Josephson Junction based Quantum Control Erick Ulin-Avila Seth Saltiel.

Josephson Junction based Quantum Control

Erick Ulin-Avila

Seth Saltiel

Control SystemsThe dynamics of an energetic system can be modeled as a Mass-Spring-Damper system. Control Theory is very well understood in many regimes, i.e. Linear, Non-linear, Deterministic, Stochastic, Analog, Digital

Open Loop

Closed-Loop or Feedback Loop

The quantum-classical transition The process of measurement

• Why is this “transition” between two very different theories so robust?– “we take this really small

fuzzy globs that are evolving in an orderly fashion, and when we put enough of them together, for some reason everything crystallizes and becomes sharp while its dynamics becomes chaotic”.

Hideo Mabuchi

quantum classical

uncertainty certainty

Simple orderly

linear

Complex

nonlinear

The quantum classical transition on trial: Is the whole more than the sum of the parts?by Hideo Mabuchi, Engineering and Science, No 2, (2002)

Quantum measurement is important to understandthe theory of decoherence

Real-time quantum feedback

• Feedback generally complicated – Wavefunction collapse– Measurement Back-action

• For understanding and designing feedback control

– Continuous measurement on Open-loop systems• Being able to determine the state of a quantum

system conditioned on actual measurement results is essential

• “Quantum Trajectory Theory– a quantum version of Kalman filtering

• Quantum feedback requires– Broadband quantum-noise limited measurement – Fast digital signal processing (state space

methods) (FPGA’s)

“Real time quantum feedback is of interest for

closed-loop control adaptive measurement

state preparation quantum error correction”

Mabuchi, Hideo (2003) Experiments in real-time quantum feedback. In: IEEE Conference on Decision and Control, 41st, (CDC 2002)

Why the Josephson Junction

• Dissipative Quantum Dynamics of Nonlinear systems is an exciting new area where the frontier between classical and quantum mechanics may be carefully investigated.

• The nonlinearity of the Josephson junction provides anharmonic oscillators, so the quantum states have varying energy-level spacings and two-level can be conveniently manipulated in isolation.

• In addition, the Josephson Junction represents a very important fundamental piece in the study of Classical Nonlinear Control Systems. – very nonlinear chaotic behavior can be observed for single JJ device

or coupled JJ devices due to changes in parameters related to its fabrication.

Berggren, Proceedings of the IEEE, Vol. 92, no10, Oct. 2004

Josephson Junction Physics.

Superconductors

• BCS Theory– Pairs of electrons (Cooper pair) with opposite

spins interact with each other at a sufficiently low temperature to create boson (no net spin)

– “condense” to occupy the same lowest energy state wavefunction, which cannot be scattered by imperfections

– Without charge carrier scattering there is no resistance

– macroscopic quantum mechanics!Kasap, S.O. Principles of Electronic Materials and Devices. McGraw-Hill 2006

Josephson Effect

• Thin Layer of Insulator between two Superconductors– Pairs’ wavefunctions overlap, tunnel barrier– This current from pair tunneling happens

when there is no voltage across the junction– When there is an applied voltage across the

junction, oscillating current:

I = IC sin (Φ)

- Φ is phase angle between wavefunctions: dΦ/dt = 4πeV/h

Feynman, R.P, Leighton, R.B, Sands, M. The Feynman Lectures on Physics, Vol. III

AC Josephson Effect• Integrating dΦ/dt and solving for the time

and voltage dependence of current gives: I = I0 sin (2πft), current is oscillating with frequency f = 2eV/h, which is exceedingly fast given the large value of e/h (4.1 x 10^33)– One Volt defined by the 483,597.9GHz it generates

• You also get a current if you apply a high frequency voltage in addition to the dc voltage– Like Laramor procession in NMR this happens

at a resonance frequency:w = 2πqV/h

Feynman, R.P, Leighton, R.B, Sands, M. The Feynman Lectures on Physics, Vol. III

I-V curve for Josephson Junction– No current w/ applied dc voltages less than Va

that breaks pairs and restores normal current– Supercurrent without any voltage– Hysteretic bistable I-V curve with ~10ps

switching time, limited by junction capacitance

Kasap, S.O. Principles of Electronic Materials and Devices. McGraw-Hill 2006

Quantum Interference• Two parallel Josephson junctions in loop

– Each path gives different phase of current depending on voltage across junction

