hamiltonian can be found in both representations, e.g. by numerically

Measurement Setup

microwave electronics 20 mK cryostat

cold stage

sample mount


BY JOHN JEFFREY JOSEPHSON

Hamiltonian Equations

Underdamped Second Order Systemsglennc/e220/e220l16b.pdf · Underdamped Second Order Systems ... case results in complex numbers • This generates a decaying oscillating case. •

Josephson Junctiwrwwon in SQUID

Josephson qubits

Josephson Junction Qubits

Direct comparison of a 1 V Josephson arbitrary waveform ......AC Josephson voltage standards based on pulse-driven Josephson arrays (Josephson arbitrary waveform synthesizer—JAWS)

Underdamped Unstable · Time to reach first peak (undamped or underdamped only). … % of in excess of . … Time to reach and stay within 2% of . … Time to rise from 10% to 90%

Hamiltonian Cycle

Solution Manual for Dynamics of Structures 5th Edition by ... · underdamped systems; (b) ÷ Solution: () = + + − = − ± = + = + − = () = (+) = = () = − = +: = = = + () =

Harmonic Resonances in Nonlinear Josephson Junction ...€¦ · This thesis presents a study of resonance dynamics in circuits of superconducting Josephson junctions. The Josephson

Nonlinear dynamics of Josephson Junction Chains and ...659243/FULLTEXT01.pdf · 1.1 Josephson Junction Chains Josephson Junction (JJ) chains consist of serially connected DC SQUIDs

Quantum phase dynamics in an LC shunted Josephson junction · parallel to a Josephson junction influences the Josephson dynamics. The presence of the shell circuit introduces two

Basic Hamiltonian mechanics - CERNcds.cern.ch/record/399399/files/p1.pdfOUTLINE OF LAGRANGIAN AND HAMILTONIAN FORMALISM ... BASIC HAMILTONIAN MECHANICS. ... The Hamiltonian formulation

Josephson effect - FAU · Josephson effect . 1. General information . 1.1 Introduction . The Josephson effect is an example of a . macroscopic quantum phenomenon. There are several

InAs Josephson junctions - Physics

zhakarov hamiltonian

Underdamped Unstable - Michigan State University€¦ ·  · 2007-02-19Second order system response. Unstable Re(s) Im(s) Overdamped or Critically damped Undamped Underdamped Underdamped

Hamiltonian Cycle (3A) · to a Hamiltonian path by removing one of its edges, but a Hamiltonian path can be extended to Hamiltonian cycle only if its endpoints are adjacent. All Hamiltonian

Hamiltonian Circuits

Hamiltonian Formalism

COMPUTER ALGORITHM FOR THE ANALYSIS OF UNDERDAMPED …

Collective Josephson Vortex Dynamics in · Collective Josephson Vortex Dynamics in Long Josephson Junction Stacks Ju H. Kim and J. Pokharel DepartmentofPhysics,UniversityofNorthDakota

Advances in Josephson Quantum Circuitsqulab.eng.yale.edu/documents/talks/Devoret-APS_Tutorial_100313.pdf• Why Josephson junctions? • Main flavors of Josephson qubits • Readout

Outline - MIT › 6.763 › www › FT03 › Lectures › Lecture11.pdf · Lecture 11: Basic Josephson Junctions Outline 1. Quantum Tunneling 2. Josephson Tunneling 3. Josephson current-phase

Hamiltonian Dynamics

Efecto Josephson - materias.df.uba.ar

Hamiltonian Graphs

Hamiltonian Cycles

Physical properties Josephson · 2020. 12. 4. · basic Josephson junction. 2.1 Basic Josephson junc-tions A Josephson junction is one superconduc-tor connected to another superconductor,

Intrinsic Josephson effect

Underdamped Second-Order Systems Overshoot Controlfolk.ntnu.no/skoge/prost/proceedings/PID-12/papers/0106.pdf · Underdamped Second-Order Systems Overshoot Control ... integral and

Abductive Inference Josephson

Hamiltonian Neural Networkspapers.nips.cc/paper/9672-hamiltonian-neural-networks.pdf · Hamiltonian NN Prediction Figure 1: Learning the Hamiltonian of a mass-spring system. The variables

CV Mikael Josephson English