Josephson Junction Qubits

1. Josephson Junction Qubits Alex Hegyi Justin Ellin Andrew Chan

2. Classical Resistance (Review) • Metals • In a metal, the electrons are shared by atoms in a lattice. • This sea of electrons is free to travel along the entire lattice. • Dissipation • Caused by inter-electron/ion interactions or other atoms, resulting in heat • (dV) = (dI)R, R = pL/A (p resistivity, length, cross-sectional area) • P = IV • Prevents indefinite propagation of currents, analogous to friction

3. Superconductors • Superconductor Properties • State characterized by zero (exactly) electrical resistance • Meissner Effect – weak external fields only penetrate small distances (London skin Depth) • Type I – Superconductivity destroyed abruptly when field reaches critical value • Type II – additional critical temperature which permits magnetic flux but still no electrical resistivity • Generation of a current to cancel external field

4. BSC Theory • Fermi Energy • The lowest energy of the highest occupied quantum state at absolute zero was considered to be the Fermi Energy Where N/V is the density of fermions This can be derived by considering a 3-dimensional square box. • BSC- Bardeen, Cooper, and Schrieffer 1957 • The theory essentially accounts for an energy level even below this threshold. • The gap between this energy level and the fermi energy accounts for many of the properties of superconductors • Whereas before the electron could be excited in a continuous spectrum of possible energy interactions (and interchange/lose energy with lattice and other electrons), there is now a discrete energy gap. • The excitations become forbidden and the electron sees no “obstacles” or no resistance! But what accounts for this gap?

5. Cooper Pairs • The atoms in a lattice are not fixed • Free electrons are repulsed from other electrons but are able to attract and distort the positively charged nucleus. This distortion in turn attracts other electrons. • Coupling (on the order of fractions of an eV) usually broken by thermal energy or coulomb interaction. • When the thermal energy is low, T ~ 5K, this dominates effectively linking electrons in pairs to each other even over “large” distances . • The electrons pair up with those of opposite spin. • Exclusion principle no longer applies. All electron pairs condense into this bound state energy.

6. Two Notes on Modern Superconductors • Current Lifetime – occasionally interactions may result that do go across the gap. • Experimentally, currents on superconductors can perpetuate for upwards of tens of thousands of years. • Theoretically, could last longer than the known age of the universe. • High Temperature Superconductors –superconductors that can’t be explained by BCS because state achieved well above fermi levels • (Sn5In)Ba4Ca2Cu10Oy: superconducting at ~200K (Dry ice is about this range) • How do they work?

7. Josephson Junction • Brian David Josephson proposed (1964) sandwiching an insulator between two superconductors. • Provided separation is small, current will tunnel through the barrier • However when the current reaches a certain critical value then a voltage will develop across the junction which will in turn increase the voltage further. • The frequency of this oscillation is ~ 100 GHz • Below this critical current, no voltage. Above, oscillating voltage.

8. Some Uses of Junctions • SQUIDs (superconducting quantum interference devices) • Precise Measurements • Voltage to Frequency Converter • Single-Electron Transistors

9. Flux Qubit • Quantum state is stored in the direction of the current • |0> is counter-clockwise • |1> is clockwise

10. Manipulate State • Requires a constant external magnetic flux • Flux determines the energy difference between the two states • Apply a microwave pulse • Causes the flux qubit to oscillate between ground state (|0>) and excited state (|1>)

11. SQUID • Superconducting Quantum Interference Device Critical Current Below: Current flows without voltage Above: Oscillating current develops

12. Measurement • Apply a current pulse to SQUID • Collapses state • Magnetic flux through flux qubit determines critical current of SQUID

13. Qubit Interaction • Entanglement between two qubits is achieved by coupling their fluxes • Superconducting bus • Transfers a quantum state from one qubit to another by sending a single photon along a superconducting wire

14. “Additional” DiVincenzo Criteria Conversion of stationary, flying qubits Optical Microcavities, Cavity QED Transmission of flying qubits Fiber Optics Microwave transmission lines (Circuit QED)—way to accomplish the above in case of superconducting qubits* *Wallraff et al., Nature, 431, 9 Sept. 2004

15. Strong Coupling/Cavity QED Two-level quantum system coupled to electromagnetic cavity “Strong Coupling” characterized as coherent exchange of excitation between cavity, quantum system i.e., coherent conversion between stationary, flying qubit Model—Two SHOs connected by weak spring

16. Microwave Resonator/Qubit System *Schoelkopf and Girvin, Nature, 451, 7 Feb. 2008

17. Quantum Communication If energy difference between |0> and |1> resonant with cavity, energy exchanged (Rabi rotation) If off-resonant (dispersive) energy not exchanged Align qubits along transmission line, tune energy difference (using gate bias, flux bias) to control interaction with line

18. Microwave Resonator/Qubit System *Wallraff et al., Nature, 431, 9 Sept. 2004