Josephson Junction based Quantum Control

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

2. 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

3. 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 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)

4. 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 controladaptive measurementstate preparationquantum error correction” Mabuchi, Hideo (2003) Experiments in real-time quantum feedback. In: IEEE Conference on Decision and Control, 41st, (CDC 2002)

5. 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

6. Josephson Junction Physics.

7. 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

8. 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 = ICsin (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

9. 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

10. 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

11. 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

12. 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

13. Flux Phase Superconducting Qubits 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?)

14. 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

15. Quantum Control for JJ based devices

16. 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: 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)

17. 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

18. 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

19. 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 nano-electromechanical system Kurt Jacobs and Pavel Lougovski J.Phys. A:Math. Theor 40 (2007)

20. 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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