1. Quantum measurement and superconducting qubits Yuriy Makhlin (Landau Institute) STMP-09, St. Petersburg 2009, July 3-8

2. superconducting qubits quantum readout parametric driving and bifurcation stationary states, switching curve Outline

3. Quantum bits (qubits): 2-level quantum systems, Logic gates:unitary evolution Quantum algorithms:sequence of elementary operations Quantum computers Why quantum? computers: - high computational capacity? (R. Feynman, D. Deutsch) Shor algorithm: factorization in polynomial time, 1994 physics: - fundamental phenomena in quantum-mechanical systems - devices: new challenges

4. NMR [Chuang et al., Cory et al.] nuclear spins in various environments B x (t) BzBz Physical realizations of qubits Ion traps [Cirac & Zoller, … ] laser 2-level ions

5. 2-level quantum systems — N qubits controlled dynamics initialization - cooling or read-out coherence: read-out:quantum measurement controlled weak enough for error correction Requirements to realizations of quantum computers controlled 1-qubit gates 2-qubit gates

6. Electron spins in semiconductors Loss & DiVincenzo, 1998 +   spins >   charges + exactly 2 states Kouwenhoven et al. (Delft)

7. Josephson quantum bits quantum degree of freedom:charge or phase (magn. flux) macroscopic quantum physics (unlike in ion traps, NMR, optical resonators) artificial atoms: flexibility in fabrication scalability (many qubits) easy to integrate in el. structures combine: coherence of superconducting state advanced control techniques for single-charge and SQUID systems Quantronium (Saclay) no excitations at low T 

8. V. Bouchiat et al. (1996) Single-electron effects junctions w/ small area 10nm x 10nm typical capacitanceC ≈ 10 -15 F typical energies E C =e 2 /2C ≈ 1 K 

9. Josephson charge qubits VgVg xx N xx tunable 2 states only, e.g. for E C » E J Nakamura et al. ’99 2 qubits: Pashkin et al. ‘03   ~ 2 ns

10. Problems: decoherence noise => random phases => dephasing (T 2 ) noise => transitions => energy relaxation (T 1 ) low-frequency 1/f noise => strong dephasing 1/T 2 = 1/(2T 1 ) + 1/T 2 *

11. Quantronium operation at a saddle point Charge-phase qubit Vion et al. (Saclay) 2002

12. “Transmon” Koch et al. (Yale) 2007  (“coordinate”) EJEJ heavy particle (C B ) => flat bands => very low sensitivity to charge noise still (slightly) anharmonic => address only two states T 2  2  s( close to 2T 1 without spin echo) in a periodic potential e -  E J /E C E J /E C ~50 E(q)

13. “Topologically protected” qubits

14. Quantum measurement detector result01 state|0>|1> probability|a| 2 |b| 2 Measurement as entanglement unitary evolution of qubit + detector |M i i --- macroscopically distinct states - reliable ? single-shot? - QND? back-action - fast? quality of detection:

15. probability that m electrons passed Dynamics of current I(t) QPC

16. Qubit dynamics Reduced density matrix of qubit:  meas  ≥  

17. Mixing (relaxation)

18. Quantum readout for Josephson qubits switching readout – monitor the critical current no signal at optimal point strong back-action by voltage pulse (no QND), quasiparticles threshold readout Quantronics Ithier, 2005 U= - I c cos  – I  

19. Quantum readout for Josephson qubits dispersive readout – monitor the eigen-frequency of LC-oscillator Sillanpää et al. 2005 Wallraff et al. 2004 monitor reflection / transmission amplitude / phase

20. Quantum readout for Josephson qubits Josephson bifurcation amplifier (Siddiqi et al.) – dynamical switching exploits nonlinearity of JJ switching between two oscillating states (different amplitudes and phases) advantages: no dc voltage generated, close to QND higher repetition rate qubit always close to optimal point Siddiqi et al. 2003 cf. Ithier, thesis 2005

21. Parametric bifuraction for qubit readout I 0 =I c sin  0  =  0 +x  =  t Landau, Lifshitz, Mechanics diff. driving: Dykman et al. ‘98 w/ A.Zorin

22. Method of slowly-varying amplitudes A 2 =u 2 +v 2 Migulin et al., 1978

23. Equations of motion in polar coordinates: Equations of motion Hamiltonian

24. without quartic term stationary solutions: A=0or A2A2 detuning     =0

25. detuning

26. detuning,  -  0 driving P bistability “Phase diagram”

27. width detuning P switching 0 1 P sw = 1 – e –  |0> |1> Switching curves single-shot readout? contrast

28. Dykman et al. (1970’s – 2000’)

29. 0, A + – stable statesA - – unstable state Velocity profile and stationary oscillating states 2D Focker-Planck eq. => 1D FPE near origin:  relaxes fast, A – slow degree of freedom (at rate  )

30. Near bifurcation  -  cos 2  = (  + 3 / 2  P 2 ) / P(  + 3 / 2  P 2 ) ds/dt = - dW(s)/ds +  (s) => W s W(s) ≠ U(s) !

31. Tunneling, switching curves Focker-Planck equation for P(s) WW  ~ exp(-  W/T eff ) W = a s 2 – b s 4 width of switching curve  - ~  T

32. For a generic bifurcation(incl. JBA) controls width of switching curve For a parametric bifurcation- additional symmetry u,v -> -u,-v shift by period of the drive => should be even ! at origin

33. Another operation mode no mirror symmetry => generic case, stronger effect of cooling

34. Conclusions Period-doubling bifurcation readout: towards quantum-limited detection? low back-action rich stability diagram various regimes of operation tuning amplitude or frequency results bifurcations, tunnel rates, switching curves, stationary states for various parameters, temperature and driving dependence of the response with double period