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 adaptive measurement Being able to determine the state of a quantum state preparation system conditioned on actual measurement results is essential quantum error correction For understanding and designing feedback control 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 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 is phase angle between wavefunctions d dt 4 eV h Feynman R P Leighton R B Sands M The Feynman Lectures on Physics Vol III AC Josephson Effect Integrating d dt and solving for the time and voltage dependence of current gives I I0 sin 2 ft 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 2 qV 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 V a 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 x1 x2 as the following Planar Dynamical Which can be written with system x1 x 2 x 2 sin x1 x 2 Theodore van Duzer Superconductive Devices and Circuits Prentice hall 1999 Zhao Y Wang W Chaos