Josephson%20Junction%20based%20Quantum%20Control

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Josephson Junction based Quantum Control

• Erick Ulin-Avila
• Seth Saltiel

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Control Systems
The 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
Closed-Loop or Feedback Loop
Open Loop
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The quantum-classical transition The process of
measurement
quantum classical
uncertainty certainty
Simple orderly linear Complex nonlinear

• 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 measurement is important to
understand the theory of decoherence
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)
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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) (FPGAs)

Real time quantum feedback is of interest for
closed-loop control adaptive measurementstate
preparation quantum error correction
Mabuchi, Hideo (2003) Experiments in real-time
quantum feedback. In IEEE Conference on
Decision and Control, 41st, (CDC 2002)
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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
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Josephson Junction Physics.
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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
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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 (F)
• - F is phase angle between wavefunctions
  dF/dt 4peV/h

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

• Integrating dF/dt and solving for the time and
  voltage dependence of current gives I I0 sin
  (2pft), current is oscillating with frequency f
  2eV/h, which is exceedingly fast given the large
  value of e/h (4.1 x 1033)
• 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 2pqV/h

Feynman, R.P, Leighton, R.B, Sands, M. The
Feynman Lectures on Physics, Vol. III
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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
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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
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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
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Superconducting Qubits
Flux
Phase
Charge

• 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?)
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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-action

Berggren, K.K. Quantum Computing with
Superconductors. IEEE (2004) Kim, M.D.
Controllable Coupling of Phase-coupled Flux
Qubits. PHYSICAL REVIEW B 74, 184501 2006
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Quantum Control for JJ based devices
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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

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)
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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

K. Likharev, O. Mukhanov, and V. Semenov (then at
Moscow State University, Moscow, Russia)
O. A. Mukhanov and V. K. Semenov, A Novel Way of
Digital Information Processing in Josephson
Junctions Circuits Department of Physics, Moscow
State University, 1985.
SEMENOV AND AVERIN SFQ CONTROL CIRCUITS FOR
JOSEPHSON JUNCTION QUBITS
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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.

F. W. Strauch, PRL 91, 167005 (2003).
P.R. Johnson, Proc. of SPIE Vol. 5436
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Coherent Control of Macroscopic quantum states in
a single Cooper-pair box
Other relevant papers
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. AMath.
Theor 40 (2007)
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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.

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additional
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NEWS Spin-Optics