Josephson Junction based
Quantum Control
Erick Ulin-Avila
Seth Saltiel
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
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
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)
Real-time quantum feedback
•
Feedback generally complicated
– Wavefunction collapse
– Measurement Back-action
•
“Real time quantum feedback
is of interest for
closed-loop control
– Continuous measurement on Open-loop systems
• Being able to determine the state of a quantum
adaptive measurement
system conditioned on actual measurement results
state preparation
is essential
quantum error correction”
• “Quantum Trajectory Theory
For understanding and designing feedback
control
– a quantum version of Kalman filtering
•
Quantum feedback requires
– Broadband quantum-noise limited measurement
– Fast digital signal processing (state space
methods) (FPGA’s)
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
Phase
Flux
Superconducting Qubits
• Three different kinds depending on
dominant energy scales
Charge
– 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?)
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
Quantum Control for JJ based
devices
The JJ dynamics
•
The net current can be written as:
I  I C sin  
•
  C 


2eR
2e
If we define:

1
RC

2eIc
C
 
2eI
C
we can express it as :


      sin   

•
Which can be written, with: x1   x2  
as the following Planar Dynamical

system:
x1  x2

x 2    sin x1  x2  
Theodore van Duzer, Superconductive Devices and Circuits, Prentice hall (1999)
Zhao Y, Wang W, Chaos synchronization in a Josephson junction system via ...,
Chaos, Solitons & Fractals (2007)
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
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 readout.
F. W. Strauch, PRL 91, 167005 (2003).
P.R. Johnson, Proc. of SPIE Vol. 5436
Other relevant papers
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 nanoelectromechanical system
Kurt Jacobs and Pavel Lougovski J.Phys. A:Math. Theor 40 (2007)
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
NEWS:
Spin-Optics