– Voltage induced by flux through loop– Magnetic field present in the loop creates

current interference pattern between junctions relative phase changes

– Sensitive magnetometer

Feynman, R.P, Leighton, R.B, Sands, M. The Feynman Lectures on Physics, Vol. III

SQUIDs

• Two types of SQUIDs– Multi-junction (dc SQUIDs) use two or more

Josephson junctions to show interference with constant magnetic fields giving DC current out

– One-Junction (RF SQUIDs) uses only one Josephson junction and obtains interference due to the reaction flux of the current induced in the loop from the changing magnetic field

• RF refers to the radio frequency of oscillation

Van Dozer, T, Turner, C.W. Principles of Superconductive Devices and Circuits. Prentice Hall 1999

Superconducting Qubits

• Three different kinds depending on dominant energy scales– Charge Qubit: small junctions where energy to

charge capacitance w/ cooper pair leading– Flux Qubit: energy of inductive flux (coupling)

• Electron pairs to flow continuously around the loop (clockwise/counter), rather than tunnel discretely across the junctions (as in cooper pair box)

– Phase Qubit: energy of tunneling through junction dominates, large C and IC

• phase difference natural variable, flux negligible

Johnson, et al. ”Quantum control of superconducting phase qubits.” Quantum Information and Computation III (2004?)

Charge

FluxPhase

Coupling Qubits• Many ways to couple qubits

– Flux qubits coupled inductively can be controlled and tuned with current or phase using dc SQUIDs for read-out and control

– Phase qubits coupled through capacitor and controlled with applying microwaves tuned to transitions or changing bias current

• These methods include ways decouple qubits before and after gate operations to avoid back-actionBerggren, K.K. “Quantum Computing with Superconductors.” IEEE (2004)

Kim, M.D. “Controllable Coupling of Phase-coupled Flux Qubits.” PHYSICAL REVIEW B 74, 184501 2006

Quantum Control for JJ based devices

The JJ dynamics

• The net current can be written as:

• If we define:

we can express it as :

• Which can be written, with:

as the following Planar Dynamical

system:

e

C

eRII C 22sin

RC

1

C

eIc

2

C

eI

2

sin

1x

2x

21 xx

212 sin xxx

Zhao Y, Wang W, Chaos synchronization in a Josephson junction system via ...,Chaos, Solitons & Fractals (2007)

Theodore van Duzer, Superconductive Devices and Circuits, Prentice hall (1999)

SFQ Control Circuits for JJ Qubits

• Three types:– Magnetic pulse generators– Read-out circuits– Digital circuits controlling them

• The most natural is provided by RSFQ technology.

– Reduced power consumption– High speed– Reduced output noise– Trade offs between

• Power and speed• Shunt resistors vs critical currents

– Read-out circuits must enable a dynamical compensation of the backaction down to a level approaching SQL

SEMENOV AND AVERIN: SFQ CONTROL CIRCUITS FOR JOSEPHSON JUNCTION QUBITS

O. A. Mukhanov and V. K. Semenov, A Novel Way of Digital InformationProcessing in Josephson Junctions Circuits: Department of Physics, Moscow State University, 1985.

K. Likharev, O. Mukhanov, and V. Semenov (then at Moscow State University, Moscow, Russia)

Quantum Control of phase qubits

• Nontrivial dynamical process which requires self-consistent modeling

• A qubit quantum gate– The applied bias current

determines the tilt of the washboard potential, which in turn determines:

• the number, • energy level spacings,• effective degree of

anharmoniticity

• A two-qubit quantum gate – de-tuning the relative bias currents of

the two junctions dynamically decouples them, which is sufficient for quantum computation and state read-out.

P.R. Johnson, Proc. of SPIE Vol. 5436F. W. Strauch, PRL 91, 167005 (2003).

Coherent Control of Macroscopic quantum states in a single Cooper-pair box

J.Nakamura, NATURE Vol. 398 (1999)

Coherent Coupling of a single photon to a cooper pair box

A. Wallraff et.al., NATURE (2004)

Emergent Quantum Jumps in a nano-electromechanical system

Kurt Jacobs and Pavel Lougovski J.Phys. A:Math. Theor 40 (2007)

Other relevant papers

Conclusions

• An introduction to methods of exploration in Quantum control Systems and coherence

• Remark the Importance of the Nonlinearity of Josephson Junctions

• The use of Josephson Junctions as well as its control in Quantum Computation

• Some details on control of JJ based systems was explained, specifically the quantum control of superconductive phase qubits.

additional

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